Categories of Spaces Built from Local Models

Categories of spaces built from local models

Low Zhen Lin

Trinity Hall University of Cambridge

20th June 2016

This dissertation is submitted for the degree of Doctor of Philosophy

獻僅給以我此⽗論母⽂ This dissertation is dedicated to my parents.

Abstract

Many of the classes of objects studied in geometry are defined by first choosing a class of nice spaces and then allowing oneself to glue these local models together to construct more general spaces. The most well-known examples are manifolds and schemes. The main purpose of this thesis is to give a unified account of this procedure of constructing a category of spaces built from local models and to study the general properties of such categories of spaces. The theory developed here will be illustrated with reference to examples, including the aforementioned manifolds and schemes. For concreteness, consider the passage from commutative rings to schemes. There are three main steps: first, one identifies a distinguished class of ring homomorphisms corresponding to open immersions of schemes; second, one defines the notion of an open covering in terms of these distinguished homomorphisms; and finally, one embeds the opposite of the category of commutative ringsin an ambient category in which one can glue (the formal duals of) commutative rings along (the formal duals of) distinguished homomorphisms. Traditionally, the ambient category is taken to be the category of locally ringed spaces, but following Grothendieck, one could equally well work in the category of sheaves for the large Zariski site—this is the so-called ‘functor of points approach’. A third option, related to the exact completion of a category, is described in this thesis. The main result can be summarised thus: categories of spaces built from local models are extensive categories with a class of distinguished morphisms, sub- ject to various stability axioms, such that certain equivalence relations (defined relative to the class of distinguished morphisms) have pullback-stable quotients; moreover, this construction is functorial and has a universal property.

君⼦於其⾔︐無所苟⽽已矣。故君⼦名之必可⾔也︐⾔之必可⾏也。刑罰不中︐則民無所措⼿⾜。禮樂不興︐則刑罰不中︔事⾔不名不成不順 ︐ 正 ︐ 則 ︐ 則禮則事樂⾔不不不成興順 ︔ ︔ ︔ If names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success. When affairs cannot be carried on to success, proprieties and music will not flourish. When proprieties and music do not flourish, punishments will not be properly awarded. When punishments are not properly awarded, the people do not know how to move hand or foot. Therefore a superior man considers it necessary that the names he uses may be spoken ︽論語 ‧ ⼦路︾ appropriately, and also that what he speaks may be carried out appropriately. What the superior man requires is just that in his words there may be nothing incorrect. Analects, Book XIII, translated by James Legge

Contents

Preface xi

Introduction xiii Context ...... xiii Summary ...... xiv Outline ...... xix Guide for readers ...... xxi

Chapter I. Abstract topology 1 1.1. The relative point of view ...... 1 1.2. Local properties of morphisms ...... 13 1.3. Regulated categories ...... 31 1.4. Exact quotients ...... 40 1.5. Strict coproducts ...... 61

Chapter II. Charted objects 77 2.1. Sites ...... 77 2.2. Ecumenae ...... 89 2.3. Extents ...... 102 2.4. Functoriality ...... 117 2.5. Universality ...... 138

ix Contents

Chapter III. Specificities 145 3.1. Compactness ...... 145 3.2. Discrete fibrations ...... 153 3.3. Manifolds ...... 161 3.4. Topological spaces ...... 166 3.5. Schemes ...... 174

Appendix A. Generalities 199 a.1. Presheaves ...... 199 a.2. Coverages ...... 209 a.3. Sheaves ...... 223

Bibliography 233

Index 235

x Preface

Declaration of originality

This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared in the preface and specified in the text. It is not substantially the same as any that I have submitted,or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other university or similar institution except as declared in the preface and specified in the text. I further state that no substan- tial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma, or other qualification at the University of Cambridge or any other university or similar institution except as declared in the preface and specified in the text.

xi Preface

Acknowledgements

I am very grateful to my supervisor, Peter Johnstone, for taking me as a student and allowing me to chart my own course through the last three and a half years. I would also like to thank all the other members of the category theory research group, past and present, whose company I have had the pleasure of having: Mar- tin Hyland, Julia Goedecke, Olivia Caramello, Ignacio López Franco, Nathan Bowler, Rory Lucyshyn-Wright, Guilherme Lima de Carvalho e Silva, Achilleas Kryftis, Christina Vasilakopoulou, Tamara von Glehn, Sori Lee, Filip Bár, Paige North, Enrico Ghiorzi, Adam Lewicki, Sean Moss, Nigel Burke, Georgios Char- alambous, and Eric Faber. Their presence contributed to a stimulating working environment in Cambridge, and I am glad to have been part of it. In this context, I must also mention Jonas Frey and Ohad Kammar, whose regular participation in the category theory seminar surely warrants their inclusion in the preceding list as honorary members of our research group. The internet has been a great boon to researchers all over the world. I have benefited from the insights gleaned from electronic correspondence with fellow mathematicians: Michael Shulman, Aaron Mazel-Gee, Emily Riehl, Qiaochu Yuan, Lukas Brantner, Geoffroy Horel, David White, Martin Brandenburg, and many others besides—primarily users of 푛Forum, MathOverflow, and Mathe- matics StackExchange—who are too numerous to list. In addition, TEX Stack- Exchange has been a very helpful resource for resolving TEXnical diﬃculties. The research presented in this thesis was conducted mostly during the course of a Ph.D. studentship funded by the Cambridge Commonwealth, European and International Trust and the Department of Pure Mathematics and Mathematical Statistics. I gratefully acknowledge their financial support. Last, but not least, I thank all my friends and family for their moral support, particularly Saran Tunyasuvunakool and Kathryn Atwell, whose friendship I have been fortunate to enjoy over the past seven and a half years. There are no words to express the profundity of my gratitude to my parents. I dedicate this thesis to them.

xii Introduction

Context

Another manifold-like notion is the notion of sheaf. Indeed, a sheaf on a topological space is a topological space obtained by gluing together open subspaces of the base space. This heuristic can be made precise and remains valid for Gro- thendieck’s notion of sheaf on a site: a site is a category equipped with a notion of covering, and a sheaf on a site can be regarded as a formal colimit of a diagram in the base. Thus, it should come as no surprise that manifold-like objects can be represented by sheaves on the category of local models, in the sense that there is a fully faithful functor from the category of manifold-like objects to the category of sheaves.

There is an alternative definition of scheme based on the aforementioned sheaf representation: this is called ‘the functor-of-points approach to algebraic geo-

xiii Introduction

metry’ and can be found in e.g. [Demazure and Gabriel, 1970]. The great advant- age of the definition in terms of functors over the traditional one in terms oflocally ringed spaces is that one no longer needs to explicitly model affine schemes— the local models—as geometric objects; instead, one can just glue together affine schemes formally. In exchange, one has to give up a certain sense of concreteness, but it is precisely the high level of abstraction that makes the functor-of-points approach so amenable to generalisation: one can mimic the functorial definition of ‘scheme’ in any reasonable geometric situation to get a manifold-like notion. Of course, one should say what one means by ‘reasonable geometric situation’ here. For the purposes of defining manifold-like notions, not much is needed: it would suffice to have a class of well-behaved morphisms—akin to local homeomorphisms of topological spaces or étale morphisms of schemes—and a compatible notion of covering. Joyal and Moerdijk [JM] have previously investigated this idea, albeit without discussing the problem of defining manifold-like notions. In a sense, this thesis is a fulfilment of a suggestion of Shulman [2012]:

It should be possible to axiomatize further “open map structure”, along the lines of [JM] and [DAG 5], enabling the identification of a general class of “schemes” in [the exact completion] as the congru- ences where gluing happens only along “open subspaces.”

Summary

The primary goal of this thesis is to give a uniform account of manifold-like notions. Although the concept is straightforward enough, as ever, the devil is in the details. In essence, the diﬃculty is the tradeoff between having a bad category of nice objects and having a nice category of bad objects.[1] It is not very hard to develop an elegant theory of manifold-like notions if one assumes that all the categories and functors involved behave nicely with respect to limits of finite diagrams. Unfortunately, the category of manifolds is the classical example of a bad category of nice objects—not all pullbacks exist—so such a theory would not account for manifolds. Similarly, one cannot easily account for the functor

[1] Of course, this is only an empirical observation, sometimes attributed to Grothendieck. xiv Summary

that sends a scheme to its underlying topological space because this functor does not preserve pullbacks. For better or worse, the theory that is developed in this work can accommodate the two examples mentioned above. While the theory may not be as elegant as one may have hoped for at first, it is at least general enough to include well- known examples. What follows is a summary of this theory; some details have been changed or omitted in order to simplify the exposition. First, let us fix what it means to have a category of local models. An admissible ecumene consists of the following data:

[2] • An extensive category, u�.

• A class of morphisms in u�, 햤, with the following properties:

– Every isomorphism in u� is a member of 햤.

– 햤 is closed under composition.

– 햤 is closed under coproduct.

– 햤 is a quadrable class of morphisms in u�, i.e. u� has pullbacks of members of 햤 along arbitrary morphisms and, for every pullback square in u� of the form below, 푋′ 푋 푓 ′ 푓 푌 ′ 푌 if 푓 : 푋 → 푌 is a member of 햤, then 푓 ′ : 푋′ → 푌 ′ is also a member of 햤.

– Every member of 햤 is an effective epimorphism in u�. [3] – Every finite diagram in u� has an 햤-weak limit.

• A class of morphisms in u�, u�, with the following properties:

– Every isomorphism in u� is a member of u�.

– u� is closed under composition.

– u� is a quadrable class of morphisms in u�. [2] See definition 1.5.5. [3] See definition 1.4.9.

xv Introduction

– For every object 푋 in u� and every small set 퐼, the codiagonal morphism

∇ : ∐푖∈퐼 푋 → 푋 is a member of u�. – u� is closed under coproduct. – Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if both 푔 : 푌 → 푍 and 푔 ∘ 푓 : 푋 → 푍 are members of u�, then 푓 : 푋 → 푌 is also a member of u�. – Given a member 푓 : 푋 ↠ 푌 of 햤 and a morphism 푔 : 푌 → 푍 in u�, if both 푓 : 푋 ↠ 푌 and 푔 ∘ 푓 : 푋 → 푍 are members of u�, then 푔 : 푌 → 푍 is also a member of u�. – If 푓 : 푋 → 푌 is a member of u�, then there is a small set Φ of objects in

u�∕푋 with the following properties:

* The induced morphism ∐(푈,푥)∈Φ 푈 → 푋 in u� is a member of 햤. * For every (푈, 푥) ∈ Φ, both 푥 : 푈 → 푋 and 푓 ∘ 푥 : 푈 → 푌 are monomorphisms in u� that are members of u�. – Every member of 햤 is also a member of u�. For example, the following data define admissible ecumenae:

(a) u� is the category of Hausdorff spaces, u� is the class of local homeomorphisms, and 햤 is the class of the class of surjective local homeomorphisms.

(b) u� is the category of disjoint unions of small families of aﬃne schemes, u� is the class of local isomorphisms, and 햤 is the class of surjective local isomorphisms.

(c) u� is the category of disjoint unions of small families of open subspaces of euclidean spaces, u� is the class of local diffeomorphisms, and 햤 is the class of surjective local diffeomorphisms.

Next, we make precise what it means to have a category of spaces built from local models. We say that an admissible ecumene as above is effective if the following additional condition is satisfied:

[4] • For every object 푋 in u� and every tractable equivalence relation (푅, 푑0, 푑1) on 푋 in u�, there is a morphism 푓 : 푋 ↠ 푌 in u� such that 푓 : 푋 ↠ 푌 is a

member of 햤 and (푅, 푑0, 푑1) is a kernel pair of 푓 : 푋 ↠ 푌 . [4] See definition 2.2.14. xvi Summary

The main result of this thesis says that every admissible ecumene can be embed- ded in an effective admissible ecumene that is universal with respect to admissible functors.[5] The category of charted objects is (the underlying category of) this universal effective admissible ecumene. For example:

(a) For (u�, u�, 햤) as in example (a), the category of charted objects is (equivalent to) the category of locally Hausdorff spaces.

(b) For (u�, u�, 햤) as in example (b), the category of charted objects is (equivalent to) the category of schemes.

(c) For (u�, u�, 햤) as in example (c), the category of charted objects is (equivalent to) the category of manifolds, possibly neither second-countable nor Hausdorff.

As one might expect, every charted object can be obtained as a quotient (in the category of charted objects) of an object in u� by a tractable equivalence relation (not necessarily in u�). This fact is easily deduced from the explicit construction of the category of charted objects as a full subcategory of the exact completion of u� relative to 햤, which is an exact category that u� embeds into and is universal with respect to functors that preserve limits of finite diagrams and send members of 햤 to effective epimorphisms. Since u� is an extensive category and 햤 is closed under coproduct, the exact completion is a pretopos.[6] This is very convenient for technical purposes: recalling the tradeoff discussed in the first paragraph, what we are doing is embedding a bad category of nice objects in a nice category of bad objects, which means that many proofs boil down to showing that certain constructions on nice objects—which are guaranteed to exist in the nice category —yield nice objects. To make a tighter connection with the previously mentioned work of Joyal and Moerdijk [JM], we propose the following definition. A gros pretopos consists of the following data:

• A pretopos, u�.

[5] See definition 2.5.5 and theorem 2.5.7. [6] See proposition 1.5.13.

xvii Introduction

• A class of morphisms in u�, u�, with the following properties:

– Every isomorphism in u� is a member of u�.

– u� is closed under composition.

– u� is a quadrable class of morphisms in u�.

– For every object 푋 in u� and every small set 퐼, the codiagonal morphism

∇ : ∐푖∈퐼 푋 → 푋 is a member of u�. – u� is closed under coproduct.

– Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if both 푔 : 푌 → 푍 and 푔 ∘ 푓 : 푋 → 푍 are members of u�, then 푓 : 푋 → 푌 is also a member of u�.

– Given an effective epimorphism 푓 : 푋 ↠ 푌 in u�, a morphism 푔 : 푌 → 푍

in u� and a kernel pair (푅, 푑0, 푑1) of 푓 : 푋 → 푌 in u�, if 푑0, 푑1 : 푅 → 푋 and 푔 ∘ 푓 : 푋 → 푍 are all members of u�, then both 푓 : 푋 ↠ 푌 and 푔 : 푌 → 푍 are also members of u�.

The axioms on u� are almost the same as Joyal’s axioms for a class of étale morphisms—the difference being that we omit the descent axiom and strengthen the quotient axiom. Here, ‘gros’ is used in opposition to ‘petit’: given an object 푋

in u�, the petit pretopos u�∕푋 is the full subcategory of the slice category u�∕푋 spanned by the objects (퐹 , 푝) such that 푝 : 퐹 → 푋 is a member of u�. Of course, a pretopos equipped with a class of étale morphisms as defined in [JM, §1] is a gros pretopos, but we require a bit more generality. For instance, the pretopos associated with an admissible ecumene admits the structure of a gros pretopos such that the intersection of the class of distinguished morphisms in the pretopos with the original category is the original class of distinguished morphisms—in fact, we will see two different constructions: one that yields a class of étale morphisms and one that does not.[7] It turns out that the latter is what we need to construct the category of charted objects. Given a full subcategory u� ⊆ u�, a (u�, u�)-atlas of an object 푌 in u� is an effective epimorphism 푓 : 푋 ↠ 푌 in u� where 푋 is an object in u� and 푓 : 푋 ↠ 푌 is a member of u�, and a (u�, u�)-extent in u� is an object that admits a (u�, u�)-atlas. [7] See proposition 2.3.2 and paragraph 2.5.3. xviii Outline

Under certain hypotheses on u� and u�, the category of (u�, u�)-extents in u� admits the structure of an effective admissible ecumene. For instance, if u� is the exact completion of an admissible ecumene, u� is the image of the original category, and u� is the induced class of local homeomorphisms, then this is part of the statement of the main result. Although we define the category of charted objects for an admissible ecumene to be the category of extents in the exact completion with respect to a certain gros pretopos structure, we should distinguish between ‘charted object’ and ‘extent’ because there exist gros pretoposes that do not arise in this way. Indeed, whereas the petit pretopos over a charted object is guaranteed to be localic,[8] the petit pretopos over an extent can fail to be localic. The extra generality afforded by defining extents in the setting of a general gros pretopos makes it possible tofit algebraic spaces—generalised schemes—into our framework, but exploring that possibility will be left for future work.

Outline

Abstract topology

In the first chapter, we study various aspects of what it means to be a category of spaces.

• In §1.1, we discuss the relative point of view of Grothendieck, i.e. the idea that a morphism is a family of objects (the domain) parametrised by the base (the codomain), and we define some related terminology that will be used throughout this work.

• In §1.2, we consider what it means for a morphism to have a property locally on the domain, locally on the base, or locally, with respect to a coverage. This is partially a generalisation of earlier work by Joyal and Moerdijk [JM, §§1 and 5].

• In §1.3, we study categories with a class of morphisms that have good properties with regards to pullbacks and images, such as regular categories.

[8] See definition 2.1.4 and lemma 2.3.12(a).

xix Introduction

• In §1.4, we consider the problem of adding exact quotients—i.e. coequalisers of equivalence relations that behave well under pullback—to a category with a class of covering morphisms. Specifically, we will see a sheaf-theoretic construction of the exact completion of a category equipped with a unary topology in the sense of [Shulman, 2012].

• In §1.5, we study extensive categories and we examine properties of their exact completions. In particular, we will see that the exact completion of an extensive category equipped with a superextensive coverage is a pretopos.

Charted objects

In the second chapter, we use the concepts introduced in the first chapter to construct categories of charted objects, i.e. spaces built from local models.

• In §2.1, we consider full subcategories of pretoposes for which the associated Yoneda representation is fully faithful and we identify the essential image of such Yoneda representations.

• In §2.2, we define various notions of categories equipped with structure making it possible to interpret basic notions of topology such as open embeddings and local homeomorphisms.

• In §2.3, we examine the properties of the category of extents—i.e. objects in a gros pretopos equipped with an étale atlas.

• In §2.4, we investigate suﬃcient conditions for a functor between gros pretoposes to preserve atlases and extents. Specifically, we will see that a functor between gros pretoposes will preserve local homeomorphisms between extents if it preserves coproducts, some coequalisers, and pullbacks of local homeomorphisms between distinguished objects.

• In §2.5, we characterise the category of extents by a universal property in a special case, namely, when étale morphisms coincide with local homeomorphisms and the coverage is generated by local homeomorphisms. xx Guide for readers

Specificities

In the final chapter, we see specific examples of the notions introduced inthe preceding chapters.

• In §3.1, we examine three classes of continuous maps of topological spaces that arise by relativising the notion of compactness.

• In §3.2, we construct a combinatorial example of a gros pretopos based on discrete fibrations of simplicial sets, which are closely related to discrete fibrations of categories.

• In §3.3, we construct an admissible ecumene for which the charted objects are the smooth manifolds of fixed dimension and cardinality.

• In §3.4, we construct admissible ecumenae from categories of topological spaces and investigate when a topological space is representable by a charted object.

• In §3.5, we see two prima facie different ways of defining schemes as extents in a gros pretopos and show that they are the same.

Guide for readers

Prerequisites

I assume the reader is familiar with category theory—at least Chapters I–V and X of [CWM]. The appendix contains some material on topics not covered in op. cit.

Conventions

Following [CWM], ‘category’ always means a category with a set of objects and a set of morphisms, whereas ‘metacategory’ refers to a category with a class of objects and a class of morphisms. Nonetheless, from time to time, it is convenient to use terminology previously only defined for categories for metacategories as well. This can be justified by adopting a suitable universe axiom, but we willnot do so.

xxi Introduction

Internal structure

The following is one possible reading order: §§1.1, a.1, a.2, 1.2, 3.1, 1.3, a.3, 1.4, 1.5, 2.1, 2.2, 2.3, 3.2, 3.3, 2.4, 2.5, 3.4, 3.5. That said, because concrete examples are deferred to chapter III, readers may find it helpful to peek ahead from time to time. An index for finding definitions appears at the end ofthe document, after the bibliography. The material within each chapter and each section is laid out linearly; readers should avoid skipping to the middle of a section as there may be local conventions in force. Each section is divided into “paragraphs”, which are identified by a label printed in the margin.

xxii Chapter I Abstract topology

1.1 The relative point of view

Synopsis. We discuss the relative point of view of Grothendieck and define related terminology.

1.1.1 ※ Throughout this section, u� is an arbitrary category.

1.1.2 ¶ The central tenet of the relative point of view of Grothendieck is to regard morphisms 푓 : 푋 → 푌 in u� as objects (푋, 푓) in the slice category

u�∕푌 . In turn, objects in u�∕푌 are to be regarded as “families” of objects in u� indexed (or parametrised) by 푌 . This can be made precise in special

cases: for instance, there is a canonical equivalence between Set∕퐼 and Set퐼 .[1] Of course, from the relative point of view, pullback of morphisms is reindexing (or reparametrisation), so we should focus our attention on those properties and constructions which are compatible with pullback.

1.1.2(a) Definition. A morphism 푓 : 푋 → 푌 in u� is quadrable if it has the following property:

• For every morphism 푦 : 푌 ′ → 푌 in u�, there is a pullback square in u� of the form below: 푋′ 푋 푓 ′ 푌 푦 푌 We write qdbl u� for the set of all quadrable morphisms in u�.

[1] Depending on the definition of Set퐼 , this may depend on the axiom of replacement.

1 Abstract topology

1.1.2(b) Definition. A subset ℱ ⊆ mor u� is closed under pullback in u� if it has the following property: • For every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of ℱ, then 푓 ′ : 푋′ → 푌 ′ is also a member of ℱ.

1.1.2(c) Definition.A quadrable class of morphisms in u� is a subset ℱ ⊆ mor u� with the following properties: • Every member of ℱ is a quadrable morphism in u�. • ℱ is closed under pullback in u�.

1.1.3 Lemma. Let ℱ be a quadrable class of morphisms in u�. Given a morphism 푓 : 푋 → 푌 in u� and a monomorphism 푔 : 푌 → 푍 in u�, if the composite 푔 ∘ 푓 : 푋 → 푍 is a member of ℱ, then the morphism 푓 : 푋 → 푌 is also a member of ℱ.

Proof. The following is a pullback square in u�:

푋 푋 푓 푔∘푓

푌 푔 푍

Thus the left vertical arrow is a member of ℱ if the right vertical arrow is a member of ℱ. ■

1.1.4(a) Definition.A class of fibrations in u� is a subset ℱ ⊆ mor u� that satisfies the following axioms:

• ℱ is a quadrable class of morphisms in u�.

• For every object 푋 in u�, id : 푋 → 푋 is a member of ℱ.

• ℱ is closed under composition.

2 1.1. The relative point of view

1.1.4(b) Definition. Let ℱ be a class of fibrations in u� and let 푆 be an object in

u�. An object (푋, 푓) in u�∕푆 is ℱ-fibrant if the morphism 푓 : 푋 → 푆 is a member of ℱ.

We write ℱ(푆) for the full subcategory of u�∕푆 spanned by the ℱ-fibrant objects.

1.1.4(c) Example. iso u�, the class of isomorphisms in u�, is the smallest class of fibrations in u�.

1.1.4(d) Example. The class of quadrable morphisms in u� is the largest class of fibrations in u�. (Note that the pullback pasting lemma implies that the class of quadrable morphisms in u� is closed under composition.)

1.1.4(e) Example. The class of quadrable split epimorphisms in u� is a class of fibrations in u�.

1.1.5 ¶ Let ℱ be a class of fibrations in u�. Consider a commutative diagram in u� of the form below,

푓 푔 푋 푍 푌 푎 푐 푏 퐴 퐶 퐵 ℎ 푘 where the vertical arrows are members of ℱ. Suppose u� has pullbacks. It

is not true that the induced morphism 푎 ×푐 푏 : 푋 ×푍 푌 → 퐴 ×퐶 퐵 is a member of ℱ in general. Rather:

If is a member of , then Pullbacks Lemma. ⟨푎, 푓⟩ : 푋 → 퐴×퐶 푍 ℱ 푎×푐 푏 : 푋×푍 푌 → and fibrations is also a member of . 퐴 ×퐶 퐵 ℱ

Proof. By considering the following commutative diagram in u�,

푎×푐id푌 푋 ×푍 푌 퐴 ×퐶 푌 푌 id 푔 퐴×id퐶 푔 푋 퐴 × 푍 푍 ⟨푎,푓⟩ 퐶 푐

퐴 퐶 ℎ

3 Abstract topology

we see that 푎 ×푐 id푌 : 푋 ×푍 푌 → 퐴 ×퐶 푌 is a member of ℱ. On the other hand, by the pullback pasting lemma, we also have the following pullback square in u�,

퐴 ×퐶 푌 푌 id 퐴×id퐶 푏 푏

퐴 ×퐶 퐵 퐵

so id is also a member of . Hence, 퐴 ×id퐶 푏 : 퐴 ×퐶 푌 → 퐴 ×퐶 퐵 ℱ

푎 ×푐 푏 : 푋 ×푍 푌 → 퐴 ×퐶 퐵 is indeed a member of ℱ. ■

1.1.6 ¶ Let ℱ be a quadrable class of morphisms in u�. The following can be regarded as a generalisation of the Hausdorff separation axiom, particularly when ℱ is regarded as a class of closed embeddings.

Definition. An object 푋 in u� is ℱ-separated if, for every object 푇 in u�

and every parallel pair 푥0, 푥1 : 푇 → 푋 in u�, there is an equaliser diagram in u� of the form below,

푥 푡 0 푇 ′ 푇 푋 푥1

where 푡 : 푇 ′ → 푇 is a member of ℱ.

Example. An object in u� is subterminal if and only if it is (iso u�)-separated.

Diagonal Lemma. Let 푋 be an object in u�. Assuming 푋 × 푋 exists in u�, the fol- criterion for lowing are equivalent: separatedness (i) 푋 is an ℱ-separated object in u�. The diagonal is a member of . (ii) Δ푋 : 푋 → 푋 × 푋 ℱ

Proof. (i) ⇒ (ii). The following is an equaliser diagram in u�,

Δ 푋 푋 푋 × 푋 푋

where the parallel pair of arrows are the two projections. Thus the diag-

onal Δ푋 : 푋 → 푋 × 푋 is a member of ℱ.

4 1.1. The relative point of view

(ii) ⇒ (i). Given any parallel pair 푥0, 푥1 : 푇 → 푋 in u�, we have the following pullback square in u�,

푇 ′ 푋

푡 Δ푋 푇 푋 × 푋 ⟨푥0,푥1⟩

and given any such pullback square,

푥 푡 0 푇 ′ 푇 푋 푥1

is an equaliser diagram in u�. Note that 푡 : 푇 ′ → 푇 is a member of ℱ, as required. ■

1.1.7 ¶ Let ℱ be a quadrable class of morphisms in u�. Given an object 푌 in

u�, let ℱ푌 be the class of morphisms in u�∕푌 whose underlying morphism

in u� is a member of ℱ. It is not hard to see that ℱ푌 is a quadrable class of

morphisms in u�∕푌 . This allows us to extend the definition of ‘separated’ from objects in u� to morphisms in u�.

Definition. A morphism 푓 : 푋 → 푌 in u� is ℱ-separated if the object

(푋, 푓) in u�∕푌 is ℱ푌 -separated.

Remark. Assuming 1 is a terminal object in u�, an object 푋 in u� is ℱ- separated if and only if the unique morphism 푋 → 1 in u� is ℱ-separated.

Properties Proposition. of separated (i) Assuming every isomorphism in u� is a member of ℱ, every monomorphisms morphism in u� is ℱ-separated. (ii) Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if the composite 푔 ∘ 푓 : 푋 → 푍 is ℱ-separated, then the morphism 푓 : 푋 → 푌 is also ℱ-separated. (iii) The class of ℱ-separated morphisms in u� is closed under pullback in u�. (iv) Assuming ℱ is closed under composition, the class of ℱ-separated morphisms in u� is closed under composition.

5 Abstract topology

Proof. (i)–(ii). Straightforward.

(iii). Consider a pullback square in u�:

푥 푋′ 푋 푓 ′ 푓 푌 ′ 푌 ℎ Suppose 푓 : 푋 → 푌 is an ℱ-separated morphism in u�. We must verify that 푓 ′ : 푋′ → 푌 ′ is also an ℱ-separated morphism in u�. Let ′ ′ ′ be morphisms in such that ′ ′ ′ ′ . 푥0, 푥1 : 푇 → 푋 u� 푓 ∘ 푥0 = 푓 ∘ 푥1 Then ′ ′ , and since is -separated, there 푓 ∘ 푥 ∘ 푥0 = 푓 ∘ 푥 ∘ 푥1 푓 : 푋 → 푌 ℱ is an equaliser diagram in u� of the form below,

푥∘푥′ 푡 0 푇 ′ 푇 푋 ′ 푥∘푥1

where 푡 : 푇 ′ → 푇 is a member of ℱ. Note that the universal property of pullbacks implies that ′ ′ . Thus, 푥0 ∘ 푡 = 푥1 ∘ 푡

푥′ 푡 0 푇 ′ 푇 푋′ ′ 푥1 is also an equaliser diagram in u�, and this completes the proof.

(iv). Let 푥0, 푥1 : 푇 → 푋 be morphisms in u� such that 푔∘푓 ∘푥0 = 푔∘푓 ∘푥1. Since 푔 : 푌 → 푍 is ℱ-separated, there is an equaliser diagram in u� of the form below, 푓∘푥 푡 0 푇 ′ 푇 푌 푓∘푥1 where 푡 : 푇 ′ → 푇 is a member of ℱ. Since 푓 : 푋 → 푌 is ℱ-separated, there is an equaliser diagram in u� of the form below,

푥 ∘푡 푡′ 0 푇 ″ 푇 ′ 푋 푥1∘푡

where 푡′ : 푇 ″ → 푇 ′ is a member of ℱ. It is straightforward to verify that the following is an equaliser diagram in u�,

푥 푡∘푡′ 0 푇 ″ 푇 푋 푥1

6 1.1. The relative point of view

and since ℱ is closed under composition, this completes the proof that 푓 : 푋 → 푌 is ℱ-separated. ■

1.1.8 ¶ Let ℱ be a quadrable class of morphisms.

Properties of Proposition. separated objects (i) Assuming every isomorphism in u� is a member of ℱ, every subterminal object in u� is ℱ-separated. (ii) Given a morphism 푓 : 푋 → 푌 in u�, if 푋 is a ℱ-separated object in u�, then 푓 : 푋 → 푌 is a ℱ-separated morphism in u�. (iii) Given objects 푋 and 푌 in u�, if 푋 × 푌 exists in u� and 푌 is ℱ- separated, then the projection 푋 × 푌 → 푋 is ℱ-separated. (iv) Assuming ℱ is closed under composition, if 푓 : 푋 → 푌 is an ℱ- separated morphism in u� and 푌 is an ℱ-separated object in u�, then 푋 is also an ℱ-separated object in u�.

Proof. Omitted. (The arguments are similar to those in the proof of proposition 1.1.7.) ◊

1.1.9 ¶ Let ℱ be a class of fibrations in u�.

Lemma. Let 푓 : 푋 → 푌 and 푔 : 푌 → 푍 be morphisms in u�. If 푔 : 푌 → 푍 is ℱ-separated and 푔 ∘ 푓 : 푋 → 푍 is a member of ℱ, then 푓 : 푋 → 푌 is also a member of ℱ.

Proof. Consider a pullback square in u� of the form below: 푞 푋 ×푍 푌 푌 푝 푔 푋 푍 푔∘푓

By proposition 1.1.7, the projection 푝 : 푋 ×푍 푌 → 푋 is ℱ-separated.

The following is an equaliser diagram in u�∕푋, id ⟨id푋,푓⟩ (푋, id푋) (푋 ×푍 푌 , 푝) (푋 ×푍 푌 , 푝) ⟨푝,푓∘푝⟩

thus ⟨id푋, 푓⟩ : 푋 → 푋 ×푍 푌 is a member of ℱ. Since 푔 ∘ 푓 : 푋 → 푍 is

a member of ℱ, the projection 푞 : 푋 ×푍 푌 → 푌 is also a member of ℱ.

Thus, 푓 = 푞 ∘ ⟨id푋, 푓⟩ : 푋 → 푌 is indeed a member of ℱ. ■

7 Abstract topology

1.1.10 Definition.A class of separated fibrations in u� is a subset ℱ ⊆ mor u� that satisfies the following axioms:

• ℱ is a class of fibrations in u�.

• Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, assuming 푔 : 푌 → 푍 is a member of ℱ, the composite 푔 ∘ 푓 : 푋 → 푍 is a member of ℱ if and only if the morphism 푓 : 푋 → 푌 is a member of ℱ.

Recognition Lemma. Let ℱ be a class of fibrations in u�. The following are equivalent: principles for (i) ℱ is a class of separated fibrations in u�. classes of separated fibrations (ii) Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if 푓 : 푋 → 푌 is a monomorphism in u� and both 푔 ∘ 푓 : 푋 → 푍 and 푔 : 푌 → 푍 are members of ℱ, then 푓 : 푋 → 푌 is also a member of ℱ. Given morphisms and in , if (iii) 푓 : 푋 → 푌 푔 : 푌 → 푋 u� 푔 ∘ 푓 = id푋 and 푔 : 푌 → 푋 is a member of ℱ, then 푓 : 푋 → 푌 is also a member of ℱ. If is a member of , then the relative diagonal (iv) 푔 : 푌 → 푍 ℱ Δ푔 : is a member of . 푌 → 푌 ×푍 푌 ℱ (v) Every member of ℱ is an ℱ-separated morphism in u�.

Proof. (i) ⇒ (ii), (ii) ⇒ (iii), (iii) ⇒ (iv). Immediate.

(iv) ⇔ (v). Apply the diagonal criterion for separatedness (lemma 1.1.6).

(iv) ⇒ (i). Let 푓 : 푋 → 푌 and 푔 : 푌 → 푍 be morphisms in u�. Suppose both 푔 ∘ 푓 : 푋 → 푍 and 푔 : 푌 → 푍 are members of ℱ. We must show that 푓 : 푋 → 푌 is also a member of ℱ. Since ℱ is a quadrable class of morphisms in u�, we have the following commutative diagram in u�,

푓 푋 푌 id ⟨id푋,푓⟩ Δ푔

푋 ×푍 푌 푌 ×푍 푌 푌 푔

푋 푌 푍 푓 푔

8 1.1. The relative point of view

where every square and every rectangle is a pullback diagram in u�. By

hypothesis, Δ푔 : 푌 → 푌 ×푍 푌 is a member of ℱ, so ⟨id푋, 푓⟩ : 푋 →

푋 ×푍 푌 is also a member of ℱ. Similarly, 푔 ∘ 푓 : 푌 → 푍 is a member of

ℱ, so the projection 푋 ×푍 푌 → 푌 is also a member of ℱ. But ℱ is closed under composition, so 푓 : 푋 → 푌 is indeed a member of ℱ. ■

Remark. Let ℱ be a class of fibrations and let ℱmono be the class of

monomorphisms in u� that are members of ℱ. Then ℱmono is a class of separated fibrations in u�.

1.1.11 ¶ Let ℱ be a class of fibrations in u�.

Definition. A morphism in u� is ℱ-perfect if it is ℱ-separated and also a member of ℱ.

Properties Proposition. of perfect (i) Every monomorphism in u� that is a member of ℱ is ℱ-perfect. morphisms (ii) The class of ℱ-perfect morphisms in u� is a class of separated fibrations.

Proof. Straightforward. (Use proposition 1.1.7 and lemma 1.1.10.) ⧫

1.1.12 ¶ Let 푆 be an object in u� and let ℱ be a class of separated fibrations.

The inclusion creates limits of all finite dia- Limits of finite Proposition. ℱ(푆) ↪ u�∕푆 diagrams of grams. separated fibrations Proof. It is clear that (푆, id푆) is a terminal object in both ℱ(푆) and u�∕푆. Similarly, since ℱ is a class of fibrations, ℱ(푆) has binary products and

the inclusion ℱ(푆) ↪ u�∕푆 preserves binary products. Thus, it suﬃces to

show that ℱ(푆) has equalisers and the inclusion ℱ(푆) ↪ u�∕푆 preserves equalisers, but this is a corollary of lemma 1.1.10. ■

1.1.13 ¶ We briefly recall the notion of orthogonality.

9 Abstract topology

Definition. An object 푆 in u� is right orthogonal to a morphism 푓 : 푋 → 푌 in u� if the induced map

u�(푓, 푆) : u�(푌 , 푆) → u�(푋, 푆)

is a bijection. More generally, 푆 is right orthogonal to a subset ℒ ⊆ mor u� if 푆 is right orthogonal to every member of ℒ.

Limits of objects Proposition. Let ℒ be a subset of mor u�. The full subcategory of u� right ortho- spanned by the objects that are right orthogonal to ℒ is closed under gonal to a class limits of all diagrams. of morphisms

Proof. Straightforward. ⧫

Remark. In particular, every terminal object in u� is right orthogonal to every morphism in u�. (However, u� may not have any terminal objects at all.)

1.1.14 ¶ In consideration of the relative point of view, it behoves us to extend the notion of orthogonality from objects to morphisms.

Definition. A morphism 푝 : 푍 → 푆 in u� is right orthogonal to a morphism 푓 : 푋 → 푌 in u� if the following is a pullback square in Set:

u�(푌 ,푝) u�(푌 , 푍) u�(푌 , 푆)

u�(푓,푍) u�(푓,푆) u�(푋, 푍) u�(푋, 푆) u�(푋,푝)

More generally, 푝 : 푍 → 푆 is right orthogonal to a subset ℒ ⊆ mor u� if 푝 : 푍 → 푆 is right orthogonal to every member of ℒ. We write ℒ⊥ for the set of all morphisms in u� that are right orthogonal to ℒ.

Remark. Assuming 1 is a terminal object in u�, an object 푋 in u� is right orthogonal to ℒ if and only if the unique morphism 푋 → 1 is a member of ℒ⊥.

10 1.1. The relative point of view

Alternative Lemma. Let 푝 : 푍 → 푆 and 푓 : 푋 → 푌 be morphisms in u�. The criteria for right following are equivalent: orthogonality (i) The morphism 푝 : 푍 → 푆 in u� is right orthogonal to the morphism 푓 : 푋 → 푌 in u�. (ii) The object (푍, 푆, 푝) in (u� ↓ u�) is right orthogonal to the morphism in . (푓, id푌 ) : (푋, 푌 , 푓) → (푌 , 푌 , id푌 ) (u� ↓ u�) For every morphism in , the object in is (iii) 푞 : 푌 → 푆 u� (푍, 푝) u�∕푆 right orthogonal to the morphism in . 푓 : (푋, 푞 ∘ 푓) → (푌 , 푞) u�∕푆

Proof. Straightforward. ⧫

Properties of Proposition. Let ℒ be a subset of mor u�. morphisms (i) Every isomorphism in u� is a member of ℒ⊥. right ortho- (ii) The full subcategory of corresponding to ⊥ is closed under gonal to a class (u� ↓ u�) ℒ of morphisms limits of all diagrams. (iii) ℒ⊥ is closed under pullback in u�. (iv) Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, assuming 푔 : 푌 → 푍 is a member of ℒ⊥, 푓 : 푋 → 푌 is a member of ℒ⊥ if and only if 푔 ∘ 푓 : 푋 → 푍 is a member of ℒ⊥. (v) Given a morphism 푓 : 푋 → 푌 in u�, assuming 푌 is right orthogonal to ℒ, 푋 is right orthogonal to ℒ if and only if 푓 : 푋 → 푌 is a member of ℒ⊥.

Proof. (i) and (ii). Apply proposition 1.1.13 to lemma 1.1.14.

(iii) and (iv). Use the pullback pasting lemma.

(v). This is a consequence of the fact that the pullback of a bijection is again a bijection. ■

1.1.15 Remark. In particular, for any subset ℒ ⊆ mor u�, if ℒ⊥ ⊆ qdbl u�, then ℒ⊥ is a class of separated fibrations in u�.

1.1.16 ¶ It is sometimes useful to weaken the notion of right orthogonality by replacing ‘bijection’ with ‘injection’.

11 Abstract topology

Right orthogon- Lemma. Let 푓 : 푋 → 푌 be a morphism in u� and let 푆 be an object in u�. ality of diagonals Assuming the product 푆 × 푆 exists in u�, the following are equivalent: (i) The induced map

u�(푓, 푆) : u�(푌 , 푆) → u�(푋, 푆)

is injective. The diagonal is right orthogonal to . (ii) Δ푆 : 푆 → 푆 × 푆 푓 : 푋 → 푌

Proof. Consider the diagram below:

Δu�(푌 ,푆) u�(푌 , 푆) u�(푌 , 푆) × u�(푌 , 푆)

u�(푓,푆) u�(푓,푆)×u�(푓,푆) u�(푋, 푆) u�(푋, 푆) × u�(푋, 푆) Δu�(푋,푆)

Clearly, this is a pullback square if and only if Δ푆 : 푆 → 푆 × 푆 is right orthogonal to 푓 : 푋 → 푌 . On the other hand, this is a pullback square if and only if u�(푓, 푆) : u�(푌 , 푆) → u�(푋, 푆) is injective, so we are done. ■

12 1.2. Local properties of morphisms

1.2 Local properties of morphisms

Synopsis. We consider variations on what it means for a morphism to have a property locally with respect to a coverage.

Prerequisites. §§1.1, a.1, a.2.

1.2.1 ¶ In general, given a topological space 푋 and a property 푃 of topological spaces, we say that 푋 has property 푃 locally if, for every open subspace 푈 ⊆ 푋, there is a cover Φ of 푈 such that every element of Φ has property 푃 . On the other hand, in the relative setting, there are at least two possible ways to interpret ‘locally’. For instance, given a continuous map 푓 : 푋 → 푌 and a property 푃 of continuous maps, we may say that 푓 : 푋 → 푌 has property 푃 locally on the domain if, for every open subspace 푈 ⊆ 푋, there is a cover Φ of 푈 such that, for every 푈 ′ ∈ Φ, the restriction 푓 : 푈 ′ → 푌 has property 푃 . Or, we may say that 푓 : 푋 → 푌 has property 푃 locally on the base if, for every open subspace 푉 ⊆ 푌 , there is a cover Ψ of 푉 such that, for every 푉 ′ ∈ Ψ, the restriction 푓 : 푓 −1푉 ′ → 푉 ′ has property 푃 . We could go even further by combining the two interpretations. In this section, we study generalisations of these ideas in the abstract setting of a category with a coverage. It should be noted that the notion of having a property locally on the base is straightforwardly generalised to objects in a fibred category, as is the notion of having a property locally on the domain, but combining the two is diﬃcult. As such, we will only discuss the case of morphisms. This is partially a generalisation of earlier work by Joyal and Moerdijk [JM, §§1 and 5].

1.2.2 ※ Throughout this section:

• u� is a category.

• 햩 is a coverage on u�.

• ℬ is a set of morphisms in u� containing all identity morphisms and closed under composition.

13 Abstract topology

• u� is a set of morphisms in u�.

1.2.3(a) Definition. The class u� is ℬ-sifted if it has the following property:

• If 푓 : 푋 → 푌 is a member of ℬ and 푔 : 푌 → 푍 is a member of u�, then the composite 푔 ∘ 푓 : 푋 → 푍 is also a member of u�.

1.2.3(b) Definition. The class u� is ℬ-cosifted if it has the following property:

• If 푓 : 푋 → 푌 is a member of u� and 푔 : 푌 → 푍 is a member of ℬ, then the composite 푔 ∘ 푓 : 푋 → 푍 is also a member of u�.

1.2.3(c) Definition. The class u� is ℬ-bisifted if it is both ℬ-sifted and ℬ-cosifted.

Example. ℬ itself is ℬ-bisifted.

1.2.4 Definition. A morphism 푓 : 푋 → 푌 in u� is of u�-type (ℬ, 햩)-semilocally on the domain if it has the following property:

• There is a 햩-covering ℬ-sink Φ on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is a member of u�.

A morphism in u� is of u�-type 햩-semilocally on the domain if it is of u�-type (mor u�, 햩)-semilocally on the domain.

Properties of Proposition. Let u�̂be the class of morphisms in u� of u�-type (ℬ, 햩)-semi- morphisms of locally on the domain. a given type (i) We have .̂ semilocally u� ⊆ u� on the domain (ii) If u� is ℬ-cosifted, then u�̂ is also ℬ-cosifted. (iii) Assuming ℬ is a quadrable class of morphisms in u�, if u� is ℬ-sifted, then u�̂ is also ℬ-sifted. (iv) Assuming both ℬ and u� are quadrable classes of morphisms in u�, for every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of u�,̂ then 푓 ′ : 푋′ → 푌 ′ is also a member of u�.̂

14 1.2. Local properties of morphisms

(v) Every morphism in u� of u�-typê (ℬ, 햩)-semilocally on the domain is a member of u�.̂

Proof. Straightforward. (For (iii)–(v), use proposition a.2.14.) ⧫

1.2.5 Definition. A morphism 푓 : 푋 → 푌 in u� is of u�-type (ℬ, 햩)-locally on the domain if it has the following property:

• For every object (푈, 푥) in u�∕푋, if 푥 : 푈 → 푋 is a member of ℬ, then 푓 ∘ 푥 : 푈 → 푌 is of u�-type (ℬ, 햩)-semilocally on the domain.

A morphism in u� is of u�-type 햩-locally on the domain if it is of u�-type (mor u�, 햩)-locally on the domain.

Properties of Proposition. Let u�̂be the class of morphisms in u� of u�-type (ℬ, 햩)-locally morphisms of a on the domain. given type locally (i) Every member of ̂ is a morphism in of -type -semilocally on the domain u� u� u� (ℬ, 햩) on the domain.

(ii) u�̂ is ℬ-sifted.

(iii) If u� is ℬ-sifted, then u� ⊆ u�.̂

(iv) If u� is ℬ-cosifted, then u�̂ is also ℬ-cosifted.

(v) Assuming u� is ℬ-sifted and both ℬ and u� are quadrable classes of morphisms in u�, for every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of u�,̂ then 푓 ′ : 푋′ → 푌 ′ is also a member of u�.̂

(vi) Assuming both ℬ and u� are quadrable classes of morphisms in u�, if u� is closed under composition, then the class of morphisms in u� of u�-type (ℬ, 햩)-locally on the domain is also closed under composition.

(vii) Every morphism in u� of u�-typê (ℬ, 햩)-locally on the domain is a member of u�.̂

15 Abstract topology

Proof. (i)–(iv). Straightforward.

(v). Use the pullback pasting lemma to reduce to proposition 1.2.4.

(vi). Let 푓 : 푋 → 푌 and 푔 : 푌 → 푍 be morphisms in u� of u�-type (ℬ, 햩)-locally on the domain. Then there is a 햩-covering ℬ-sink Φ on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘푥 : 푈 → 푌 is a member of u�, and there is a 햩-covering ℬ-sink Ψ on 푌 with the same property mutatis mutandis. For each (푈, 푥) ∈ Φ and each (푉 , 푦) ∈ Ψ, we may choose a pullback square in u� of the form below, 푢 푇 푈 푣 푓∘푥

푉 푦 푌 where 푢 : 푇 → 푈 is a member of ℬ and 푣 : 푇 → 푉 is a member of u�; thus, by hypothesis, 푥 ∘ 푢 : 푇 → 푋 is a member of ℬ and 푔 ∘ 푓 ∘ 푥 ∘ 푢 = 푔 ∘ 푦 ∘ 푢 : 푇 → 푍 is a member of u�. Hence, by proposition a.2.14, 푔 ∘ 푓 : 푋 → 푍 is of u�-type (ℬ, 햩)-locally on the domain.

(vii). Let u�2 be the class of morphisms in u� that are of u�-type (ℬ, 햩)-semi-

locally on the domain. By (i), u�̂ ⊆ u�2, and we know that every morphism

in u� of u�2-type (ℬ, 햩)-semilocally on the domain is a member of u�2, so every morphism in u� of u�-typê (ℬ, 햩)-semilocally on the domain is also a

member of u�2. Thus, every morphism in u� of u�-typê (ℬ, 햩)-locally on the domain is a member of u�.̂ ■

1.2.6 Lemma. Let 푓 : 푋 → 푌 be a morphism in u�. If ℬ is a quadrable class When semi- of morphisms in u� and u� is a ℬ-sifted class of morphisms in u�, then the locally on following are equivalent: the domain implies locally (i) 푓 : 푋 → 푌 is of u�-type (ℬ, 햩)-locally on the domain. on the domain (ii) 푓 : 푋 → 푌 is of u�-type (ℬ, 햩)-semilocally on the domain.

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). Let 푓 : 푋 → 푌 be a member of ℬ and let 푔 : 푌 → 푍 be a morphism in u� of u�-type (ℬ, 햩)-semilocally on the domain. We must show that 푔 ∘ 푓 : 푋 → 푍 is of u�-type (ℬ, 햩)-semilocally on the domain.

16 1.2. Local properties of morphisms

There is a 햩-covering ℬ-sink Ψ on 푌 such that, for every (푉 , 푦) ∈ Ψ, 푔 ∘ 푦 : 푉 → 푍 is a member of u�. For each (푉 , 푦) ∈ Ψ, choose a pullback square in u� of the form below:

푓 ∗푉 푉

푓 ∗푦 푦 푋 푌 푓

Since u� is ℬ-sifted, 푔 ∘ 푓 ∘ 푓 ∗푦 : 푓 ∗푉 → 푍 is a member of u�. Moreover, ∗ ∗ by proposition a.2.14, {(푓 푉 , 푓 푦) | (푉 , 푦) ∈ Ψ} is a 햩-covering ℬ-sink on 푋. The claim follows. ■

1.2.7 Definition. A morphism 푓 : 푋 → 푌 in Fam(u�) is of u�-type if it has the following property:

• For every (푗, 푖) ∈ idx 푓, the morphism 푓(푗, 푖) : 푋(푖) → 푌 (푗) is a member of u�.

Properties of Proposition. matrices of (i) If every isomorphism in u� is a member of u�, then every coproduct morphisms of injection in Fam(u�) is of u�-type. a given type (ii) If u� is closed under composition, then the class of morphisms in Fam(u�) of u�-type is also closed under composition.

(iii) If u� is a quadrable class of morphisms in u�, then the class of morphisms in Fam(u�) of u�-type is a quadrable class of morphisms in Fam(u�).

Proof. Straightforward. ⧫

1.2.8 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is familially of u�-type if there is a pair (Φ, Ψ) with the following properties:

• Φ is a familial representation of 퐴.

• Ψ is a familial representation of 퐵.

• For each (푋, 푎) ∈ Φ, there exist a unique (푌 , 푏) ∈ Ψ and a unique morphism 푓 : 푋 → 푌 in u� such that ℎ(푎) = 푏 ⋅ 푓 and 푓 : 푋 → 푌 is a member of u�.

17 Abstract topology

Recognition prin- Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). The following are ciple for morph- equivalent: isms familially of a given type (i) The morphism ℎ : 퐴 → 퐵 is familially of u�-type.

(ii) There is a morphism 푓 : 푋 → 푌 in Fam(u�) of u�-type such that (⨿푋, ⨿푌 , ⨿푓) ≅ (퐴, 퐵, ℎ) (in (Psh(u�) ↓ Psh(u�))).

Proof. Straightforward. ⧫

1.2.9 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is of u�-type 햩-semilocally on the base if there is a 햩-local generating set Ψ of elements of 퐵 that satisfies the following condition:

• For each (푌 , 푏) ∈ Ψ, there is a representation (푋, (푓, 푎)) of Pb(푏 ⋅ −, ℎ) such that the morphism 푓 : 푋 → 푌 in u� is a member of u�.

Example. If 푓 : 푋 → 푌 is a member of u�, then h푓 : h푋 → h푌 is of u�-type semilocally on the base.

Recognition Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). The following are principle for equivalent: morphisms of presheaves of (i) The morphism ℎ : 퐴 → 퐵 is of u�-type semilocally on the base. a given type There exist a family where each is a morphism (ii) (푓푖 | 푖 ∈ 퐼) 푓푖 푋푖 → semilocally 푌 in u� that is a member of u� and a pullback square in Psh(u�) of the on the base 푖 form below, h ∐푖∈퐼 푋푖 퐴

∐푖∈퐼 h푓 ℎ h ∐푖∈퐼 푌푖 퐵 where h is -locally surjective. ∐푖∈퐼 푌푖 ↠ 퐵 햩

Proof. Straightforward. ⧫

18 1.2. Local properties of morphisms

Properties of Proposition. morphisms of (i) The class of morphisms in Psh(u�) of u�-type 햩-semilocally on the presheaves of base is closed under (possibly infinitary) coproduct in Psh(u�). a given type semilocally (ii) Given a pullback square in Psh(u�) of the form below, on the base 퐴̃ 퐴

ℎ̃ ℎ 퐵̃ 퐵

where 퐵̃ ↠ 퐵 is 햩-locally surjective, if ℎ̃ : 퐴̃ → 퐵̃ is of u�-type 햩- semilocally on the base, then ℎ : 퐴 → 퐵 is also of u�-type 햩-semilocally on the base.

Proof. Straightforward. ⧫

1.2.10 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is of u�-type 햩-locally on the base if it has the following property:

• For every element (푌 , 푏) of 퐵, the projection Pb(푏 ⋅ −, ℎ) → h푌 is of u�-type 햩-semilocally on the base.

Example. Assuming u� is a quadrable class of morphisms in u�, if 푓 :

푋 → 푌 is a member of u�, then h푓 : h푋 → h푌 is of u�-type 햩-locally on the base.

Properties of Proposition. morphisms of (i) Every morphism in Psh(u�) that is of u�-type 햩-locally on the base is presheaves of a also of u�-type 햩-semilocally on the base. given type locally on the base (ii) The class of morphisms in Psh(u�) of u�-type 햩-locally on the base is a quadrable class of morphisms in Psh(u�).

(iii) The class of morphisms in Psh(u�) of u�-type 햩-locally on the base is closed under (possibly infinitary) coproduct in Psh(u�).

(iv) If every identity morphism in u� is a member of u�, then every isomorphism in Psh(u�) is of u�-type 햩-locally on the base.

19 Abstract topology

(v) Given a pullback square in Psh(u�) of the form below,

퐴̃ 퐴

ℎ̃ ℎ 퐵̃ 퐵

where 퐵̃ ↠ 퐵 is 햩-locally surjective, if ℎ̃ : 퐴̃ → 퐵̃ is of u�-type 햩- locally on the base, then ℎ : 퐴 → 퐵 is also of u�-type 햩-locally on the base.

Proof. (i). Apply lemma 1.2.13, proposition a.2.14, and the pullback pasting lemma.

(ii)–(iv). Straightforward.

(v). Let (푌 , 푏) be an element of 퐵 and u� be the sieve on 푌 consisting

of the objects (푉 , 푦) in u�∕푌 such that 푏 ⋅ 푦 is in the image of 퐵̃ → 퐵. By construction, u� is a 햩-covering sieve on 푌 . Since ℎ̃ : 퐴̃ → 퐵̃ is of u�-type 햩-locally on the base, we may then apply the pullback pasting

lemma and proposition 1.2.9 to deduce that Pb(푏 ⋅ −, ℎ) → h푌 is of u�-type 햩-semilocally on the base. ■

1.2.11 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). If u� is a quadrable When semiloc- class of morphisms in u�, then the following are equivalent: ally on the base The morphism is of -type -locally on the base. implies locally (i) ℎ : 퐴 → 퐵 u� 햩 on the base (ii) The morphism ℎ : 퐴 → 퐵 is of u�-type 햩-semilocally on the base.

(iii) The subpresheaf 퐵′ ⊆ 퐵 is 햩-dense, where 퐵′ consists of the elements (푇 , 푏) of 퐵 such that there exist a morphism 푡 : 푈 → 푇 in u� that is a member of u� and a pullback square in Psh(u�) of the form below:

h푈 퐴 푡∘− ℎ h 퐵 푇 푏⋅−

Proof. Apply proposition a.2.14 and the pullback pasting lemma. ■

20 1.2. Local properties of morphisms

1.2.12 Definition. A morphism 푓 : 푋 → 푌 in u� is of u�-type 햩-locally on

the base (resp. of u�-type 햩-semilocally on the base) if h푓 : h푋 → h푌 is a morphism in Psh(u�) that is of u�-type 햩-locally on the base (resp. of u�-type 햩-semilocally on the base).

Properties of Proposition. Let u�̂ be the class of morphisms in u� of u�-type 햩-locally on morphisms of a the base. given type locally If every isomorphism in is a member of , then every isomorphism on the base (i) u� u� in u� is also a member of u�.̂

(ii) If u� is a quadrable class of morphisms in u�, then u� ⊆ u�.̂

(iii) For every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of u�,̂ then 푓 ′ : 푋′ → 푌 ′ is also a member of u�.̂

(iv) Every morphism in Psh(u�) that is of u�-typê 햩-locally on the base is also of u�-type 햩-locally on the base.

Proof. Apply proposition 1.2.10. ■

1.2.13 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is 햩-semilocally of u�- type if there is a 햩-local generating set Ψ of elements of 퐵 that satisfies the following condition:

• For each (푌 , 푏) ∈ Ψ, there is a 햩-local generating set Φ(푌 ,푏) of elements

of Pb(푏 ⋅ −, ℎ) such that, for every (푋, (푓, 푎)) ∈ Φ(푌 ,푏), the morphism 푓 : 푋 → 푌 in u� is a member of u�.

Example. If 푓 : 푋 → 푌 is a member of u�, then h푓 : h푋 → h푌 is 햩-semilocally of u�-type.

21 Abstract topology

Recognition Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). The following are principle for equivalent: morphisms (i) The morphism is -semilocally of -type. of presheaves ℎ : 퐴 → 퐵 햩 u� semilocally of (ii) There is a 햩-weak pullback square in Psh(u�) of the form below, a given type ⨿푋 퐴 ⨿푓 ℎ ⨿푌 퐵

where 푓 : 푋 → 푌 is a morphism in Fam(u�) of u�-type and ⨿푌 ↠ 퐵 is 햩-locally surjective.

Proof. Straightforward. ⧫

Properties of Proposition. morphisms (i) Every morphism in Psh(u�) that is familially of u�-type is also 햩- of presheaves semilocally of u�-type. semilocally of a given type (ii) The class of morphisms in Psh(u�) 햩-semilocally of u�-type is closed under (possibly infinitary) coproduct in Psh(u�).

(iii) Given a 햩-weak pullback square in Psh(u�) of the form below,

퐴̃ 퐴

ℎ̃ ℎ 퐵̃ 퐵

where 퐵̃ ↠ 퐵 is 햩-locally surjective, if ℎ̃ : 퐴̃ → 퐵̃ is 햩-semilocally of u�-type, then ℎ : 퐴 → 퐵 is also 햩-semilocally of u�-type.

Proof. Straightforward. (Use lemma 1.2.13 and proposition a.2.14; for (iii), also use the weak pullback pasting lemma (lemma a.2.19).) ⧫

1.2.14 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is 햩-locally of u�-type if it has the following property:

• For every element (푌 , 푏) of 퐵, the projection Pb(푏 ⋅ −, ℎ) → h푌 is 햩- semilocally of u�-type.

22 1.2. Local properties of morphisms

Example. Assuming u� is a quadrable class of morphisms in u�, if 푓 :

푋 → 푌 is a member of u�, then h푓 : h푋 → h푌 is 햩-locally of u�-type.

Properties of Proposition. morphisms (i) Every morphism in Psh(u�) that is 햩-locally of u�-type is also 햩- of presheaves semilocally of u�-type. locally of a given type (ii) The class of morphisms in Psh(u�) 햩-locally of u�-type is a quadrable class of morphisms in Psh(u�).

(iii) The class of morphisms in Psh(u�) 햩-locally of u�-type is closed under (possibly infinitary) coproduct in Psh(u�).

(iv) Assuming every identity morphism in u� is a member of u�, for every presheaf on and every set , the codiagonal is - 퐴 u� 퐼 ∐푖∈퐼 퐴 → 퐴 햩 locally of u�-type.

(v) If every identity morphism in u� is a member of u�, then every isomorphism in Psh(u�) is 햩-locally of u�-type.

(vi) Given a 햩-weak pullback square in Psh(u�) of the form below,

퐴̃ 퐴

ℎ̃ ℎ 퐵̃ 퐵

where 퐵̃ ↠ 퐵 is 햩-locally surjective, if ℎ̃ : 퐴̃ → 퐵̃ is 햩-locally of u�-type, then ℎ : 퐴 → 퐵 is also 햩-locally of u�-type.

(vii) If every identity morphism in u� is a member of u�, then every 햩- locally bijective morphism in Psh(u�) is 햩-locally of u�-type.

Proof. (i). Apply lemma 1.2.13, proposition a.2.14, and the weak pullback pasting lemma (lemma a.2.19).

(ii)–(v). Straightforward.

(vi). Let (푌 , 푏) be an element of 퐵 and u� be the sieve on 푌 consisting

of the objects (푉 , 푦) in u�∕푌 such that 푏 ⋅ 푦 is in the image of 퐵̃ → 퐵. By construction, u� is a 햩-covering sieve on 푌 . Since ℎ̃ : 퐴̃ → 퐵̃ is 햩- locally of u�-type, we may then apply the weak pullback pasting lemma

23 Abstract topology

and proposition 1.2.13 to deduce that Pb(푏 ⋅ −, ℎ) → h푌 is 햩-semilocally of u�-type.

(vii). Let ℎ : 퐴 → 퐵 be a 햩-locally bijective morphism of presheaves on u�. Then the following is a 햩-weak pullback square in Psh(u�):

id 퐴 퐴 id ℎ 퐴 퐵 ℎ

Thus, by (vi), the claim reduces to (v). ■

1.2.15 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). If u� is a quadrable When semi- class of morphisms in u�, then the following are equivalent: locally implies locally (i) The morphism ℎ : 퐴 → 퐵 is 햩-locally of u�-type.

(ii) The morphism ℎ : 퐴 → 퐵 is 햩-semilocally of u�-type.

Proof. Apply proposition a.2.14 and the weak pullback pasting lemma (lemma a.2.19). ■

1.2.16 ¶ The following definition is a variation on the collection axiom introduced in [JM, §1].

Definition. The 햩-local collection axiom for u� is the following: • Given a morphism 푓 : 푋 → 푌 in u� and a 햩-covering sieve u� on 푋, if 푓 : 푋 → 푌 is a member of u�, then there is a 햩-covering sink Ψ on 푌

such that, for each (푇 , 푦) ∈ Ψ, there is a 햩-local generating set Φ(푇 ,푦)

of elements of Pb(푦 ∘ −, 푓 ∘ −) such that, for each (푈, (푡, 푥)) ∈ Φ(푇 ,푦), 푡 : 푈 → 푇 is a member of u� and (푈, 푥) is in u� .

Remark. Assuming u� is closed under composition, if every 햩-covering sink contains a 햩-covering u�-sink, then u� satisfies the 햩-local collection axiom.

24 1.2. Local properties of morphisms

Alternative Lemma. The following are equivalent: criteria for the (i) u� satisfies the 햩-local collection axiom. local collec- (ii) For every morphism in , if is a member of tion axiom 푓 : 푋 → 푌 u� 푓 : 푋 → 푌 u�, then for every 햩-covering sieve u� on 푋, there is a 햩-weak pullback square in Psh(u�) of the form below,

′ 푝 ⨿푋 h푋 ⨿푓 ′ 푓∘− ′ ⨿푌 푞 h푌

where ′ is -locally surjective, ′ ′ ′ is a 푞 : ⨿푌 → h푌 햩 푓 : 푋 → 푌 ′ ′ morphism in Fam(u�) of u�-type, and for every element (푇 , 푥 ) of ⨿푋 , (푇 , 푝(푥′)) is in u� .

Proof. Straightforward. ⧫

1.2.17 Proposition. If u� is a quadrable class of morphisms in u� that satisfies Composition the 햩-collection axiom and is closed under composition, then the class of of morphisms morphisms in Psh(u�) 햩-locally of u�-type is also closed under composition. of presheaves locally of a Proof. In view of proposition 1.2.14, it suﬃces to prove the following given type statement:

• Given morphisms ℎ : 퐴 → 퐵 and 푘 : 퐵 → 퐶 in Psh(u�), if ℎ : 퐴 → 퐵 is 햩-locally of u�-type and 푘 : 퐵 → 퐶 is 햩-semilocally of u�-type, then the composite 푘 ∘ ℎ : 퐴 → 퐶 is also 햩-semilocally of u�-type.

So suppose ℎ : 퐴 → 퐵 is 햩-locally of u�-type and 푘 : 퐵 → 퐶 is 햩- semilocally of u�-type. By lemma 1.2.13, there is a 햩-weak pullback square in Psh(u�) of the form below,

⨿푌 퐵 ⨿푔 푘 ⨿푍 퐶

where ⨿푍 ↠ 퐶 is 햩-locally surjective and 푔 : 푌 → 푍 is a morphism in

Fam(u�) of u�-type. The projection (⨿푌 )×퐵 퐴 → ⨿푌 is also 햩-semilocally

25 Abstract topology

of u�-type, so there is a 햩-weak pullback square in Psh(u�) of the form below, ′ ⨿푋 (⨿푌 ) ×퐵 퐴 ⨿푓 ′

⨿푌 ′ ⨿푌 satisfying the conditions mutatis mutandis. Then, using lemma 1.2.16 and the hypothesis that u� satisfies the 햩-local collection axiom, we may find a 햩-weak pullback square in Psh(u�) of the form below,

⨿푌 ″ ⨿푌 ⨿푔″ ⨿푔 ⨿푍″ ⨿푍 where ⨿푍″ ↠ ⨿푍 is 햩-locally surjective, 푌 ″ → 푌 factors as a morphism 푌 ″ → 푌 ′ in Fam(u�) followed by the 햩-covering morphism 푌 ′ ↠ 푌 in Fam(u�), and 푔″ : 푌 ″ → 푍″ is a morphism in Fam(u�) of u�-type. Since u� is a quadrable class of morphisms in u�, by proposition 1.2.7, there is a pullback square in Psh(u�) of the form below,

⨿푋″ ⨿푋′ ⨿푓 ″ ⨿푓 ′ ⨿푌 ″ ⨿푌 ′ where 푓 ″ : 푋″ → 푌 ″ is a morphism in Fam(u�) of u�-type. Hence, we obtain a commutative diagram in Psh(u�) of the form below,

⨿푋″ 퐴 ⨿푔″ ℎ ⨿푌 ″ 퐵 ⨿푓 ″ 푘 ⨿푍″ 퐶 where, by the weak pullback pasting lemma (lemma a.2.19), both squares and the outer rectangle are all 햩-weak pullback diagrams in Psh(u�). Since the horizontal arrows are 햩-locally surjective, it follows from the hypothesis that u� is closed under composition that 푘 ∘ ℎ : 퐴 → 퐶 is 햩- semilocally of u�-type, as claimed. ■

26 1.2. Local properties of morphisms

1.2.18 Definition. A morphism 푓 : 푋 → 푌 in u� is 햩-locally of u�-type (resp.

햩-semilocally of u�-type) if h푓 : h푋 → h푌 is a morphism in Psh(u�) that is 햩-locally of u�-type (resp. 햩-semilocally of u�-type).

Lemma. Let 푟 : 퐹 → 퐴 be a 햩-locally surjective morphism in Psh(u�) and let ℎ : 퐴 → 퐵 be a morphism in Psh(u�) 햩-locally of u�-type. Assuming u� satisfies the 햩-local collection axiom, there is a 햩-weak pullback square in Psh(u�) of the form below,

푝 퐴̃ 퐴

ℎ̃ ℎ ̃ 퐵 푞 퐵

where 푞 : 퐵̃ ↠ 퐵 is 햩-locally surjective, 푝 : 퐴̃ → 퐴 factors through 푟 : 퐹 → 퐴, and ℎ̃ : 퐴̃ → 퐵̃ is familially of u�-type.

Proof. By lemma 1.2.13 and proposition 1.2.14, there is a 햩-weak pullback square in Psh(u�) of the form below,

⨿푋 퐴 ⨿푓 ℎ ⨿푌 퐵

where ⨿푌 ↠ 퐵 is 햩-locally surjective and 푓 : 푋 → 푌 is a morphism in

Fam(u�) of u�-type. By proposition a.2.14, the projection ⨿푋×퐴퐹 → ⨿푋 is 햩-locally surjective, so we may apply lemma 1.2.16 to obtain a 햩-weak pullback square in Psh(u�) of the form below,

⨿푋′ ⨿푋 ⨿푓 ′ ⨿푓 ⨿푌 ′ ⨿푌

where ⨿푌 ′ ↠ ⨿푌 is 햩-locally surjective, 푓 ′ : 푋′ → 푌 ′ is a morphism in Fam(u�) of u�-type, and ⨿푋′ ↠ ⨿푋 factors through the projection ′ ′ ′ ⨿푋 ×퐴 퐹 → ⨿푋. We can then take 퐴̃ = ⨿푋 , 퐵̃ = ⨿푌 , and ℎ̃ = ⨿푓 and use the weak pullback pasting lemma (lemma a.2.19) to complete the proof. ■

27 Abstract topology

Properties of Proposition. Let u�̂ be the class of morphisms in u� 햩-locally of u�-type. morphisms (i) If every isomorphism in u� is a member of u�, then every isomorphism locally of a in is also a member of .̂ given type u� u� (ii) If u� is a quadrable class of morphisms in u�, then u� ⊆ u�.̂

(iii) u�̂ is closed under pullback in u�.

(iv) If u� satisfies the 햩-local collection axiom, then u�̂ also satisfies the 햩-local collection axiom.

(v) If u� is a quadrable class of morphisms in u� that satisfies the 햩-local collection axiom and is closed under composition, then u�̂is also closed under composition.

(vi) Every morphism in Psh(u�) that is 햩-locally of u�-typê is also 햩- locally of u�-type.

Proof. Apply propositions 1.2.14 and 1.2.17, lemma 1.2.18, and the weak pullback pasting lemma (lemma a.2.19). ■

1.2.19 ¶ The following is a generalisation of Proposition 1.9 in [JM].

1.2.19(a) Lemma. Let 푓 : 푋 → 푌 be a monomorphism in u�. Assuming ℬ is a Recognition quadrable class of morphisms in u�, the following are equivalent: principle for (i) 푓 : 푋 → 푌 is of ℬ-type (ℬ, 햩)-semilocally on the domain. monomorphisms of a given (ii) 푓 : 푋 → 푌 is of ℬ-type 햩-semilocally on the domain. type semilocally on the domain Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). Let Φ be a 햩-covering sink on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is a member of ℬ. Then, by lemma 1.1.3, 푥 : 푈 → 푋 itself is a member of ℬ. Hence, 푓 : 푋 → 푌 is indeed of ℬ-type (ℬ, 햩)-semilocally on the domain. ■

1.2.19(b) Lemma. Let 푓 : 푋 → 푌 be a quadrable monomorphism in u� and let ℬ̂ be Recognition the class of morphisms in u� of ℬ-type (ℬ, 햩)-semilocally on the domain. principle for The following are equivalent: quadrable mono- is -semilocally of -type. morphisms (i) 푓 : 푋 → 푌 햩 ℬ semilocally of 28 a given type 1.2. Local properties of morphisms

(ii) 푓 : 푋 → 푌 is of ℬ-typê 햩-semilocally on the base.

Proof. (i) ⇒ (ii). Apply lemma 1.2.19(a).

(ii) ⇒ (i). Immediate. ■

1.2.20 ¶ The following is a generalisation of Proposition 1.10 in [JM].

Recognition Lemma. Let 푓 : 푋 → 푌 be a ℬ-separated morphism in u�. Assuming ℬ principle for is a class of fibrations in u�, the following are equivalent: separated morph- (i) 푓 : 푋 → 푌 is of ℬ-type (ℬ, 햩)-semilocally on the domain. isms of a given type semilocally (ii) 푓 : 푋 → 푌 is of ℬ-type 햩-semilocally on the domain. on the domain Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). Let Φ be a 햩-covering sink on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is a member of ℬ. Then, by lemma 1.1.9, 푥 : 푈 → 푋 itself is a member of ℬ. Hence, 푓 : 푋 → 푌 is indeed of ℬ-type (ℬ, 햩)-semilocally on the domain. ■

Remark. Since monomorphisms are always ℬ-separated, lemma 1.2.20 can be regarded as a generalisation of lemma 1.2.19(a), at least in the case where ℬ is a class of fibrations.

Perfect morph- Proposition. Suppose u� is the class of morphisms in u� 햩-semilocally of isms and locality ℬ-type. Assume the following hypotheses: • ℬ is a class of fibrations in u�. • Every morphism in u� of ℬ-type (ℬ, 햩)-semilocally on the domain is a member of ℬ. • Every quadrable morphism in u� of ℬ-type 햩-semilocally on the base is a member of ℬ. Let 푓 : 푋 → 푌 be a quadrable morphism in u� such that the relative diagonal is also a quadrable morphism in . The Δ푓 : 푋 → 푋 ×푌 푋 u� following are equivalent:

(i) 푓 : 푋 → 푌 is a ℬ-perfect morphism in u�. (ii) 푓 : 푋 → 푌 is a u�-perfect morphism in u�.

29 Abstract topology

(iii) 푓 : 푋 → 푌 is a member of u� and is ℬ-separated.

Proof. (i) ⇒ (ii). By proposition 1.2.18, we have ℬ ⊆ u�; thus, every ℬ-perfect morphism in u� is also u�-perfect.

(ii) ⇒ (iii). The relative diagonal Δ푓 : 푋 → 푋 ×푌 푋 is a quadrable monomorphism, so we may apply lemma 1.2.19(b) to deduce that it is a member of ℬ. Thus, by lemma 1.1.6, 푓 : 푋 → 푌 is indeed ℬ-separated.

(iii) ⇒ (i). Suppose 푓 : 푋 → 푌 is a member of u�. Then there is a 햩- covering sink Ψ on 푌 such that, for every (푉 , 푦) ∈ Ψ, we have a pullback square in u� of the form below,

푥 푈 푋 푣 푓

푉 푦 푌

where 푣 : 푈 → 푉 is of ℬ-type 햩-semilocally on the domain. In addition, suppose 푓 : 푋 → 푌 is ℬ-separated. By proposition 1.1.7, 푣 : 푈 → 푉 is also ℬ-separated. Thus, by lemma 1.2.20, 푣 : 푈 → 푉 is of ℬ-type (ℬ, 햩)-semilocally on the domain, so 푣 : 푈 → 푉 is a member of ℬ. Hence, 푓 : 푋 → 푌 is of ℬ-type 햩-semilocally on the base, so 푓 : 푋 → 푌 is a member of ℬ. ■

30 1.3. Regulated categories

1.3 Regulated categories

Synopsis. We study categories with a class of morphisms that have good properties with regards to pullbacks and images, such as regular categories.

Prerequisites. §1.1.

1.3.1 ※ Throughout this section, u� is a category and ℱ is a class of separated fibrations in u�.

1.3.2(a) Definition. An ℱ-embedding in u� is a monomorphism in u� that is a member of ℱ.

1.3.2(b) Definition. An ℱ-subobject of an object 푌 in u� is an object (푋, 푓) in

u�∕푌 where 푓 : 푋 → 푌 is an ℱ-embedding in u�.

Properties Proposition. The class of ℱ-embeddings in u� is a class of separated of fibrant fibrations in u�. embeddings Proof. Straightforward. ⧫

[1] 1.3.3 Definition. An ℱ-calypsis in u� is a morphism 푓 : 푋 → 푌 in u� with the following property:

• If 푓 = 푚∘푒 for some ℱ-embedding 푚 : 푌 ′ → 푌 in u�, then 푚 : 푌 ′ → 푌 is an isomorphism in u�.

Remark. If every object in u� is ℱ-separated, then every ℱ-calypsis in u� is an epimorphism.

Properties Proposition. of calypses (i) Every extremal epimorphism in u� is an ℱ-calypsis in u�. (ii) Every ℱ-embedding in u� is right orthogonal to every ℱ-calypsis in u�. (iii) The class of ℱ-calypses in u� is closed under composition.

Proof. Straightforward. ⧫

[1] — from Greek «κάλυψις», covering.

31 Abstract topology

1.3.4(a) Definition. An ℱ-calypsis 푓 : 푋 → 푌 in u� is quadrable if it has the following properties:

• 푓 : 푋 → 푌 is a quadrable morphism in u�. • For every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌 the morphism 푓 ′ : 푋′ → 푌 ′ is an ℱ-calypsis in u�.

1.3.4(b) Definition. An exact ℱ-image of a quadrable morphism 푓 : 푋 → 푌 in u� is an ℱ-embedding im(푓) : Im(푓) → 푌 in u� with the following property: quadrable • There is a (necessarily unique) ℱ-calypsis 휂푓 : 푋 → Im(푓)

in u� such that im(푓) ∘ 휂푓 = 푓.

Remark. Exact images are unique up to unique isomorphism, if they exist.

Example. If 푓 : 푋 → 푌 is an ℱ-embedding in u�, then 푓 : 푋 → 푌 is its own exact ℱ-image.

[2] 1.3.5(a) Definition. A quadrable morphism 푓 : 푋 → 푌 in u� is ℱ-eucalyptic if it has the following property:

• For every ℱ-embedding 푚 : 푋′ → 푋 in u�, 푓 ∘ 푚 : 푋′ → 푌 admits an exact ℱ-image.

1.3.5(b) Definition. A quadrable morphism 푓 : 푋 → 푌 in u� is quadrably ℱ-eucalyptic if it has the following property: • For every pullback square in u� of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌 the morphism 푓 ′ : 푋′ → 푌 ′ is ℱ-eucalyptic.

[2] — from Greek «εὖ», well, and «καλύπτω», I cover.

32 1.3. Regulated categories

Properties of Proposition. eucalyptic (i) Every ℱ-embedding in u� is quadrably ℱ-eucalyptic. morphisms (ii) The class of ℱ-eucalyptic morphisms in u� is closed under composition.

(iii) The class of quadrably ℱ-eucalyptic morphisms in u� is a class of fibrations in u�.

Proof. Straightforward. (For (ii), use proposition 1.3.3.) ⧫

[3] 1.3.6 Definition. A quadrable morphism in u� is ℱ-agathic if it is both ℱ- separated and quadrably ℱ-eucalyptic.

Properties Proposition. of agathic (i) Every ℱ-embedding in u� is ℱ-agathic. morphisms (ii) Assuming every quadrable ℱ-calypsis is an extremal epimorphism in u�, every ℱ-agathic monomorphism in u� is an ℱ-embedding.

(iii) The class of ℱ-agathic morphisms in u� is a class of separated fibrations.

Proof. (i). By proposition 1.1.7, monomorphisms in u� are ℱ-separated; and by proposition 1.3.5, ℱ-embeddings in u� are quadrably ℱ-eucalyptic.

(ii). Let 푓 : 푋 → 푌 be an ℱ-agathic monomorphism in u�. Then it factors as a quadrable ℱ-calypsis 푒 : 푋 → Im(푓) followed by an ℱ-embedding im(푓) : Im(푓) → 푌 . But 푒 : 푋 → Im(푓) is both a monomorphism and an extremal epimorphism, so it must be an isomorphism. Hence 푓 : 푋 → 푌 is also an ℱ-embedding.

(iii). We know that the class of ℱ-agathic morphisms in u� is a class of fibrations in u�, so we may apply lemma 1.1.10 to (i) to deduce the claim. ■

1.3.7 ¶ Let 푌 be an object in u�.

[3] — from Greek «ἀγαθικός», good.

33 Abstract topology

Definition. An exact ℱ-union of a set Φ of ℱ-subobjects of 푌 is an ℱ-subobject (푋,̄ 푓)̄ of 푌 with the following property: −1 ̄ • For every object (푇 , 푦) in u�∕푌 , 푦 (푋,̄ 푓) is a coproduct of

{푦−1(푋, 푓) | (푋, 푓) ∈ Φ}

in the category of ℱ-subobjects of 푇 , where 푦−1 denotes pullback along 푦 : 푇 → 푌 in u�.

Remark. Exact unions are unique up to unique isomorphism, if they exist.

1.3.8 ¶ There are numerous variations on the definition of ‘regular category’; we shall use the following.

Definition.A regular category is a cartesian monoidal category u� with pullbacks of monomorphisms and exact ℳ-images of every morphism, where ℳ is the class of monomorphisms in u�.

Recognition Lemma. Let 푓 : 푋 → 푌 be a morphism in u�. Assuming u� is a regular principle for category, the following are equivalent: calypses in (i) is an effective epimorphism in . regular categories 푓 : 푋 → 푌 u� (ii) 푓 : 푋 → 푌 is a regular epimorphism in u�. (iii) 푓 : 푋 → 푌 is a strong epimorphism in u�. (iv) 푓 : 푋 → 푌 is an extremal epimorphism in u�. (v) 푓 : 푋 → 푌 is an ℳ-calypsis in u�, where ℳ is the class of monomorphisms in u�.

Proof. (i) ⇒ (ii), (ii) ⇒ (iii), (iii) ⇒ (iv), (iv) ⇒ (v). Straightforward.

(v) ⇒ (i). See Proposition 1.3.4 in [Johnstone, 2002, Part A]. □

Remark. Thus, every kernel pair in a regular category is also a kernel pair of some effective epimorphism.

34 1.3. Regulated categories

1.3.9 Lemma. Consider a pullback square in u�: Descent of mono- 푝 morphisms 푋̃ 푋 푓 ̃ 푓 ̃ 푌 푞 푌

If 푞 : 푌̃ → 푌 is a quadrable morphism in u� such that every pullback of 푞 : 푌̃ → 푌 is an epimorphism in u�, then the following are equivalent:

(i) 푓̃ : 푋̃ → 푌̃ is a monomorphism in u�.

(ii) 푓 : 푋 → 푌 is a monomorphism in u�.

Proof. (i) ⇒ (ii). Let 푥0, 푥1 : 푇 → 푋 be a parallel pair of morphisms in

u�. Suppose 푓 ∘ 푥0 = 푓 ∘ 푥1. Since 푓̃ : 푋̃ → 푌̃ is a monomorphism, the pullback pasting lemma implies that there exist morphisms ̃푥: 푇̃ → 푋̃ and 푡 : 푇̃ → 푇 in u� such that both of the following are pullback squares in u�:

푡 푡 푇̃ 푇 푇̃ 푇

̃푥 푥0 ̃푥 푥1 ̃ ̃ 푋 푝 푋 푋 푝 푋

But 푡 : 푇̃ → 푇 is an epimorphism in u�, so we have 푥0 = 푥1.

(ii) ⇒ (i). Straightforward. ■

1.3.10 Definition. A functor 퐹 : u� → u� is regular if u� is a regular category and 퐹 : u� → u� preserves limits of finite diagrams and extremal epimorphisms.

Criteria for a Lemma. Let 퐹 : u� → u� be a regular functor. The following are equival- regular functor to ent: be conservative (i) 퐹 : u� → u� reflects extremal epimorphisms.

(ii) 퐹 : u� → u� reflects extremal epimorphisms and monomorphisms.

(iii) 퐹 : u� → u� is conservative.

35 Abstract topology

Proof. (i) ⇒ (ii). Since u� has kernel pairs and 퐹 : u� → u� preserves kernel pairs, if 퐹 : u� → u� reflects extremal epimorphisms, then 퐹 : u� → u� also reflects monomorphisms.

(ii) ⇒ (iii). A morphism (in any category) is an isomorphism if and only if it is both a monomorphism and an extremal epimorphism.

(iii) ⇒ (i). Let 푓 : 푋 → 푌 be a morphism in u�. Since 퐹 : u� → u� is a regular functor, 퐹 푓 : 퐹 푋 → 퐹 푌 is an extremal epimorphism in u� if and only if 퐹 im(푓) : 퐹 Im(푓) → 퐹 푌 is an isomorphism in u�; and since 퐹 : u� → u� is conservative, 퐹 im(푓) : 퐹 Im(푓) → 퐹 푌 is an isomorphism in u� if and only if 푓 : 푋 → 푌 is an extremal epimorphism in u�. ■

1.3.11 ¶ Let 휅 be a regular cardinal.

Definition.A 휅-ary coherent category is a regular category u� with exact ℳ-unions of every 휅-small set of ℳ-subobjects of every object, where ℳ is the class of monomorphisms in u�.

Initial objects Lemma. Let u� be a 휅-ary coherent category. in coherent (i) u� has an initial object 0. categories For every object in , the unique morphism in is (ii) 푌 u� ⊥푌 : 0 → 푌 u� a monomorphism.

(iii) For every object 푋 in u�, every morphism 푋 → 0 in u� is an isomorphism.

Proof. See Lemma 1.4.1 in [Johnstone, 2002, Part A]. □

1.3.12 Definition.A regulated category is a pair (u�, u�) where u� is a category [4] and u� is a (not necessarily full) subcategory of u� with the following properties:

• u� is a class of separated fibrations in u�.

• Every morphism in u� is an u�-agathic morphism in u�.

[4] However, abusing notation, we will also regard u� as a subset of mor u�.

36 1.3. Regulated categories

Given such, a regulated morphism in u� is a morphism in u�. We will often abuse notation by referring to u� itself as a regulated category, omitting u�.

1.3.12(a) Example. Every category is a regulated category in which the regulated morphisms are the isomorphisms.

1.3.12(b) Example. Every regular category is a regulated category in which every morphism is regulated.

1.3.12(c) Example. If every ℱ-agathic monomorphism in u� is an ℱ-embedding in u�, then (u�, u�) is a regulated category, where u� is the subcategory of u� consisting of the ℱ-agathic morphisms in u�.

Recognition prin- Lemma. Let u� be a subcategory of u�. Assuming u� is a class of separated ciple for regu- fibrations in u�, the following are equivalent: lated categories (i) (u�, u�) is a regulated category.

(ii) Every morphism in u� factors as a quadrable u�-calypsis in u� followed by a u�-embedding in u�.

Proof. Straightforward. ⧫

Remark. In particular, if (u�, u�) is a regulated category, then (u�, u�) is also a regulated category.

1.3.13 ※ For the remainder of this section, (u�, u�) is a regulated category.

1.3.14 Proposition. The category For every object in , the slice category is a regular cat- (i) 푋 u� u�∕푋 of regulated egory in which the extremal epimorphisms are the morphisms that are objects in a regu- quadrable -calypses in . lated category u� u� (ii) For every morphism 푓 : 푋 → 푌 in u�, the pullback functor 푓 ∗ : is a regular functor. u�∕푌 → u�∕푋

37 Abstract topology

Proof. (i). By hypothesis, every morphism in u�∕푋 is u�-agathic as a morphism in u�, so it factors as a quadrable u�-calypsis followed by a u�-embedding in u�. Furthermore, by proposition 1.1.12, the inclusion

u�∕푋 ↪ u�∕푋 creates limits. But every monomorphism in u�∕푋 is a u�-

embedding in u�, so it follows that every morphism in u�∕푋 has an exact

ℳ푋-image, where ℳ푋 is the class of monomorphisms in u�∕푋. Thus,

the extremal epimorphisms in u�∕푋 are indeed the morphisms that are quadrable u�-calypses in u�.

∗ (ii). It is clear that 푓 : u�∕푌 → u�∕푋 preserves limits of finite diagrams, and the argument above implies that extremal epimorphisms are also preserved. ■

1.3.15 Proposition. Let 푓 : 푋 → 푌 be a quadrable morphism in u�. The Recognition prin- following are equivalent: ciple for quadrable calypses (i) 푓 : 푋 → 푌 is a quadrable u�-calypsis in u�. (ii) 푓 : 푋 → 푌 admits an exact u�-image and the pullback functor ∗ is conservative. 푓 : u�∕푌 → u�∕푋

Proof. (i) ⇒ (ii). Consider a commutative diagram in u� of the form below,

푓 ″ 푋″ 푌 ″ 푥′ 푦′ 푓 ′ 푋′ 푌 ′ 푥 푦 푋 푌 푓

where the vertical arrows are morphisms in u� and both squares are pull- ′ ″ ′ ′ back squares in u�. Suppose 푥 : (푋 , 푥 ∘ 푥 ) → (푋 , 푥) is an extremal ″ ′ epimorphism in u�∕푋. Then, by proposition 1.3.14, 푥 : 푋 → 푋 is a u�-calypsis in u�. Since 푓 : 푋 → 푌 is a quadrable u�-calypsis in u�, 푓 ′ : 푋′ → 푌 ′ is a u�-calypsis in u�, hence 푦′ : 푌 ″ → 푌 ′ is also an u�-calypsis in u�. But 푦′ : 푌 ″ → 푌 ′ is u�-eucalyptic, so it follows that ′ ″ ′ ′ 푦 : (푌 , 푦 ∘ 푦 ) → (푌 , 푦) is an extremal epimorphism in u�∕푌 . Thus,

38 1.3. Regulated categories

∗ we see that 푓 : u�∕푌 → u�∕푋 reflects extremal epimorphisms. We may then apply lemma 1.3.10.

∗ (ii) ⇒ (i). Observe that 푓 : u�∕푌 → u�∕푋 sends the object (Im(푓), im(푓)) ∗ in u�∕푌 to a terminal object in u�∕푋. Since 푓 : u�∕푌 → u�∕푋 is conservative, it follows that im(푓) : Im(푓) → 푌 is an isomorphism in u�, so 푓 : 푋 → 푌 is indeed a quadrable u�-calypsis in u�. ■

1.3.16 ¶ Let (u�0, u�0) and (u�1, u�1) be regulated categories.

Definition.A regulated functor (u�0, u�0) → (u�1, u�1) is a functor

퐹 : u�0 → u�1 with the following properties:

• 퐹 preserves regulated morphisms, i.e. 퐹 sends morphisms in u�0 to

morphisms in u�1. • 퐹 preserves pullbacks along regulated morphisms, i.e. given a pull-

back square in u�0, say

푇 ′ 푇 푥 푋′ 푋

if 푥 : 푇 → 푋 is in u�0, then 퐹 preserves this pullback square. • 퐹 preserves exact images of regulated morphisms, i.e. given a morph-

ism 푓 : 푋 → 푌 in u�0, 퐹 im(푓) : 퐹 Im(푓) → 퐹 푌 is an exact u�1- image of 퐹 푓 : 퐹 푋 → 퐹 푌 . Let be a functor. Assuming preserves regulated Recognition prin- Lemma. 퐹 : u�0 → u�1 퐹 ciple for regu- morphisms and pullbacks along regulated morphisms, the following are lated functors equivalent: is a regulated functor. (i) 퐹 : (u�0, u�0) → (u�1, u�1) For every object in , the evident functor (ii) 푋 u�0

퐹푋 : (u�0)∕푋 → (u�1)∕퐹 푋

given on objects by (푇 , 푥) ↦ (퐹 푇 , 퐹 푥) is a regular functor.

Proof. This is a consequence of proposition 1.3.14. ■

39 Abstract topology

1.4 Exact quotients

Synopsis. We consider the problem of freely adding exact quotients to a category with a class of covering morphisms.

Prerequisites. §§1.1, 1.3, a.1, a.2, a.3.

1.4.1 ¶ Let u� be a category.

Definition.A strict epimorphism (resp. universally strict epimorphism) in u� is a morphism 푓 : 푋 → 푌 in u� such that the principal sieve ↓⟨푓⟩ is strict-epimorphic (resp. universally strict-epimorphic).

Recognition prin- Lemma. Let 푓 : 푋 → 푌 be a morphism in u�. The following are equival- ciple for strict ent: epimorphisms (i) The morphism 푓 : 푋 → 푌 is a strict epimorphism in u�. For every morphism in , if for all (ii) ℎ : 푋 → 푍 u� ℎ ∘ 푥0 = ℎ ∘ 푥1 elements of , then there is a unique morphism (푇 , (푥0, 푥1)) Kr(푓 ∘ −) 푔 : 푌 → 푍 in u� such that 푔 ∘ 푓 = ℎ.

Proof. This is a special case of lemma a.2.6. ■

Properties Proposition. of strict (i) Every strict epimorphism in u� is an epimorphism in u�. epimorphisms (ii) Every regular epimorphism in u� is a strict epimorphism in u�. (iii) A strict epimorphism in u� is an effective epimorphism in u� if (and only if) it admits a kernel pair in u�.

Proof. Straightforward. (Use lemma 1.4.1.) ⧫

1.4.2 ¶ Let 푋 be an object in a category u�.

1.4.2(a) Definition. An equivalence relation on 푋 in u� is a tuple (푅, 푑0, 푑1) where: • h , h : h → h × h is a monomorphism in Psh(u�). ⟨ 푑1 푑0⟩ 푅 푋 푋 • The image of h , h : h → h × h is an equivalence relation on ⟨ 푑1 푑0⟩ 푅 푋 푋 h푋.

40 1.4. Exact quotients

Remark. In other words, an equivalence relation on 푋 is a representation

of some equivalence relation on h푋.

1.4.2(b) Definition. An exact quotient of an equivalence relation (푅, 푑0, 푑1) on 푋 in u� is a quadrable morphism 푞 : 푋 → 푋̄ in u� with the following properties:

• The following is a pullback square in u�:

푑 푅 0 푋

푑1 푞 ̄ 푋 푞 푋

• Every pullback of 푞 : 푋 → 푋̄ is an effective epimorphism in u�.

1.4.3 ¶ Let u� be a category. The following is a special case of the notion of coverage (paragraph a.2.8) and generalises the notion of weakly unary topology in the sense of [Shulman, 2012, §3].

1.4.3(a) Definition.A unary coverage on u� is a subset 햤 ⊆ mor u� with the following properties:

• For every object 푋 in u�, id : 푋 → 푋 is a member of 햤. • For every morphism 푓 : 푋 → 푌 in u� and every morphism 푦 : 푇 → 푌 in u�, if 푦 : 푇 → 푌 is a member of 햤, then there is a commutative diagram in u� of the form below,

푈 푇 푥 푦 푋 푌 푓

where 푥 : 푈 → 푋 is also a member of 햤.

1.4.3(b) Definition. A unary coverage 햤 on u� is upward-closed if it has the following property:

• Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if 푔 ∘ 푓 : 푋 → 푍 is a member of 햤, then 푔 : 푌 → 푍 is also a member of 햤.

41 Abstract topology

Remark. If 햤 is an upward-closed unary coverage, then: (i) Every split epimorphism in u� is a member of 햤. (ii) The full subcategory of (u� ↓ u�) spanned by the members of 햤 is replete.

1.4.3(c) Definition. A unary coverage 햤 on u� is saturated if it has the following property: • For every commutative square in u� of the form below,

푈 푇 푥 푦 푋 푌 푓

if both 푥 : 푈 → 푋 and 푓 : 푋 → 푌 are members of 햤, then 푦 : 푇 → 푌 is also a member of 햤.

Example. If u� is a category with pullbacks and 햤 is the class of universally strict epimorphisms in u�, then 햤 is a saturated unary coverage on u� (by lemma a.2.3(c) and corollary a.2.3(d)).

1.4.4 ※ For the remainder of this section, u� is a category and 햤 is a unary coverage on u�. Abusing notation, we will conflate 햤 with the associated coverage on u�.

1.4.5 Remark. Recalling paragraph a.2.13, 푓 : 푋 → 푌 is a 햤-covering morphism in u� if and only if there is a morphism 푥 : 푇 → 푋 in u� such that 푓 ∘ 푥 : 푇 → 푌 is a composite of some finite (composable) sequence of members of 햤.

Recognition Lemma. Let 햤 be a unary coverage on u�. The following are equivalent: principles (i) 햤 is saturated. for saturated unary coverages (ii) 햤 is upward-closed and closed under composition.

(iii) Every 햤-covering morphism in u� is a member of 햤.

Proof. Straightforward. ⧫

42 1.4. Exact quotients

Properties Proposition. The class of 햤-covering morphisms in u� is the smallest of covering saturated unary coverage on u� that contains 햤. morphisms Proof. Straightforward. ⧫

1.4.6 ¶ Let 퐴 be a presheaf on u�.

1.4.6(a) Definition. An 햤-local generator of 퐴 is an element (푋, 푎) of 퐴 such that {(푋, 푎)} is an 햤-local generating set of elements of 퐴. The presheaf 퐴 is 햤-locally 1-generable if it has an 햤-local generator.

1.4.6(b) Definition. An 햤-local presentation of 퐴 is tuple (푋, 푃 , 푎, 푑0, 푑1) with the following properties: • (푋, 푎) is an 햤-local generator of 퐴.

• (푃 , (푑1, 푑0)) is an 햤-local generator of the kernel relation Kr(푎 ⋅ (−)),

where 푎 ⋅ (−) : h푋 → 퐴 is the unique morphism that sends id푋 to 푎. The presheaf 퐴 is 햤-locally 1-presentable if it admits an 햤-local presentation.

Example. For every object 푋 in u�, (푋, 푋, id푋, id푋, id푋) is an 햤-local

presentation of h푋.

1.4.7 ¶ Let 퐵 be a presheaf on u� and let (푌 , 푏) be an 햤-local generator of 퐵.

Elements of Lemma. Let (푋, 푎) be an element of 퐵. There exist an 햤-covering morph- locally 1- ism 푝 : 푋̃ → 푋 in u� and a morphism 푓 : 푋̃ → 푌 in u� such that generable 푏 ⋅ 푓 = 푎 ⋅ 푝. presheaves Proof. Straightforward. (Recall remark 1.4.5 and paragraph a.2.13.) ⧫

Locally 1- Corollary. Let 퐴 be an 햤-locally 1-generable subpresheaf of 퐵. There generable subpre- is a morphism 푓 : 푋̃ → 푌 in u� such that (푋,̃ 푏 ⋅ 푓) is an 햤-local gener- sheaves of locally ator of 퐴. 1-generable presheaves Proof. Let (푋, 푎) be an 햤-local generator of 퐴. By lemma 1.4.7, we have an 햤-covering morphism 푝 : 푋̃ → 푋 in u� and a morphism 푓 : 푋̃ → 푌 in u� such that 푏 ⋅ 푓 = 푎 ⋅ 푝. On the other hand, (푋,̃ 푎 ⋅ 푝) is also an 햤-local ′ ′ generator of 퐴: for every 햤-closed subpresheaf 퐴 ⊆ 퐴, if 푎 ⋅ 푝 ∈ 퐴 (푋̃), then 푎 ∈ 퐴′(푋), so 퐴′ = 퐴. ■

43 Abstract topology

1.4.8 Lemma. Let 퐴 be an 햤-locally 1-generable presheaf on u�. For every 햤- Local generating ′ ′ local generating set Φ of elements of 퐴, there is (푋 , 푎 ) ∈ Φ such that sets of locally ′ ′ (푋 , 푎 ) is an 햤-local generator of 퐴. 1-generable presheaves Proof. Let (푋, 푎) be an 햤-local generator of 퐴. Recalling remark 1.4.5 and paragraph a.2.13, since Φ is an 햤-local generating set of elements of ′ ′ 퐴, there exist an element (푋 , 푎 ) ∈ Φ, an 햤-covering morphism 푝 : 푋̃ → 푋 in u�, and a morphism 푓 : 푋̃ → 푋′ in u� such that 푎 ⋅ 푝 = 푎′ ⋅ 푓. Thus, ′ ′ every 햤-closed subpresheaf of 퐴 containing (푋 , 푎 ) must also contain ′ ′ (푋, 푎), so (푋 , 푎 ) itself is an 햤-local generator of 퐴. ■

1.4.9 ¶ Let 푋 : u� → u� be a diagram.

Definition. An 햤-weakly limiting cone on 푋 is a cone 휆 : Δ푋̃ ⇒ 푋 in u� with the following property: • For every object 푇 in u� and every cone 휉 : Δ푇 ⇒ 푋 in u�, there exist morphisms ̃푥: 푈 → 푋̃ and 푡 : 푈 → 푇 in u� such that 푡 : 푈 → 푇 is an

햤-covering morphism in u� and, for every object 푗 in u� , 휆푗 ∘ ̃푥= 휉푗 ∘ 푡. We will often abuse notation by referring to the object 푋̃ as an 햤-weak limit of 푋, omitting 휆.

Example. Every limiting cone on 푋 is also an 햤-weakly limiting cone.

Recognition Lemma. Let 푋̃ be an object in u� and let 휆 : Δ푋̃ ⇒ 푋 be a cone in u�. principle for The following are equivalent: weak limits (i) 휆 : Δ푋̃ ⇒ 푋 is an 햤-weakly limiting cone.

(ii) (푋,̃ 휆) is an 햤-local generator of the presheaf [u� , u�](Δ−, 푋).

Proof. Straightforward. ⧫

1.4.10 ¶ The following is due to Shulman [2012].

Definition. The Shulman condition on (u�, 햤) is the following: • Every finite diagram in u� has a 햤-weak limit.

44 1.4. Exact quotients

Remark. A (strongly) unary topology in the sense of [Shulman, 2012, §3] is precisely a saturated unary coverage that satisfies the Shulman condition.

Proposition. Let 푋 : u� → u� be a diagram in u�. Assuming the Shulman condition on (u�, 햤), the presheaf lim h is 햤-locally 1-presentable. ←−−u� 푋 Proof. By lemma 1.4.9, there is an 햤-local generator 푋,̃ 휆 of lim h . ( ) ←−−u� 푋 Let 푅 = Kr(휆 ⋅ (−)) ⊆ h푋̃ × h푋̃ . It is not hard to see that 푅 is isomorphic to the presheaf of cones over the diagram in u� obtained by attaching two copies of the cone 휆 over the given diagram 푋. Thus, by hypothesis, 푅 is 햤-locally 1-generable. Hence, lim h is 햤-locally 1-presentable. ■ ←−−u� 푋 1.4.11 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). Given 햤-local gen- Morphisms erators (푋, 푎) and (푌 , 푏) of 퐴 and 퐵, respectively, there is an element of locally 1- of such that is an -covering morphism (푇 , (푥, 푦)) h푋 × h푌 푥 : 푇 → 푋 햤 generable in u� and 푏 ⋅ 푦 = ℎ(푎) ⋅ 푥. In particular, there is a commutative square in presheaves Psh(u�) of the form below, 푦∘− h푇 h푌 푎⋅(푥∘−) 푏⋅(−) 퐴 퐵 ℎ where the vertical arrows are 햤-locally surjective.

Proof. Straightforward. (Recall paragraph a.2.13.) ⧫ Let and be morphisms in 1.4.12 Proposition. ℎ0 : 퐴0 → 퐵 ℎ1 : 퐴1 → 퐵 Pullbacks of . Assuming the Shulman condition on , if both and are Psh(u�) (u�, 햤) 퐴0 퐴1 morphisms -locally 1-generable and is -locally 1-presentable, then 햤 퐵 햤 Pb(ℎ0, ℎ1) of presheaves is also 햤-locally 1-generable. with locally 1-presentable Proof. Choose any 햤-local presentation of 퐵, say (푌 , 푄, 푏, 푑0, 푑1). By codomain lemma 1.4.11, there is a commutative diagram in Psh(u�) of the form below, h 푓0∘− h 푓1∘− h 푋0 푌 푋1

푎0⋅(−) 푏⋅(−) 푎1⋅(−)

퐴0 퐵 퐴1 ℎ0 ℎ1

45 Abstract topology

where the vertical arrows are 햤-locally surjective. We have the following commutative diagrams in Psh(u�),

푅 Pb(ℎ0, ℎ1) 퐵

Δ퐵 h h 푋0 × 푋1 퐴0 × 퐴1 퐵 × 퐵 (푎0⋅−)×(푎1⋅−) ℎ0×ℎ1

푅 Kr(푏 ⋅ −) 퐵

Δ퐵 h h h h 푋0 × 푋1 푌 × 푌 퐵 × 퐵 (푓0∘−)×(푓1∘−) (푏⋅−)×(푏⋅−) where every square is a pullback square in Psh(u�); in particular,

푅 = Pb(ℎ0 ∘ (푎0 ⋅ −), ℎ1 ∘ (푎1 ⋅ −)) = Pb(푏 ⋅ (푓0 ∘ −), 푏 ⋅ (푓1 ∘ −)) as subpresheaves of h h . Note that Pb is -locally 푋0 × 푋1 푅 ↠ (ℎ0, ℎ1) 햤 surjective, by proposition a.2.14. By definition, we have an 햤-locally

surjective morphism h푄 → Kr(푏 ⋅ −) in Psh(u�), so there is a pullback square in Psh(u�) of the form below,

푅̃ h푄

푅 Kr(푏 ⋅ −)

where 푅̃ ↠ 푅 is 햤-locally surjective. On the other hand, the pullback pasting lemma implies that 푅̃ is (isomorphic to) the limit of the following diagram in Psh(u�), h 푋1

푓1∘−

h푄 h푌 푑0∘−

푑1∘−

h푋 h푌 0 푓0∘− so by lemma 1.4.9, the Shulman condition on (u�, 햤) implies that 푅̃ is 햤-

locally 1-generable. But the composite 푅̃ ↠ 푅 ↠ Pb(ℎ0, ℎ1) is 햤-locally

surjective, so Pb(ℎ0, ℎ1) is also 햤-locally 1-generable. ■

46 1.4. Exact quotients

Generators Corollary. Let 퐴 be a presheaf on u� and let (푋, 푎) be an element of 퐴. of locally 1- Assuming the Shulman condition on (u�, 햤), the following are equivalent: presentable is an -locally 1-presentable presheaf on and is an - presheaves (i) 퐴 햤 u� (푋, 푎) 햤 local generator of 퐴. There is an element of such that (ii) (푃 , 푑0, 푑1) h푋×h푋 (푋, 푃 , 푎, 푑0, 푑1) is an 햤-local presentation of 퐴.

Proof. This is a special case of proposition 1.4.12. ■

1.4.13 Lemma. Let 퐵 be an 햤-locally 1-presentable presheaf on u� and let 퐴 Recognition prin- be a subpresheaf of 퐵. Assuming the Shulman condition on (u�, 햤), the ciple for locally following are equivalent: 1-presentable subpresheaves (i) 퐴 is an 햤-locally 1-presentable presheaf on u�.

(ii) 퐴 is an 햤-locally 1-generable presheaf on u�.

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). Choose any 햤-local generator of 퐴, say (푋, 푎). We must show that Kr(푎 ⋅ −) is 햤-locally 1-generable. But we have the following pullback square in Psh(u�),

Kr(푎 ⋅ −) h푋

푎⋅−

h푋 푎⋅− 퐵

so we may apply proposition 1.4.12. ■

1.4.14 Lemma. Let ℎ : 퐴 → 퐵 be an 햤-locally surjective morphism in Psh(u�). Quotients Assuming the Shulman condition on (u�, 햤), if 퐴 is 햤-locally 1-presentable, of locally 1- then the following are equivalent: presentable presheaves (i) 퐵 is an 햤-locally 1-presentable presheaf on u�.

(ii) Kr(ℎ) is an 햤-locally 1-generable presheaf on u�.

Proof. (i) ⇒ (ii). This is a special case of proposition 1.4.12.

47 Abstract topology

(ii) ⇒ (i). Choose any 햤-local generator of 퐴, say (푋, 푎). We have the following commutative diagram in Psh(u�),

Kr(ℎ(푎) ⋅ −) 푅1 h푋 푎⋅−

푅0 Kr(ℎ) 퐴 ℎ h 퐴 퐵 푋 푎⋅− ℎ

where every square is a pullback square in Psh(u�). Since h푋 and Kr(ℎ)

are both 햤-locally 1-generable and 퐴 is 햤-locally 1-presentable, 푅0 must also be 햤-locally 1-generable. But we also have a pullback square in Psh(u�) of the form below,

Kr(ℎ(푎) ⋅ −) h푋 푎⋅−

푅0 퐴

so Kr(ℎ(푎) ⋅ −) has an 햤-local generator, say (푄, (푥0, 푥1)). It follows that

(푋, 푄, ℎ(푎), 푥1, 푥0) is an 햤-local presentation of 퐵. ■

1.4.15 ¶ Let 퐴 and 퐵 be presheaves on u�.

1.4.15(a) Proposition. Assuming the Shulman condition on (u�, 햤), if both 퐴 and Products of 퐵 are 햤-locally 1-generable, then 퐴 × 퐵 is also 햤-locally 1-generable. locally 1- generable Proof. Let (푋, 푎) and (푌 , 푏) be 햤-local generators of 퐴 and 퐵, respect- presheaves ively. Then, by proposition a.2.14, (푎 ⋅ −) × (푏 ⋅ −) : h푋 × h푌 → 퐴 × 퐵

is 햤-locally surjective. But lemma 1.4.9 implies that h푋 × h푌 is 햤-locally 1-generable, so it follows that 퐴 × 퐵 is also 햤-locally 1-generable. ■

1.4.15(b) Proposition. Assuming the Shulman condition on (u�, 햤), if both 퐴 and Products of 퐵 are 햤-locally 1-presentable, then 퐴×퐵 is also 햤-locally 1-presentable. locally 1- presentable Proof. Let (푋, 푎) and (푌 , 푏) be 햤-local generators of 퐴 and 퐵, respect- presheaves ively. By corollary 1.4.12, both Kr(푎 ⋅ −) and Kr(푏 ⋅ −) are 햤-locally 1-

generable. Let ℎ = (푎 ⋅ −) × (푏 ⋅ −) : h푋 × h푌 → 퐴 × 퐵. Clearly, Kr(ℎ) ≅

48 1.4. Exact quotients

Kr(푎 ⋅ −) × Kr(푏 ⋅ −), so by proposition 1.4.15(a), Kr(ℎ) is 햤-locally 1-

generable. But proposition a.2.14 implies that ℎ : h푋 × h푌 → 퐴 × 퐵 is

햤-locally surjective, and by proposition 1.4.10, h푋 × h푌 is 햤-locally 1- presentable, so we may apply lemma 1.4.14 to deduce that 퐴 × 퐵 is also 햤-locally 1-presentable. ■

1.4.16 Theorem. The following are equivalent: Limits of (i) (u�, 햤) satisfies the Shulman condition. diagrams of locally 1- (ii) The full submetacategory of Psh(u�) spanned by the 햤-locally 1- presentable presentable presheaves on u� is closed under limit of finite diagrams. presheaves Proof. (i) ⇒ (ii). The terminal presheaf on u� is 햤-locally 1-presentable (proposition 1.4.10), and the the product of two 햤-locally 1-presentable presheaves on u� is 햤-locally 1-presentable (proposition 1.4.15(b)), so it suﬃces to verify that the equaliser of a parallel pair of morphisms between 햤-locally 1-presentable presheaves on u� is 햤-locally 1-presentable. But this is a consequence of proposition 1.4.12 and lemma 1.4.13, so we are done.

(ii) ⇒ (i). Apply lemma 1.4.9. ■

Morphisms Corollary. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�) and let (푌 , 푏) be of locally 1- an 햤-local generator of 퐵. Assuming the Shulman condition on (u�, 햤), presentable if both 퐴 and 퐵 are 햤-locally 1-presentable, then there exist an 햤-local presheaves generator (푋, 푎) of 퐴 and a morphism 푓 : 푋 → 푌 in u� such that the diagram in Psh(u�) shown below commutes,

푓∘− h푋 h푌

푎⋅− 푏⋅− 퐴 퐵 ℎ and the induced morphism is -locally surjective. h푋 → Pb(ℎ, 푏 ⋅ −) 햤

Proof. Apply theorem 1.4.16 and proposition a.2.14. ■

49 Abstract topology

1.4.17(a) Definition.A fork in u� is a diagram in u� of the form below,

푑 1 푓 푃 푋 푌 (∗) 푑0

where 푓 ∘ 푑1 = 푓 ∘ 푑0.

1.4.17(b) Definition. The fork (∗) is mid-햤-exact if the following is an 햤-weak pullback square in u�: 푑 푃 0 푋

푑1 푓 (†) 푋 푌 푓

1.4.17(c) Definition. The fork (∗) is left-exact if (†) is a pullback square in u�.

1.4.17(d) Definition. The fork (∗) is right-햤-exact if (푋, 푃 , 푓, 푑0, 푑1) is an 햤-

local presentation of h푌 .

1.4.17(e) Definition. The fork (∗) is 햤-exact if it is both left-exact and right-햤- exact.

Remark. Clearly, every left-exact fork is also mid-햤-exact.

Recognition Lemma. The following are equivalent: principle for (i) The fork is right- -exact. right-exact forks (∗) 햤 (ii) The fork (∗) is mid-햤-exact and 푓 : 푋 → 푌 is an 햤-covering morphism in u�.

Proof. Straightforward. (Recall lemma 1.4.9.) ⧫

1.4.18 ¶ The sheaf condition with respect to 햤 can be considered to be a kind of limit preservation condition. More precisely:

50 1.4. Exact quotients

Lemma. Let 퐴 be a presheaf on u�. Assuming (u�, 햤) satisfies the Shulman condition, the following are equivalent:

(i) 퐴 is an 햤-sheaf on u�. (ii) 퐴 : u� op → Set sends right-햤-exact forks in u� to equaliser diagrams in Set. (iii) For every right-햤-exact fork in u� of the form below,

푑 0 푓 푅 푋 푌 푑1 if 푓 : 푋 → 푌 is a member of 햤, then the following is an equaliser diagram in Set:

−⋅푑 −⋅푓 0 퐴(푌 ) 퐴(푋) 퐴(푅) −⋅푑1

Proof. (i) ⇒ (ii). Consider a right-햤-exact fork in u�:

푑 0 푓 푅 푋 푌 푑1

′ Let 푎 ∈ 퐴(푋) and suppose 푎 ⋅ 푑0 = 푎 ⋅ 푑1. We wish to find 푎 ∈ 퐴(푌 ) such that 푎′ ⋅ 푓 = 푎. Since 퐴 satisfies the sheaf condition with respect to the principal sieve ↓⟨푓⟩, such an element 푎′ is necessarily unique because − ⋅ 푓 : 퐴(푌 ) → 퐴(푋) is injective. On the other hand, by lemma a.2.6, such 푎′ exists if 푎 has the following property:

• For every element (푇 , (푥0, 푥1)) of h푋 × h푋, if 푓 ∘ 푥0 = 푓 ∘ 푥1, then

푎 ⋅ 푥0 = 푎 ⋅ 푥1.

However, given an element (푇 , (푥0, 푥1)) of h푋 × h푋, if 푓 ∘ 푥0 = 푓 ∘ 푥1, there exist an 햤-covering morphism 푡 : 푈 ↠ 푇 in u� and a morphism 푟 : 푈 → 푅 in u� such that the following diagrams in u� commute,

푈 푟 푅 푈 푟 푅

푡 푑1 푡 푑0 푇 푋 푇 푋 푥0 푥1 so (푎 ⋅ 푥0) ⋅ 푡 = (푎 ⋅ 푥1) ⋅ 푡. Since − ⋅ 푡 : 퐴(푇 ) → 퐴(푈) is also injective, the claim follows.

51 Abstract topology

(ii) ⇒ (iii). Immediate.

(iii) ⇒ (i). Let 푓 : 푋 ↠ 푌 be a member of 햤. We must show that 퐴 satisfies the sheaf condition with respect to the principal sieve ↓⟨푓⟩. Since (u�, 햤) satisfies the Shulman condition, there is aright 햤-exact fork in u� of the form below:

푑 0 푓 푅 푋 푌 푑1

In particular, − ⋅ 푓 : 퐴(푌 ) → 퐴(푋) is injective. Consider a commutative square of the form below,

푠 ↓⟨푓⟩ El(퐴)

u�∕푌 u�

where u�∕푌 → u� and El(퐴) → u� are the respective projections. Let ′ (푋, 푎) = 푠(푋, 푓). Then 푎 ⋅ 푑0 = 푎 ⋅ 푑1, so there is a unique 푎 ∈ 퐴(푌 ) ′ such that 푎 ⋅ 푓 = 푎. This defines a functor u�∕푌 → El(퐴) making the evident triangles commute, and by the Yoneda lemma, it is the unique such functor. Thus 퐴 indeed satisfies the sheaf condition with respect to ↓⟨푓⟩. ■

1.4.19 ¶ The following technical results will be needed later.

1.4.19(a) Lemma. Consider a commutative diagram in u� of the form below: Covering morph- 푥1 isms and weak 푃 푋1 pullback squares 푞 푓1 푦 푥 1 0 푄 푌1

푦0 푔1

푋0 푌0 푍 푓0 푔0

If the outer square is an -weak pullback square in and both 햤 u� 푓0 : 푋0 ↠ and are -covering morphisms in , then the inner 푌0 푓1 : 푋1 ↠ 푌1 햤 u� square is also an 햤-weak pullback square in u�.

52 1.4. Exact quotients

Proof. Consider a commutative square in u� of the form below:

′ 푦1 푇 푌1 ′ 푦0 푔1 푌 푍 0 푔0 By remark 1.4.5 and proposition 1.4.5, there is a commutative diagram in u� of the form below,

′ 푥1 푆 푋1 푡 푓1 푦′ 푥′ 1 0 푇 푌1 ′ 푦0 푔1

푋0 푌0 푍 푓0 푔0 where 푡 : 푆 ↠ 푇 is an 햤-covering morphism in u�. Thus, there exist an 햤-covering morphism 푠 : 푈 ↠ 푆 in u� and a morphism 푝 : 푈 → 푃 in u� such that ′ and ′ . We then have ′ 푥0 ∘푝 = 푥0 ∘푠 푥1 ∘푝 = 푥1 ∘푠 푦0 ∘푞 ∘푝 = 푦0 ∘푡∘푠 and ′ , and is an -covering morphism in 푦1 ∘ 푞 ∘ 푝 = 푦1 ∘ 푡 ∘ 푠 푡 ∘ 푠 : 푈 ↠ 푇 햤 u�, as required. ■

1.4.19(b) Lemma. Consider a diagram in u� of the form below, Exact forks 푑 0 푝 and pullbacks 푋1 푋0 푋̄ 푑1 ̄ 푓1 푓0 푓 푑0 푌1 푌0 푞 푌 푑1 where:

• The top row is an 햤-exact fork in u�. • The bottom row is a left-exact fork in u�. • The two parallel squares on the left are pullback squares in u�. • The square on the right commutes. Then h Pb ̄ is an -locally bijective (푓0, 푝) ⋅ (−) : 푋0 → (푞 ∘ −, 푓 ∘ −) 햤 morphism in Psh(u�). In particular, if 햤 is a subcanonical unary coverage on u�, then the right square is a pullback square in u�.

53 Abstract topology

Proof. We have the following commutative diagram in u�,

푓1 푑0 푋1 푌1 푌0

푑1 푑1 푞

푋0 푌0 푌̄ 푓0 푞 where both squares are pullback squares in u�. Thus, by the pullback pasting lemma, in the commutative diagram in u� shown below,

푑0 푓0 푋1 푋0 푌0

푑1 푝 푞 푋 푋̄ 푌̄ 0 푝 푓 ̄

the outer rectangle is a pullback diagram in u�. Hence, by lemma 1.4.19(a), the right square is an -weak pullback square in , so h 햤 u� (푓0, 푝)⋅(−) : 푋0 → Pb(푞 ∘ −, 푓̄ ∘ −) is 햤-locally surjective, by lemma 1.4.9. We will now show that h Pb ̄ is a (푓0, 푝) ⋅ (−) : 푋0 → (푞 ∘ −, 푓 ∘ −)

monomorphism in Psh(u�). Let 푇 be an object in u� and let 푥0,0, 푥0,1 :

푇 → 푋0 be a parallel pair of morphisms in u� such that:

푝 ∘ 푥0,0 = 푝 ∘ 푥0,1 푓0 ∘ 푥0,0 = 푓0 ∘ 푥0,1

We then have a unique morphism 푥1 : 푇 → 푋1 such that:

푑1 ∘ 푥1 = 푥0,0 푑0 ∘ 푥1 = 푥0,1

On the other hand,

푑1 ∘ 푓1 ∘ 푥1 = 푓0 ∘ 푥0,0 푑0 ∘ 푓1 ∘ 푥1 = 푓0 ∘ 푥0,1

and (by the pullback pasting lemma) we have the following pullback square in u�, Δ푝 푋0 푋1

푓0 푓1

푌0 푌1 Δ푞

where the horizontal arrows are the respective relative diagonals, so 푥0,0 =

푥0,1, as claimed.

54 1.4. Exact quotients

Thus, h Pb ̄ is indeed -locally biject- (푓0, 푝)⋅(−) : 푋0 → (푞 ∘ −, 푓 ∘ −) 햤 ive. To complete the proof, observe that if 햤 is a subcanonical unary coverage on , then both h and Pb ̄ are -sheaves on , so, in u� 푋0 (푞 ∘ −, 푓 ∘ −) 햤 u� that case, by proposition a.3.7, h Pb ̄ is (푓0, 푝) ⋅ (−) : 푋0 → (푞 ∘ −, 푓 ∘ −) an isomorphism. ■

1.4.19(c) Lemma. Consider a commutative diagram in u� of the form below, Covering morph- 푤 푞 isms and pull- 푊̃ 푊 푌 back squares ̃푝 푝 푔

푋̃ 푋 푍 푥 푓

where:

• Both 푤 : 푊̃ ↠ 푊 and 푥 : 푋̃ ↠ 푋 are 햤-covering morphisms in u�. • Both the left square and outer rectangle are pullback diagrams in u�. Then is a -locally bijective morphism (푝, 푞)⋅(−) : h푊 → Pb(푓 ∘ −, 푔 ∘ −) 햤 in Psh(u�). In particular, if 햤 is a subcanonical unary coverage on u�, then the right square is a pullback square in u�.

Proof. By lemma 1.4.19(a), the right square is an 햤-weak pullback square

in u�, so (푝, 푞) ⋅ (−) : h푊 → Pb(푓 ∘ −, 푔 ∘ −) is 햤-locally surjective, by lemma 1.4.9.

We will now show that (푝, 푞)⋅(−) : h푊 → Pb(푓 ∘ −, 푔 ∘ −) is 햤-locally

injective. Let 푇 be an object in u� and let 푤0, 푤1 : 푇 → 푊 be a parallel pair of morphisms in u� such that:

푝 ∘ 푤0 = 푝 ∘ 푤1 푞 ∘ 푤0 = 푞 ∘ 푤1

Since 푥 : 푋̃ ↠ 푋 is an 햤-covering morphism in u�, there is a commutative square in u� of the form below,

푡 푇̃ 푇

̃푥 푝∘푤0 ̃ 푋 푥 푋

55 Abstract topology

where 푡 : 푇̃ ↠ 푇 is also an 햤-covering morphism in u�. Thus, there exist

unique morphisms 푤̃0, 푤̃1 : 푇̃ → 푊̃ such that:

̃푝∘ 푤̃0 = ̃푥 푤 ∘ 푤̃0 = 푤0 ∘ 푡

̃푝∘ 푤̃1 = ̃푥 푤 ∘ 푤̃1 = 푤1 ∘ 푡

But ⟨ ̃푝∘ −, 푞 ∘ 푤 ∘ −⟩ : h푊̃ → h푋̃ × h푌 is a monomorphism, so 푤̃0 = 푤̃1,

and therefore 푤0 ∘ 푡 = 푤1 ∘ 푡.

Thus, (푝, 푞)⋅(−) : h푊 → Pb(푓 ∘ −, 푔 ∘ −) is indeed 햤-locally bijective. To complete the proof, observe that if 햤 is a subcanonical unary coverage

on u�, then both h푊 and Pb(푓 ∘ −, 푔 ∘ −) are 햤-sheaves on u�, so by pro-

position a.3.7, (푝, 푞) ⋅ (−) : h푊 → Pb(푓 ∘ −, 푔 ∘ −) is an isomorphism in that case. ■

1.4.20 Definition. An exact category is a regular category u� with the following additional data:

• For each object 푋 in u� and each equivalence relation (푅, 푑0, 푑1) on

푋, an exact quotient 푞 : 푋 → 푋̄ of (푅, 푑0, 푑1) in u�.

Remark. In other words, an exact category is a regular category in which every equivalence relation is a kernel pair. (Recall remark 1.3.8.)

1.4.21 Definition. An 햤-local complex in u� is a tuple (푋, 푃 , 푑0, 푑1) where:

• 푋 and 푃 are objects in u�.

• 푑0 and 푑1 are morphisms 푃 → 푋 in u�.

• The 햤-closed support of h , h : h → h × h defines an equi- ⟨ 푑1 푑0⟩ 푃 푋 푋 valence relation on h푋.

Example. Let 푋 be an object in u�. Then (푋, 푋, id푋, id푋) is an 햤-local complex in u�, by lemma a.3.4.

56 1.4. Exact quotients

Let be a parallel pair of morphisms in . Recognition Lemma. 푑0, 푑1 : 푃 → 푋 u� principle for Assuming u� has 햤-weak pullback squares, the following are equivalent: local complexes is an -local complex. (i) (푋, 푃 , 푑0, 푑1) 햤 (ii) All of the following conditions are satisfied:

• There exist an 햤-covering morphism 푥 : 푋̃ ↠ 푋 in u� and a morphism in such that and . 푝 : 푋̃ → 푃 u� 푑0 ∘ 푝 = 푥 푑1 ∘ 푝 = 푥 • There exist -covering morphisms and in 햤 푝0 : 푃̃ ↠ 푃 푝1 : 푃̃ → 푃 such that and . u� 푑0 ∘ 푝0 = 푑1 ∘ 푝1 푑1 ∘ 푝0 = 푑0 ∘ 푝1 • There is an 햤-weak pullback square in u� of the form below,

푑 푄 0 푃

푑2 푑0 푃 푋 푑1

and there exist an 햤-covering morphism 푞 : 푄̃ ↠ 푄 in u� and a morphism in such that and 푝 : 푄̃ → 푃 u� 푑0 ∘ 푝 = 푑0 ∘ 푑0 ∘ 푞 . 푑1 ∘ 푝 = 푑1 ∘ 푑2 ∘ 푞

Proof. Straightforward. ⧫

1.4.22 ¶ Let (푋, 푃 , 푑0, 푑1) be an 햤-local complex.

Definition. The 햤-sheaf presented by (푋, 푃 , 푑0, 푑1) is the 햤-sheaf com- [1] pletion of the quotient presheaf h푋/푅 where 푅 is the 햤-closed support

of ⟨푑1, 푑0⟩: h푃 → h푋 × h푋. Let be the -sheaf presented by The sheaf Lemma. 푄(푋, 푃 , 푑0, 푑1) 햤 (푋, 푃 , 푑0, 푑1) presented by a and let be the image of the universal element in . 푎 (푋, id푋) 푄(푋, 푃 , 푑0, 푑1) local complex Then is an -local presentation of . (푋, 푃 , 푎, 푑0, 푑1) 햤 푄(푋, 푃 , 푑0, 푑1)

Proof. By lemmas a.3.3 and a.3.6, 푅 = Kr(푎 ⋅ −), so (푋, 푃 , 푎, 푑0, 푑1) is

indeed an 햤-local presentation of 푄(푋, 푃 , 푑0, 푑1). ■

[1] Recall proposition a.3.8(d).

57 Abstract topology

1.4.23 Definition. The exact completion of (u�, 햤) is the category Ex(u�, 햤) defined as follows:

• The objects are the 햤-local complexes in u�.

• The morphisms (푋, 푃 , 푑0, 푑1) → (푌 , 푄, 푒0, 푒1) are the morphisms

푄(푋, 푃 , 푑0, 푑1) → 푄(푌 , 푄, 푒0, 푒1) in Sh(u�, 햤). • Composition and identities are inherited from Sh(u�, 햤).

The insertion functor 휄 : u� → Ex(u�, 햤) is the evident functor that sends

each object 푋 in u� to the 햤-local complex (푋, 푋, id푋, id푋).

Remark. In view of lemmas 1.4.14 and 1.4.22, the evident functor 푄 : Ex(u�, 햤) → Sh(u�, 햤) is fully faithful and essentially surjective onto the full subcategory of 햤-locally 1-presentable 햤-sheaves on u�.

The exact Proposition. If (u�, 햤) satisfies the Shulman condition, then: completion is (i) Ex(u�, 햤) is an exact category. an exact category (ii) The insertion functor 휄 : u� → Ex(u�, 햤) preserves limits of finite diagrams and sends 햤-covering morphisms in u� to effective epimorphisms in Ex(u�, 햤).

Proof. (i). By theorem 1.4.16, Ex(u�, 햤) has limits of finite diagrams. Moreover, lemma 1.4.14 and theorem a.3.9 imply that every equivalence relation in Ex(u�, 햤) is a kernel pair and that the class of regular epimorphisms in Ex(u�, 햤) is quadrable. Thus, Ex(u�, 햤) is indeed an exact category.

(ii). The preservation of limits of finite diagrams is a consequence of theorem a.3.9. For the remainder of the claim, apply lemmas a.2.18 and a.3.10 to the fact that the Yoneda embedding u� → Psh(u�) sends 햤-covering morphisms in u� to 햤-locally surjective morphisms in Psh(u�). ■

1.4.24 ¶ Let u� be a category, let 햩 be a unary coverage on u�, and assume both (u�, 햤) and (u�, 햩) satisfy the Shulman condition.

58 1.4. Exact quotients

Definition. An admissible functor 퐹 : (u�, 햤) → (u�, 햩) is a functor 퐹 : u� → u� with the following properties: • 퐹 : (u�, 햤) → (u�, 햩) is a pre-admissible functor. • For every 햤-locally 1-presentable 햤-sheaf 퐴 on u�, there exist a 햩-

locally 1-presentable 햩-sheaf 퐹!퐴 on u� and a morphism 휎퐴 : 퐴 → ∗ 퐹 퐹!퐴 in Sh(u�, 햤) such that, for every 햩-locally 1-presentable 햩-sheaf 퐵 on u�, the following is a bijection:

∗ HomSh(u�,햩)(퐹!퐴, 퐵) → HomSh(u�,햤)(퐴, 퐹 퐵) ∗ ℎ ↦ 퐹 ℎ ∘ 휎퐴

The functor Lemma. Let 퐹 : (u�, 햤) → (u�, 햩) be an admissible functor. Assuming 햤 is between exact a subcanonical unary coverage on u�: completions (i) There exist a functor 퐹̄ : Ex(u�, 햤) → Ex(u�, 햩) and an isomorphism induced by ̄ of functors such that ̄ sends right-exact an admiss- 휂 : 휄퐹 ⇒ 퐹 휄 u� → Ex(u�, 햩) 퐹 ible functor forks in Ex(u�, 햤) to coequaliser diagrams in Ex(u�, 햩).

(ii) Moreover, any such (퐹̄ , 휂) is a pointwise left Kan extension of 휄퐹 : u� → Ex(u�, 햩) along 휄 : u� → Ex(u�, 햤).

Proof. By proposition a.3.13, the restriction functor 퐹 ∗ : Sh(u�, 햩) →

Sh(u�, 햤) has a left adjoint, say 퐹! : Sh(u�, 햤) → Sh(u�, 햩). Moreover, for every object 퐴 in Ex(u�, 햤), there exist an object 퐹̄ 퐴 in Ex(u�, 햩) and ∗ an isomorphism 휄 h퐹̄ 퐴 ≅ 퐹!h퐴 in Sh(u�, 햩). This defines a functor 퐹̄ : Ex(u�, 햤) → Ex(u�, 햩). By definition, we have the a natural bijection of the form below:

∗ ∗ ∗ HomEx(u�,햩)(퐹̄ 퐴, 퐵) ≅ HomSh(u�,햤)(휄 h퐴, 퐹 휄 h퐵)

In particular, taking 퐴 = 휄푋 and applying the Yoneda lemma, the functor 휄 : u� → Ex(u�, 햤) induces a natural map

HomEx(u�,햩)(퐹̄ 휄푋, 퐵) → HomEx(u�,햩)(휄퐹 푋, 퐵)

and hence a natural transformation 휂 : 휄퐹 ⇒ 퐹̄ 휄. The pair (퐹̄ , 휂) is then a pointwise left Kan extension of 휄퐹 : u� → Ex(u�, 햩) along 휄 : u� →

59 Abstract topology

Ex(u�, 햤). Furthermore, because 휄 : u� → Ex(u�, 햤) is fully faithful, 휂 : 휄퐹 ⇒ 퐹̄ 휄 is an isomorphism. Since the Yoneda representation Ex(u�, 햤) → Sh(u�, 햩) preserves right- exact forks, 퐹̄ : Ex(u�, 햤) → Ex(u�, 햩) sends right-exact forks in Ex(u�, 햤) to coequaliser diagrams in Ex(u�, 햩). It is clear that any pair (퐹̄ , 휂) as in (i) is determined (up to isomorphism) by 퐹̄ 휄 : u� → Ex(u�, 햩), so any such (퐹̄ , 휂) must be a pointwise left Kan extension as constructed above. ■

Remark. We will later see a converse to the above result, i.e. that the restriction of an appropriate functor between the exact completions is admissible.

1.4.25 ¶ Assume (u�, 햤) satisfies the Shulman condition. Let u� be an exact category, let 햪 be the class of effective epimorphisms in u�, and let 퐹 : u� → u� be a functor with the following properties.

• 퐹 : u� → u� sends 햤-local complexes in u� to 햪-local complexes in u�.

• 퐹 : u� → u� sends right-햤-exact forks in u� to right-햪-exact forks in u�.

Example. If u� has limits of finite diagrams and 퐹 : u� → u� is a functor that preserves limits of finite diagrams and sends members of 햤 to effective epimorphisms in u�, then 퐹 : u� → u� has the above properties.

Lemma. Under the above hypotheses, 퐹 : (u�, 햤) → (u�, 햪) is an admissible functor.

Proof. Recalling lemma 1.4.18, it is not hard to see that 퐹 : (u�, 햤) → (u�, 햪) is a pre-admissible functor. Thus, by proposition a.3.13, 퐹 ∗ :

Sh(u�, 햪) → Sh(u�, 햤) has a left adjoint, say 퐹! : Sh(u�, 햤) → Sh(u�, 햪). The Yoneda embedding u� → Sh(u�, 햪) preserves right-exact forks, and

퐹! : Sh(u�, 햤) → Sh(u�, 햪) preserves coequalisers, so for every 햤-local

complex (푋, 푃 , 푑0, 푑1) in u�, 퐹! : Sh(u�, 햤) → Sh(u�, 햪) sends the 햤-

sheaf presented by (푋, 푃 , 푑0, 푑1) to a representable 햪-sheaf on u�, as required. ■

60 1.5. Strict coproducts

1.5 Strict coproducts

Synopsis. We study extensive categories, i.e. categories in which coproducts behave like disjoint unions, and we examine the properties of exact completions of extensive categories.

Prerequisites. §§1.1, 1.3, 1.4, a.1, a.2, a.3.

1.5.1 ¶ Let u� be a category and let 휅 be a regular cardinal.

Definition. The 휅-ary canonical coverage on u� is the coverage 햩 where 햩(푋) is the set of 휅-small sinks Φ on 푋 with the following property:

• For every object (푇 , 푥) in u�∕푋, there is a 휅-small strict-epimorphic sink Θ on 푇 such that ↓(Θ) ⊆ 푥∗ ↓(Φ).

Remark. In general, the 휅-ary canonical coverage may fail to be upward- closed, but it is always composition-closed (by lemma a.2.3(c) and corollary a.2.3(d)).

1.5.2 ¶ Let u� be a category, let 푌 be an object in u� and let Φ be a set of objects

in u�∕푌 .

1.5.2(a) Definition. The object 푌 is the disjoint union of Φ if the following conditions are satisfied:

• The cocone (푓 | (푋, 푓) ∈ Φ) is a coproduct cocone in u�. • For every (푋, 푓) ∈ Φ, the morphism 푓 : 푋 → 푌 is a monomorphism in u�. • For every commutative square in u� of the form below,

푥1 푇 푋0 푥 0 푓0

푋1 푌 푓1

if (푋0, 푓0) and (푋1, 푓1) are in Φ, then either (푋0, 푓0) = (푋1, 푓1) or 푇 is an initial object in u� (or both).

61 Abstract topology

1.5.2(b) Definition. The cocone (푓 | (푋, 푓) ∈ Φ) is a strict coproduct cocone in u� if it has the following properties: • For every (푋, 푓) ∈ Φ, 푓 : 푋 → 푌 is a quadrable morphism in u�.

• For every object (푇 , 푦) in u�∕푌 , given pullback squares in u� of the form below 푆(푋,푓) 푋

푡(푋,푓) 푓

푇 푦 푌 for every (푋, 푓) ∈ Φ, the object 푇 is a disjoint union of the following set:

{(푆(푋,푓), 푡(푋,푓)) | (푋, 푓) ∈ Φ}

Remark. In particular, when Φ = ∅, this reduces to the notion of strict initial object. More explicitly, 푌 is a strict initial object if and only if, for every morphism 푦 : 푇 → 푌 in u�, 푇 is an initial object in u�.

Pullbacks along Proposition. Assuming (푓 | (푋, 푓) ∈ Φ) is a strict coproduct cocone, strict coprod- the functor uct cocones u� → u� ∕푌 ∏ ∕푋 (푋,푓)∈Φ defined by pullback is fully faithful.

Proof. Straightforward. ⧫

1.5.3 ※ For the remainder of this section, 휅 is a regular cardinal.

1.5.4 ¶ Let u� be a 휅-ary coherent category.

Recognition Lemma. Let 푌 be an object in u� and let Φ be a 휅-small set of subobjects principle for of 푌 . The following are equivalent: disjoint unions (i) 푌 is a disjoint union of Φ. in coherent is a coproduct of in the category of subobjects of and, categories (ii) (푌 , id푌 ) Φ 푌 for every commutative square in u� of the form below,

푇 푋1

푓1

푋0 푌 푓0

62 1.5. Strict coproducts

if and are both in , then either (푋0, 푓0) (푋1, 푓1) Φ (푋0, 푓0) = (푋1, 푓1) or 푇 is an initial object in u� (or both).

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). For the case where Φ has two elements, see the proof of Propos- ition 1.4.3 in [Johnstone, 2002, Part A]. The general case is similar. □

1.5.5 Definition.A 휅-ary extensive category is a category u� with the following data:

[1] • For every family (푋푖 | 푖 ∈ 퐼) of objects in u� where 퐼 is a 휅-small set,

an object ∐푖∈퐼 푋푖 in u� and a strict coproduct cocone (푓푖 | 푖 ∈ 퐼) in u�

where dom 푓푖 = 푋푖 and codom 푓푖 = ∐푖∈퐼 푋푖.

Remark. In particular, a 휅-ary extensive category has a (chosen) strict initial object, say 0.

1.5.6 Definition.A 휅-ary pretopos is an exact category that is also a 휅-ary extensive category.

1.5.6(a) Lemma. Every 휅-ary pretopos is a 휅-ary coherent category.

Proof. Straightforward. (Define the union of a set of subobjects tobethe exact image of their coproduct.) ⧫

1.5.6(b) Lemma. Let u� be a 휅-ary coherent category and let ℬ be a set of objects Criterion for in u�. Assume the following hypotheses: existence of • is an exact category. coproducts u� in coherent • For every object 푋 in u�, there is an effective epimorphism 푝 : 푋̃ ↠ 푋 categories in u� where 푋̃ is in ℬ.

• Strict coproducts of 휅-small families of objects in ℬ exist in u�.

Then u� is a 휅-ary pretopos.

[1] Since there is a proper class of 휅-small sets, strictly speaking, one should restrict to e.g. hereditarily 휅-small sets here.

63 Abstract topology

Proof. Let (푋푖 | 푖 ∈ 퐼) be a family of objects in u�. For each 푖 ∈ 퐼, choose

an effective epimorphism 푝푖 : 푋̃푖 ↠ 푋푖 in u� with 푋̃푖 in ℬ, and then choose a pullback square in u� of the form below:

푑0 푅푖 푋̃푖

푑1 푝푖 푋̃ 푋 푖 푝푖 푖 ̃ ̃ Let 푋 = ∐푖∈퐼 푋푖. It is not hard to verify that the canonical morphism

푋̃ × 푋̃ → 푋̃ × 푋̃ ∐ 푗 푘 (푗,푘)∈퐼×퐼

in u� is an isomorphism. Thus, for each 푖 ∈ 퐼, the composite 푅푖 →

푋̃푖 × 푋̃푖 ↣ 푋̃ × 푋̃ is a monomorphism in u�. Let ⟨푑1, 푑0⟩ : 푅 → 푋̃ × 푋̃

be the union. Then (푅, 푑0, 푑1) is an equivalence relation on 푋̃, so there is an exact fork in u� of the form below:

푑 1 푝 푅 푋̃ 푋 푑0

Moreover, for each 푖 ∈ 퐼, there is a unique morphism 푚푖 : 푋푖 → 푋 in u� such that the following diagram in u� commutes,

푝푖 푋̃푖 푋푖 ( ) 푚푖 ∗ ̃ 푋 푝 푋

where 푋̃푖 ↣ 푋̃ is the coproduct injection. Consider a commutative diagram in u� of the form below,

푅푗,푘 • 푋̃푘

푑 • 푅 0 푋̃

푑1 푝 ̃ ̃ 푋푗 푋 푝 푋

where each square is a pullback square in u�. Then, either 푗 = 푘 or 푅푗,푘 is an initial object in u� (or both). In view of lemma 1.3.11, it follows that,

64 1.5. Strict coproducts

for every pullback square in u� of form below,

푇 푋푘

푚푘 푋 푋 푗 푚푗 either 푗 = 푘 or 푇 is an initial object in u� (or both). Hence, (∗) is a

pullback square in u�, and by lemmas 1.3.8 and 1.3.9, 푚푖 : 푋푖 → 푋 is a monomorphism. We then apply lemma 1.5.4 to deduce that 푋 is a disjoint

union of {(푋푖, 푚푖) | 푖 ∈ 퐼}, as required. ■

1.5.7 ¶ Let u� be a category and let Fam휅(u�) be full subcategory of Fam(u�) spanned by those families 푋 such that idx 푋 is a hereditarily 휅-small set. is a -ary extensive category. Proposition. Fam휅(u�) 휅

Proof. Straightforward. ⧫

The free coprod- Theorem. Let u� be a category with 휅-ary coproducts, let 퐹 : u� → u� be uct completion a functor, and let be defined as in proposition . 훾 : u� → Fam휅(u�) a.1.8 There exist a functor that preserves -ary co- (i) 퐹̄ : Fam휅(u�) → u� 휅 products and an isomorphism 휂 : 퐹 ⇒ 퐹̄ 훾 of functors u� → u�. (ii) Moreover, any such (퐹̄ , 휂) is a left Kan extension of 퐹 : u� → u� along . 훾 : u� → Fam휅(u�) Proof. (i). Straightforward.

(ii). Let 퐺 : Fam휅(u�) → u� be a functor and let 휑 : 퐹 ⇒ 퐺훾 be a

natural transformation. For every object 푋 in Fam휅(u�), there is a unique

morphism 휃푋 : 퐹̄(푋) → 퐺(푋) making the diagram in u� shown below commute for every 푖 ∈ idx 푋, 푣 퐹 (푋(푖)) 푖 퐹̄(푋)

휑푋(푖) 휃푋 퐺(훾(푋)) 퐺(푋) 퐺(푢푖)

where 푢푖 : 훾(푋(푖)) → 푋 and 푣푖 : 퐹 (푋(푖)) → 퐹̄(푋) are the respective coproduct insertions. It is clear that we obtain a natural transformation 휃 : 퐹̄ ⇒ 퐺, and it is unique one such that 휃훾 ∙ 휂 = 휑. ■

65 Abstract topology

1.5.8 ¶ Let u� be a category with a strict initial object 0 and let 햤 be a unary coverage on u�.

Strict initial Lemma. Let 퐴 be a 햤-locally 1-generable presheaf on u�. Then 퐴(0) has objects and a unique element. locally 1- generable Proof. Let (푋, 푎) be an 햤-local generator of 퐴 and let ⊥푋 be the unique presheaves morphism 0 → 퐴 in u�. Of course, 푎 ⋅ ⊥푋 ∈ 퐴(0); it remains to be shown that it is the unique element of 퐴(0). Suppose 푎′ ∈ 퐴(0). Since every morphism in u� with codomain 0 is an isomorphism, by lemma 1.4.7, there ′ ′ ′ ′ is a morphism 푓 : 0 → 푋 in u� such that 푎 = 푎 ⋅ 푓 . But 푓 = ⊥푋, so ′ we indeed have 푎 = 푎 ⋅ ⊥푋. ■

The representable presheaf is an initial object in the The initial locally Proposition. h0 1-generable metacategory of 햤-locally 1-generable presheaves on u�. presheaf Proof. It is clear that h0 is 햤-locally 1-generable, and the Yoneda lemma reduces the claim to lemma 1.5.8. ■

1.5.9 ※ For the remainder of this section, u� is a 휅-ary extensive category.

1.5.10 Definition.A complemented monomorphism in u� is a monomorph- ′ ′ ism 푓 : 푋 → 푌 in u� for which there is an object (푋 , 푓 ) in u�∕푌 such ′ ′ that 푌 is a disjoint union of {(푋, 푓), (푋 , 푓 )}.

Properties of Proposition. complemented (i) Every isomorphism in u� is a complemented monomorphism in u�. monomorphisms (ii) The class of complemented monomorphisms in u� is closed under composition.

(iii) The class of complemented monomorphisms in u� is a quadrable class of morphisms in u�.

(iv) The class of complemented monomorphisms in u� is closed under 휅-ary coproduct in u�.

Proof. Straightforward. ⧫

66 1.5. Strict coproducts

1.5.11 Definition. A unary coverage 햤 on u� is 휅-summable if it has the following property:

• For every family (푓푖 | 푖 ∈ 퐼) of 햤-covering morphisms in u�, if 퐼 is a 휅-small set, then the coproduct

dom codom ∐ 푓푖 : ∐ 푓푖 → ∐ 푓푖 푖∈퐼 푖∈퐼 푖∈퐼 is also an 햤-covering morphism in u�.

Remark. The above is a condition on the whole class of 햤-covering morphisms in u�, not just the members of 햤.

1.5.12 ¶ Let 햤 be a 휅-summable unary coverage on u�.

1.5.12(a) Lemma. Let 퐵 be a 햤-locally 1-generable presheaf on u�. The set of 햤- Joins of locally 1- locally 1-generable 햤-closed subpresheaves of 퐵 (partially ordered by generable closed inclusion) has 휅-ary joins. subpresheaves Proof. Let (푌 , 푏) be an 햤-local generator of 퐵 and let Φ be a 휅-small set of elements of 퐵. We must show that the set of 햤-closed subpresheaves of 퐵 generated by the members of Φ admit a join in the set of 햤-locally 1-generable 햤-closed subpresheaves of 퐵. By corollary 1.4.7, we may assume that, for each (푋, 푎) ∈ Φ, there is some morphism 푓 : 푋 → 푌 ̄ ̄ ̄ in u� such that 푏 ⋅ 푓 = 푎. Let 푋 = ∐(푋,푎)∈Φ 푋, let 푓 : 푋 → 푌 be the induced morphism in u�, let ̄푎= 푏⋅푓,̄ and let 퐴̄ be the 햤-closed subpresheaf of 퐵 햤-locally generated by (푋,̄ ) ̄푎 . Clearly, for every (푋, 푎) ∈ Φ, we have 푎 ∈ 퐴(̄ 푋). It remains to be shown that 퐴̄ is the smallest 햤-locally 1-generable 햤-closed subpresheaf with this property. ′ ′ ′ Let 퐴 be an 햤-closed subpresheaf of 퐵 햤-locally generated by (푋 , 푎 ) and suppose, for every (푋, 푎) ∈ Φ, we have 푎 ∈ 퐴′(푋). As before, we may assume that 푎′ = 푏 ⋅ 푓 ′ for some morphism 푓 ′ : 푋′ → 푌 in u�. By lemma 1.4.7, for each (푋, 푎) ∈ Φ, we have an 햤-covering morphism ′ in and a morphism ′ in such that ′ ′ ′ . Let 푝(푋,푎) u� 푓(푋,푎) u� 푎 ⋅푓(푋,푎) = 푎⋅푝(푋,푎) ̃ dom ′ and let ′ ̃ be the induced morphism 푋 = ∐(푋,푎)∈Φ 푓(푋,푎) 푓 : 푋 → 푌 ′ ′ ′ ̄ ′ ̄ in u�. Then 푎 ⋅ 푓 = ̄푎⋅ ∐(푋,푎) 푓(푋,푎), so ̄푎∈ 퐴 (푋). Thus, 퐴 ⊆ 퐴 , so 퐴 is the desired join. ■

67 Abstract topology

1.5.12(b) Lemma. Assuming (u�, 햤) satisfies the Shulman condition, 휅-ary joins of Pullbacks of 햤-locally 1-generable 햤-closed subpresheaves of 햤-locally 1-presentable joins of locally 1- presheaves on u� are preserved by pullback. generable closed subpresheaves Proof. Let ℎ : 퐴 → 퐵 be a morphism of 햤-locally 1-presentable presheaves on u�. By corollary 1.4.16, we may choose 햤-local generators (푋, 푎) and (푌 , 푏) of 퐴 and 퐵 (respectively) and a morphism 푓 : 푋 → 푌

in u� such that 푏⋅푓 = ℎ(푎) and the induced morphism h푋 → 퐴×퐵 h푌 is 햤- locally surjective. Note that, for every 햤-locally 1-generable subpresheaf 퐵′ ⊆ 퐵, lemma 1.4.13 and theorem 1.4.16 imply that ℎ−1퐵′ is a 햤-locally 1-generable subpresheaf of 퐴. On the other hand, by corollary 1.4.7, there is a morphism 푦 : 푌 ′ → 푌 in u� such that 퐵′ is 햤-locally generated by ′ (푌 , 푏 ⋅ 푦). We may then choose an 햤-weak pullback square in u� of the form below, 푓 ′ 푋′ 푌 ′ 푥 푦 푋 푌 푓

′ and by the weak pullback pasting lemma (lemma a.2.19), (푋 , 푎 ⋅ 푥) is an 햤-local generator of 퐴′ = ℎ−1퐵′. ′ Now, let 퐼 be a 휅-small set, and for each 푖 ∈ 퐼, let 푥푖 : 푋푖 → 푋, ′ ′ ′ ′ 푦푖 : 푌푖 → 푌 , and 푓푖 : 푋푖 → 푌푖 be morphisms in u� such that:

′ • 푓 ∘ 푥푖 = 푦푖 ∘ 푓푖 .

′ ′ −1 ′ ′ • (푋푖 , 푎 ⋅ 푥푖) is an 햤-local generator of 퐴푖 = ℎ 퐵푖 , where 퐵푖 is the ′ 햤-closed subpresheaf of 퐵 햤-locally generated by (푌푖 , 푏 ⋅ 푦푖).

′ We will show that the join of {퐴푖 | 푖 ∈ 퐼} is the pullback of the join of ′ ′ ′ ′ ′ ′ {퐵푖 | 푖 ∈ 퐼}. Let 푋 = ∐푖∈퐼 푋푖 , let 푌 = ∐푖∈퐼 푌푖 , let 푥 : 푋 → 푋, 푦 : 푌 ′ → 푌 , 푓 ′ : 푋′ → 푌 ′ be the induced morphisms in u�, let 퐴′ be ′ the 햤-closed subpresheaf of 퐴 햤-locally generated by (푋 , 푎 ⋅ 푥), and let ′ ′ 퐵 be the 햤-closed subpresheaf of 퐵 햤-locally generated by (푌 , 푏 ⋅ 푦). ′ ′ ′ By lemma 1.5.12(a), 퐴 is the join of {퐴푖 | 푖 ∈ 퐼} and 퐵 is the join of ′ ′ −1 ′ −1 ′ ′ {퐵푖 | 푖 ∈ 퐼}, so 퐴 ⊆ ℎ 퐵 . It remains to be shown that ℎ 퐵 ⊆ 퐴 .

68 1.5. Strict coproducts

As in the first paragraph, choose an 햤-weak pullback square in u� of the form below. 푓 ′̃ 푋̃′ 푌 ′ ̃푥 푦 푋 푌 푓 ′ −1 ′ Note that (푋̃ , 푎 ⋅ ̃푥) is an 햤-local generator of ℎ 퐵 . To complete the ″ ′ proof, we must verify that 푎 ⋅ ̃푥∈ 퐴 (푋̃ ). Since u� is a 휅-ary extensive ̃′ ̃′ ′̃ ′̃ category, we may assume that 푋 = ∐푖∈퐼 푋푖 and 푓 = ∐푖∈퐼 푓푖 where ′̃ ̃′ ′ ̃′ each 푓푖 is a morphism 푋푖 → 푌푖 in u�. Let ̃푥푖 : 푋푖 → 푋 be the composite ̃′ ̃ ̃ of the coproduct injection 푋푖 → 푋 and ̃푥: 푋 → 푋. Then, ′̃ ℎ(푎 ⋅ ̃푥푖) = 푏 ⋅ (푓 ∘ ̃푥푖) = 푏 ⋅ (푦푖 ∘ 푓푖 ) ′ ̃′ −1 ′ ̃′ so we have 푎 ⋅ ̃푥푖 ∈ 퐴푖 (푋푖 ) = ℎ 퐵푖 (푋푖 ). Now, (by the pullback pasting lemma) there exist 햤-covering morph- ′ ̃″ ̃′ ′ ̃″ ′ ′ ′ ′̃ ′ isms ̃푥푖 : 푋푖 → 푋푖 and 푥푖 : 푋푖 → 푋푖 in u� such that 푓푖 ∘ 푥푖 = 푓푖 ∘ ̃푥푖 . ̃″ ̃″ ′ ′ ′ ′ Let 푋 = ∐푖∈퐼 푋푖 , let ̃푥 = ∐푖∈퐼 ̃푥푖 , and let 푥 = ∐푖∈퐼 푥푖 . Since 햤 is 휅-summable, 푥′ : 푋̃″ → 푋′ is an 햤-covering morphism. Moreover, by ′ ′ ′ ′ ″ ′ construction, 푥∘푥 = ̃푥∘ ̃푥 , so 푎⋅( ̃푥∘ ̃푥 ) ∈ 퐴 (푋̃ ). But 퐴 is 햤-closed, ′ ′ so 푎 ⋅ ̃푥∈ 퐴 (푋̃ ), as required. ■ 1.5.13 ¶ The following is a partial generalisation of Proposition 2.1 in [Gran and Vitale, 1998]. (Note that an extensive category satisfying the Shul- man condition with respect to the trivial coverage is weakly lextensive, by Proposition 1.2 in op. cit.)

The exact Proposition. If 햤 is a 휅-summable unary coverage on u� and (u�, 햤) sat- completion isfies the Shulman condition, then: is a pretopos (i) Ex(u�, 햤) is a 휅-ary pretopos. (ii) The insertion functor 휄 : u� → Ex(u�, 햤) preserves limits of finite diagrams and coproducts of 휅-small families of objects, and sends 햤- covering morphisms in u� to effective epimorphisms in Ex(u�, 햤).

Proof. By proposition 1.4.23, Ex(u�, 햤) is a regular category, and by lemmas 1.5.12(a) and 1.5.12(b), every 휅-small set of subobjects of every object has an exact union. Thus, Ex(u�, 햤) is a 휅-ary coherent category.

69 Abstract topology

We already know that 휄 : u� → Ex(u�, 햤) preserves limits of finite diagrams and sends 햤-covering morphisms in u� to effective epimorphisms in Ex(u�, 햤). On the other hand, by lemma 1.5.4, it also preserves coproducts of 휅-small families of objects. We then apply lemma 1.5.6(b) to deduce that Ex(u�, 햤) has strict coproducts of all 휅-small families of objects. ■

1.5.14 ¶ For each object 푌 in u�, let 햪(푌 ) be the set of 휅-small subsets Φ ⊆

ob u�∕푌 such that (푓 | (푋, 푓) ∈ Φ) is a (strict) coproduct cocone in u�.

Definition. The 휅-ary extensive coverage on u� is the coverage 햪 defined above.

Proposition. 햪 as defined above is a composition-closed coverage on u�.

Proof. Straightforward. ⧫

1.5.15 Lemma. Let 퐴 be a presheaf on u�. The following are equivalent: Recognition prin- (i) 퐴 is a 햪-sheaf on u�. ciple for sheaves op on extensive sites (ii) 퐴 : u� → Set sends 휅-ary coproducts in u� to 휅-ary products in Set.

Proof. (i) ⇒ (ii). Let 푌 be an object in u�, let Φ be a 휅-small set of objects

in u�∕푌 , and suppose 푌 is a disjoint union of Φ. For each (푋, 푓) ∈ Φ, let

푎(푋,푓) be an element of 퐴(푋). Since 퐴 is 햪-separated, there is at most

one element 푎 of 퐴(푌 ) such that 푎 ⋅ 푓 = 푎(푋,푓) for all (푋, 푓) ∈ Φ. Let u� be the sieve on 푌 generated by Φ. Since 퐴 : u� op → Set preserves terminal objects, for every commutative square in u� of the form below,

푥1 푇 푋0 푥 0 푓0

푋1 푌 푓1 if and are in , then . But (푋0, 푓0) (푋1, 푓1) Φ 푎(푋0,푓0) ⋅ 푥0 = 푎(푋1,푓1) ⋅ 푥1 퐴 satisfies the sheaf condition with respect to u� , so there is indeed an

element 푎 of 퐴(푌 ) such that 푎 ⋅ 푓 = 푎(푋,푓) for all (푋, 푓) ∈ Φ. Hence, the

canonical map 퐴(푌 ) → ∏(푋,푓)∈Φ 퐴(푋) is a bijection. (ii) ⇒ (i). Straightforward. ■

70 1.5. Strict coproducts

The Yoneda Theorem. embedding into (i) 햪 is a subcanonical coverage on u�, i.e. every representable presheaf sheaves on an on u� is a 햪-sheaf on u�. extensive site The Yoneda embedding preserves -ary (strict) (ii) h• : u� → Sh(u�, 햪) 휅 coproducts.

Proof. Apply lemma 1.5.15. ■

1.5.16 Definition. A coverage 햩 on u� is 휅-ary superextensive if it has the following properties:

• For every object 푋 in u�, every element of 햩(푋) is a 휅-small sink on 푋.

• For every object 푋 in u�, 햪(푋) ⊆ 햩(푋), where 햪 is the 휅-ary extensive coverage on u�.

Example. The 휅-ary extensive coverage on u� is 휅-ary superextensive, and by lemma 1.5.15 and theorem 1.5.15, the 휅-ary canonical coverage on u� is also 휅-ary superextensive.

Let be an object in , let be a -small set of objects in , Recognition prin- Lemma. 푋 u� Φ 휅 u�∕푋 ciple for covering let ̄ , and let ̄ be the induced morphism in 푈 = ∐(푈,푥)∈Φ 푈 ̄푥: 푈 → 푋 sinks in superex- u�. Assuming 햩 is a 휅-ary superextensive coverage on u�, the following are tensive coverages equivalent:

(i) Φ is a 햩-covering sink on 푋.

(ii) ̄푥: 푈̄ → 푋 is a 햩-covering morphism in u�.

Proof. Straightforward. ⧫

The unary Corollary. Let 햩 be a coverage on u� and 햤 be the class of 햩-covering coverage asso- morphisms in u�. If 햩 is a 휅-ary superextensive coverage on u�, then 햤 is a ciated with 휅-summable saturated unary coverage on u�. a superextensive coverage Proof. First, we must verify that 햤 is a unary coverage. Let 푓 : 푋 → 푌 be a morphism in u� and let 푞 : 푌̃ ↠ 푌 be a 햩-covering morphism in u�. Recalling paragraph a.2.13, we see that there is a 휅-small 햩-covering

71 Abstract topology

sink Φ on 푋 such that ↓(Φ) ⊆ 푓 ∗↓⟨푞⟩. Thus, by lemma 1.5.16, there is a commutative square in u� of the form below,

푋̃ 푌̃ 푝 푞 푋 푌 푓

where 푝 : 푋̃ ↠ 푋 is a 햩-covering morphism in u�. It is straightforward to check that 햤 is 휅-summable. Finally, by proposition a.2.14, 햤 is indeed upward-closed and composition-closed. ■

1.5.17 ¶ Let 햤 be a 휅-summable coverage on u� and, for each object 푋 in u�, let 햩(푋) be the set of 휅-small sinks Φ on 푋 such that the induced morphism

∐(푈,푥)∈Φ 푈 → 푋 in u� is a 햤-covering morphism in u�.

Properties of Proposition. the superex- (i) 햩 is a 휅-ary superextensive composition-closed coverage on u�. tensive coverage (ii) A morphism in u� is 햤-covering if and only if it is 햩-covering. generated by a summable (iii) If 햤 is a subcanonical unary coverage on u�, then 햩 is a subcanonical unary coverage coverage on u�. (iv) Assuming 햪 is a 휅-ary superextensive coverage on u�, if every 햤- covering morphism in u� is also 햪-covering, then every 햩-covering sink in u� is also 햪-covering. (v) Assuming 퐴 : u� op → Set sends 휅-ary coproducts in u� to 휅-ary products in Set, 퐴 is an 햤-sheaf on u� if and only if 퐴 is a 햩-sheaf on u�.

Proof. (i). It is clear that 햩 is a 휅-ary superextensive coverage on u�, and 햩 is composition-closed because 햤 is 휅-summable.

(ii). By construction, every 햤-covering morphism in u� is also 햩-covering. For the converse, suppose 푓 : 푋 → 푌 is a 햩-covering morphism in u�. Recalling paragraph a.2.13, since 햩 is composition-closed, there is a 휅-

small sink Φ on 푋 such that the induced morphism ∐(푈,푥)∈Φ 푈 → 푋 → 푌 in u� is a 햤-covering morphism in u�. But that implies 푓 : 푋 → 푌 itself is a 햤-covering morphism in u�, so we are done.

72 1.5. Strict coproducts

(iii) and (iv). Apply lemma 1.5.16.

(v). Combine lemma 1.5.15 and (iii). ■

Remark. Let 햪 be the 휅-ary extensive coverage on u� and let 햩′(푋) be defined as follows:

′ 햩 (푋) = 햪(푋) ∪ {(푈, 푥) ∈ ob u�∕푋 | 푥 ∈ 햤}

Then 햩′ is the smallest 휅-ary superextensive coverage on u� containing 햤. In general, 햩′ is strictly smaller than 햩, but the above proposition shows that they have the same covering sinks.

1.5.18 ¶ Let 햩 be a subcanonical 휅-ary superextensive coverage on u� and let 햪 be the 휅-ary extensive coverage on u�.

Lemma. Let ℎ : 퐴 → 퐵 be a complemented monomorphism in Sh(u�, 햩). If 퐵 is a representable 햩-sheaf, then 퐴 is also a representable 햩-sheaf.

Proof. By hypothesis, there is a (complemented) monomorphism ℎ′ : ′ ′ ′ 퐴 → 퐵 in Sh(u�, 햩) such that 퐵 is the disjoint union of {(퐴, ℎ), (퐴 , ℎ )} in Sh(u�, 햩). Thus, we have a unique morphism 푝 : 퐵 → 퐵 ⨿ 퐵 such that the following diagram in Sh(u�, 햩) commutes,

ℎ ℎ′ 퐴 퐵 퐴′ ℎ 푝 ℎ′ 퐵 퐵 ⨿ 퐵 퐵

where the bottom row is a coproduct diagram. By theorem 1.5.15, the Yoneda embedding u� → Sh(u�, 햩) preserves 휅-ary coproducts, so 퐵 ⨿ 퐵 is a representable 햩-sheaf on u�. But coproduct injections are quadrable in u� and the Yoneda embedding u� → Sh(u�, 햩) preserves pullbacks, so it follows that both 퐴 and 퐴′ are indeed representable 햩-sheaves on u�. ■

73 Abstract topology

Proposition. The inclusion Sh(u�, 햩) ↪ Sh(u�, 햪) preserves 휅-ary coproducts.

Proof. Let (퐴푖 | 푖 ∈ 퐼) be a family of 햩-sheaves u� where 퐼 is a 휅-small set and let 퐴 be their coproduct in Sh(u�, 햩). It suﬃces to show that the coproduct cocone is jointly 햪-locally surjective. Moreover, since coproduct cocones are preserved by pullback, we may assume that 퐴 is a represent-

able sheaf on u�. But then lemma 1.5.18 says that each 퐴푖 is also representable, so the coproduct cocone in question is indeed jointly 햪-locally surjective. ■

Remark. The above result is optimal in the following sense: if 휆 is a regular cardinal > 휅 such that u� is a non-trivial 휆-ary extensive category and 햫 is the 휆-ary extensive coverage on u�, then the inclusion Sh(u�, 햫) ↪ Sh(u�, 햪) preserves 휅-ary coproducts but not 휆-ary coproducts.

1.5.19 ¶ In general, an exact category may not have coequalisers of all parallel pairs. However:

Coequalisers Proposition. Let u� be a 휅-ary pretopos and let 햪 be the 휅-ary canonical in pretoposes coverage on . Assuming : u� 휅 > ℵ0 (i) u� has coequalisers of all parallel pairs. (ii) If u� is a 휅-ary pretopos and 퐹 : u� → u� is a functor that preserves limits of finite diagrams, 휅-ary coproducts, and exact quotients, then 퐹 : u� → u� preserves colimits of 휅-small diagrams. (iii) In particular, the Yoneda embedding u� → Sh(u�, 햪) preserves colimits of 휅-small diagrams.

Proof. (i) and (ii). See the proof of Lemma 1.4.19 in [Johnstone, 2002, Part A].

(iii). Since 햪 is a 휅-ary superextensive coverage on u�, the Yoneda embedding u� → Sh(u�, 햪) preserves 휅-ary coproducts, by theorem 1.5.15. The Yoneda embedding also preserves exact quotients, by lemma a.3.10. □

74 1.5. Strict coproducts

1.5.20 ¶ Let be a regular cardinal . We define a category Fam휅 as 휆 ≥ 휅 휆(u�) follows:

• The objects are as in Fam . 휆(u�) • The morphisms 푋 → 푌 are the morphisms

푋(푖) → 푌 (푗) ∐ ∐ 푖∈idx 푋 푗∈idx 푌 in Sh(u�, 햪) where 햪 is the 휅-ary extensive coverage on u�.

• Composition and identities are inherited from Sh(u�, 햪). We also define a functor Fam휅 as follows: 훾 : u� → 휆(u�) • For each object 푋 in u�, we have idx 훾(푋) = {∗} and 훾(푋)(∗) = 푋.

• For each morphism 푓 : 푋 → 푌 in u�, we have 훾(푓) = h푓 . Proposition. Fam휅 is a -ary extensive category. Furthermore, 휆(u�) 휆 훾 : Fam휅 is fully faithful and preserves -ary (strict) coproducts. u� → 휆(u�) 휅

Proof. It is clear that Fam휅 has -ary coproducts. On the other hand, 휆(u�) 휆 coproducts in Psh(u�) are always strict, so proposition a.2.18 and theorem a.3.9 imply that coproducts in Sh(u�, 햪) are also strict. Moreover, proposition 1.5.14 and lemma a.3.11 imply that any morphism 푋 → 푌 in Fam휅 must factor through the inclusion ′ for some ′ where 휆(u�) 푌 → 푌 푌 ′ |idx 푌 | ≤ max {|idx 푋|, 휅}, so coproduct injections (of 휅-ary coproducts) in Fam휅 are indeed quadrable. The remainder of the claim is a 휆(u�) consequence of theorem 1.5.15. ■

The relative Theorem. Let u� be a 휆-ary extensive category and let 퐹 : u� → u� be a coproduct functor that preserves 휅-ary coproducts. completion (i) There exist a functor ̄ Fam휅 that preserves -ary co- 퐹 : 휆(u�) → u� 휆 products and an isomorphism 휂 : 퐹 ⇒ 퐹̄ 훾 of functors u� → u�.

(ii) Moreover, any such (퐹̄ , 휂) is a left Kan extension of 퐹 : u� → u� along Fam휅 . 훾 : u� → 휆(u�)

Proof. Let 햫 be the 휆-ary extensive topology on u�. Note that the functor 퐹 ∗ : Psh(u�) → Psh(u�) sends 햫-sheaves on u� to 햪-sheaves on u�: this is

75 Abstract topology

a consequence of lemma 1.5.15. Let 퐹! : Sh(u�, 햪) → Sh(u�, 햫) be (the

functor part of) a left Kan extension of h퐹 : u� → Sh(u�, 햫) along the

Yoneda embedding h• : u� → Sh(u�, 햪). The unit of this Kan extension is automatically an isomorphism, because the Yoneda embedding is fully

faithful. Moreover, by the earlier observation, 퐹! : Sh(u�, 햪) → Sh(u�, 햫) is a left adjoint of the functor 퐹 ∗ : Sh(u�, 햫) → Sh(u�, 햪), so it preserves all coproducts. Since every object in Fam휅 is a -ary coproduct of 휆(u�) 휆 objects in the image of Fam휅 , this yields the desired left Kan 훾 : u� → 휆(u�) extension of 퐹 : u� → u�. ■

1.5.21 Proposition. With notation as in paragraph 1.5.20: Limits of finite (i) If u� has finitary products, then Fam휅(u�) also has finitary products. diagrams in the 휆 (ii) If has equalisers, then Fam휅 also has equalisers. relative coprod- u� 휆(u�) uct completion (iii) If has pullbacks, then Fam휅 also has pullbacks. u� 휆(u�)

Proof. (i). This is a consequence of the fact that binary products distribute over (possibly infinitary) coproducts in Sh(u�, 햪).

(ii). Consider an equaliser diagram in Sh(u�, 햪):

ℎ0 퐴′ 퐴 퐵 ℎ1

Suppose both 퐴 and 퐵 are coproducts (in Sh(u�, 햪)) of 휆-small families of representable 햪-sheaves on u�. We must show that 퐴′ has the same property. It suﬃces to show that 퐴′ is representable when 퐴 is representable: the general case follows by taking coproducts. But if 퐴 is representable, then proposition 1.5.20 and lemma a.3.11 imply that there is a represent- ′ able 햪-subsheaf 퐵 ⊆ 퐵 such that both ℎ0, ℎ1 : 퐴 → 퐵 factor through the inclusion 퐵′ ↪ 퐵, so 퐴′ is indeed representable.

(iii). A similar argument works. ■

76 Chapter II Charted objects

2.1 Sites

Synopsis. We consider full subcategories of pretoposes for which the associated Yonedarepresentation is fully faithful and we identify the essential image of such Yoneda representations.

Prerequisites. §§1.1, 1.3, 1.4, 1.5, a.1, a.2, a.3.

2.1.1 ※ Throughout this section, u� is a regular category.

2.1.2 ¶ Given a full subcategory u� ⊆ u�, we have the Yoneda representation

h• : u� → Psh(u�), which is not fully faithful in general. We will see that the following condition is suﬃcient.

Definition.A unary site for u� is a full subcategory u� ⊆ u� with the following property: • For every object 퐴 in u�, there is an effective epimorphism 푋 ↠ 퐴 in u� where 푋 is an object in u�.

Example. Of course, u� is a unary site for u�.

2.1.3 ¶ Let 휅 be a regular cardinal and assume u� is a 휅-ary pretopos.

Definition.A 휅-ary site for u� is a full subcategory u�0 ⊆ u� with the following property: • For every object 퐴 in u�, there is an effective epimorphism 푋 ↠ 퐴 in

u� where 푋 is a coproduct (in u�) of a 휅-small family of objects in u�0.

Example. Every unary site for u� is also a 휅-ary site for u� a fortiori.

77 Charted objects

Let be a full subcategory of and let be the full subcategory Recognition Lemma. u�0 u� u� principle for sites of u� spanned by the objects that are coproducts (in u�) of 휅-small families for pretoposes of objects in . The following are equivalent: u�0 is a -ary site for . (i) u�0 휅 u� (ii) u� is a unary site for u�.

Proof. Straightforward. ⧫

2.1.4 ¶ Let 휅 be a regular cardinal. The following is a generalisation of the notion of localic topos.

Definition.A 휅-ary pretopos u� is localic if the full subcategory of subterminal objects in u� is a 휅-ary site for u�.

Remark. The full subcategory of subterminal objects in a 휅-ary pretopos has finitary meets and 휅-ary joins, and moreover meets distribute over 휅-ary joins.

2.1.5 Proposition. Properties (i) Every isomorphism in u� is an effective epimorphism in u�. of effective The class of effective epimorphisms in is a quadrable class of epimorphisms in (ii) u� regular categories morphisms in u�.

(iii) The class of effective epimorphisms in u� is closed under composition.

(iv) Given morphisms ℎ : 퐴 → 퐵 and 푘 : 퐵 → 퐶 in u�, if 푘 ∘ ℎ : 퐴 → 퐶 is an effective epimorphism in u�, then 푘 : 퐵 → 퐶 is also an effective epimorphism in u�.

(v) The class of effective epimorphisms in u� is a saturated subcanonical unary coverage on u�.

Proof. (i). Immediate.

(ii). By lemma 1.3.8, the effective epimorphisms in u� are the same as the extremal epimorphisms in u�, which constitute a quadrable class of morphisms in u� because u� has exact images.

78 2.1. Sites

(iii). The composite of a pair of extremal epimorphisms is an extremal epimorphism (in a category with pullbacks of monomorphisms).

(iv). Straightforward.

(v). By (i)–(iv), the class of effective epimorphisms in u� is a saturated unary coverage on u�. On the other hand, by proposition 1.4.1, the effective epimorphisms in u� are also the same as the strict epimorphisms in u�. Thus, we may apply lemma a.2.11 to deduce that the class of effective epimorphisms is a subcanonical (unary) coverage. ■

2.1.6 ※ For the remainder of this section, u� is a unary site for u� and 햤 is the class of morphisms in u� that are effective epimorphisms in u�.

2.1.7 Proposition. Let 햪 be the class of effective epimorphisms in u�. Properties of the (i) Every member of 햤 is a regular epimorphism in u�. induced unary coverage on (ii) 햤 is a saturated unary coverage on u�. a unary site (iii) (u�, 햤) satisfies the Shulman condition.

(iv) If 퐹 : u� op → Set is a 햪-sheaf, then the restriction 퐹 : u� op → Set is an 햤-sheaf.

(v) In particular, 햤 is a subcanonical unary coverage on u�.

(vi) The restriction functor Psh(u�) → Psh(u�) sends 햪-locally surjective morphisms to 햤-locally surjective morphisms.

(vii) The restriction functor Sh(u�, 햪) → Sh(u�, 햤) is (half of) an equivalence of categories.

Proof. (i). Let 푓 : 푋 ↠ 푌 be an effective epimorphism in u�. Since u� is a

unary site for u�, there is an effective epimorphism ⟨푑1, 푑0⟩ : 푅 ↠ 푋×푌 푋 in u� with 푅 in u�. It is straightforward to verify that 푓 : 푋 ↠ 푌 is a

coequaliser in u� of the parallel pair 푑0, 푑1 : 푅 → 푋, so 푓 : 푋 ↠ 푌 is indeed a regular epimorphism in u�.

(ii). First, we must show that 햤 is a unary coverage on u�. Let 푓 : 푋 ↠ 푌 be an effective epimorphism in u�, let 푇 be an object in u�, and let 푦 : 푇 → 푌 be a morphism in u�. By proposition 2.1.5, the

79 Charted objects

projection 푇 ×푌 푋 → 푇 is also an effective epimorphism in u�. Since u� is

a unary site for u�, there is an effective epimorphism ⟨푡, 푥⟩ : 푆 ↠ 푇 ×푌 푋 in u� with 푆 in u�. In particular, we have a commutative square in u� of the form below, 푥 푆 푋 푡 푓

푇 푦 푌 where 푡 : 푆 ↠ 푇 is an effective epimorphism in u�, as required. Thus, 햤 is indeed a unary coverage on u�. Since the class of effective epimorphisms in u� is a saturated coverage on u�, 햤 is also saturated.

(iii). Let 푋 : u� → u� be a finite diagram. Then lim 푋 exists in u�, so we ←−−u� there is an effective epimorphism 푝 : 푈 ↠ lim 푋 in u� where 푈 is an ←−−u� object in u�. We will show that the evident cone Δ푈 ⇒ 푋 is an 햤-weak limit of 푋 in u�. Let 휑 : Δ푇 ⇒ 푋 be a cone in u�. By definition, it corresponds toa morphism 푥 : 푇 → lim 푋 in u�. As in (ii), we may find a commutative ←−−u� square in u� of the form below,

푢 푆 푈 푡 푝 푇 lim 푋 푥 ←−−u�

where 푆 is an object in u� and 푡 : 푆 ↠ 푇 is an effective epimorphism in u�. This proves the claim.

(iv). In view of proposition 1.4.1, the hypothesis implies that every object 퐴 in u� admits a strict epimorphism 푋 ↠ 퐴 in u� with 푋 in u�. The claim follows, by lemma a.2.6.

(v). Immediate, by (iv).

(vi). Straightforward.

(vii). Let 푅 : Psh(u�) → Psh(u�) be the evident functor defined on objects as follows:

푅(푃 ) = HomPsh(u�)(h•, 푃 )

80 2.1. Sites

Recalling proposition a.1.4, it is not hard to verify that 푅 is (the functor part of) a right adjoint of the restriction functor Psh(u�) → Psh(u�). Since u� is a full subcategory of u�, the Yoneda lemma implies that, for every presheaf 푃 on u�, the counit 푅(푃 ) → 푃 is an isomorphism. Moreover, by adjointness, proposition a.3.7 and (vi) imply that 푅(푃 ) is a 햪-sheaf on u� if 푃 is a 햤-sheaf on u�. In addition, for every object 퐴 in u�, there is a right-햪-exact fork in u� of the form below,

푅 푋 퐴

where 푅 and 푋 are in u�, so by lemma 1.4.18, the unit 퐹 → 푅(퐹 ) is an isomorphism. Thus, we indeed have an equivalence of categories. ■

2.1.8 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in u�. The following are equival- Effective ent: epimorph- is an -locally surjective morphism in . isms in regular (i) hℎ : h퐴 → h퐵 햤 Psh(u�) categories as (ii) is an effective epimorphism in . locally surjective ℎ : 퐴 → 퐵 u� morphisms Proof. (i) ⇒ (ii). Since u� is a unary site for u�, there is an effective epi-

morphism 푏 : 푌 ↠ 퐵 in u� with 푌 in u�. On the other hand, hℎ : h퐴 → h퐵 is 햤-locally surjective, so there is a commutative square in u� of the form below, 푎 푇 퐴 푦 ℎ 푌 퐵 푏

where 푇 is an object in u� and 푦 : 푇 ↠ 푌 is an effective epimorphism in u�. By proposition 2.1.5, 푏 ∘ 푦 : 푇 → 퐵 is an effective epimorphism in u�, so ℎ : 퐴 → 퐵 is indeed an effective epimorphism in u�.

(ii) ⇒ (i). Let ℎ : 퐴 → 퐵 be an effective epimorphism in u� and let (푌 , 푏)

be an element of h퐵. We wish to show that (푌 , 푏) is an element of the

햤-closed support of hℎ : h퐴 → h퐵.

81 Charted objects

Since u� is a unary site for u�, there is a commutative square in u� of the form below, 푎 푋 퐴 푓 ℎ 푌 퐵 푏

where 푋 is an object in u� and ⟨푎, 푓⟩ : 푋 ↠ 퐴 ×퐵 푌 is an effective epimorphism in u�. In particular, 푓 : 푋 ↠ 푌 is an effective epimorphism

in u�. Thus, (푌 , 푏) is indeed an element of the 햤-closed support of hℎ :

h퐴 → h퐵. ■

The Yoneda representation is fully faithful The Yoneda Theorem. h• : u� → Sh(u�, 햤) representation and preserves effective epimorphisms. with respect to a unary site Proof. By lemmas 2.1.8 and a.3.10, h• : u� → Sh(u�, 햤) preserves effective epimorphisms. Thus, for every object 퐴 in u�, there is a coequaliser diagram in u� of the form below,

푅 푋 퐴

where 푅 and 푋 are objects in u�, such that h• : u� → Sh(u�, 햤) preserves this coequaliser diagram. In view of the Yoneda lemma, it follows that

the Yoneda representation h• : u� → Sh(u�, 햤) is fully faithful. ■

2.1.9 ¶ The following gives a complete characterisation of the essential image of the Yoneda representation u� → Sh(u�, 햤) in the case where u� is an exact category.

Recognition Lemma. Let 퐹 be a 햤-sheaf on u�. Assuming u� is an exact category, the principle for following are equivalent: 1-presentable (i) There is an object 퐴 in u� such that h ≅ 퐹 in Sh(u�, 햤). sheaves 퐴 (ii) 퐹 is a 햤-locally 1-presentable (as a presheaf on u�).

Proof. (i) ⇒ (ii). It suﬃces to verify that h퐴 itself is 햤-locally 1-present-

able. It is clear that any 햤-sheaf on u� of the form h퐴 is 햤-locally 1-gener-

able. But u� has kernel pairs, so it follows that h퐴 is 햤-locally 1-presentable as well.

82 2.1. Sites

(ii) ⇒ (i). In view of theorem 2.1.8 and lemma a.3.10, the hypothesis implies there exist an object 푋 in u� and an effective epimorphism 푝 :

h푋 ↠ 퐹 in Sh(u�, 햤) such that Kr(푝) is 햤-locally 1-generable 햤-subsheaf

of h푋 × h푋. Thus, by lemma 1.4.13, Kr(푝) is 햤-locally 1-presentable, so

we have an equivalence relation (푅, 푑0, 푑1) on 푋 in u� such that the 햤-

closed support of ⟨푑1 ∘ −, 푑0 ∘ −⟩ : h푅 → h푋 × h푋 is Kr(푝). But u� has exact quotients of equivalence relations and the Yoneda representation u� → Sh(u�, 햤) preserves them, so it follows that 퐹 is representable by an object in u�. ■

2.1.10 ¶ As promised, we have the following converse to lemma 1.4.24.

Admissible func- Lemma. Let u� be an exact category, let u� be a unary site for u� , let 햪 be tors by restriction the class of effective epimorphisms in u�, and let 햩 be the class of morphisms in u� that are effective epimorphisms in u� . Consider a commutative square of the form below:

u� u� 퐹 퐹̄ u� u�

If u� is an exact category and 퐹̄ : u� → u� sends right-햪-exact forks in u� to coequaliser diagrams in u� , then 퐹 : (u�, 햤) → (u�, 햩) is an admissible functor.

Proof. By lemma a.3.10, the inclusion u� ↪ u� sends right-햤-exact forks in u� to right-햪-exact forks in u�, so by lemma 1.4.18, 퐹 : (u�, 햤) → (u�, 햩) is a pre-admissible functor. Moreover, by the Yoneda lemma, for every object 퐴 in u� and every object 퐵 in u� , we have the following natural bijection: ∗ u� (퐹̄ 퐴, 퐵) ≅ HomSh(u�,햪)(h퐴, 퐹̄ h퐵)

On the other hand, the restriction functor Sh(u�, 햪) → Sh(u�, 햤) is fully faithful (and essentially surjective on objects) by proposition 2.1.7, so 퐹 : (u�, 햤) → (u�, 햩) is indeed an admissible functor. ■

83 Charted objects

2.1.11 ※ For the remainder of this section:

• 휅 is a regular cardinal.

• u� is a 휅-ary pretopos.

• 햪 is the 휅-ary canonical coverage on u�.

• u�0 is a 휅-ary site for u�.

• For every object 푋 in u�0, 햩0(푋) is the set of 휅-small sinks Φ on 푋 (as

an object in u�0) such that the induced morphism ∐(푇 ,푥)∈Φ 푇 → 푋 is an effective epimorphism in u�.

2.1.12 ¶ Let 퐴 be an object in u�, let Φ be a set of objects in u�∕퐴, and let u� be

the sieve of u�∕퐴 generated by Φ.

2.1.12(a) Lemma. Suppose Φ is a 휅-small sink on 퐴. The following are equivalent: Recognition (i) Φ is a universally strict-epimorphic sink on 퐴. principle for is a strict-epimorphic sink on . universally strict- (ii) Φ 퐴 epimorphic sinks The induced morphism is an effective epimorph- (iii) ∐(푋,푎)∈Φ 푋 → 퐴 in pretoposes ism in u�.

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (iii). Let 푈 = ∐(푋,푎)∈Φ 푋 and let 푝 : 푈 → 퐴 be the induced morphism in u�. By proposition 1.3.3, it suﬃces to show that 푝 : 푈 → 퐴 is an extremal epimorphism in u�. Let 푚 : 퐴′ → 퐴 be a monomorphism in u� and suppose u� ⊆ ↓⟨푚⟩. Then we have a commutative diagram of the form below, u� ↓⟨푚⟩

u�∕퐴 id u�∕퐴 so 푚 : 퐴′ → 퐴 is an isomorphism in u�. Thus, 푝 : 푈 → 퐴 is indeed an extremal epimorphism in u�.

(iii) ⇒ (i). With notation as above, if 푝 : 푈 → 퐴 is an effective epimorphism in u�, then by propositions 1.4.1 and 2.1.5, {(푈, 푝)} is a universally

84 2.1. Sites

strict-epimorphic sink on 퐴. On the other hand, by lemma 1.5.15, 휅- ary coproduct cocones in 휅-ary extensive categories are universally strict- epimorphic sinks, so lemma a.2.3(c) implies that Φ is also a universally strict-epimorphic sink. ■

2.1.12(b) Lemma. Suppose that, for every (푋, 푎) ∈ Φ, 푋 is an object in u�. Let be the full subcategory of spanned by the objects such that u�0 u� (푋, 푎) is an object in . If is a separated presheaf on with respect to 푋 u�0 퐹 u� the 휅-ary canonical coverage, then, for every commutative square of the form below, 푠 u�0 El(퐹 )

u� u� where u� → u� and El(퐹 ) → u� are the respective projections, there is a unique functor u� → El(퐹 ) making both evident triangles commute.

Proof. It suﬃces to verify the following:

• For every commutative square in u� of the form below,

푥1 푈 푋1 푎 푥0 푎1 푋 퐴 0 푎0

where (푈, 푎) is in u� and both (푋0, 푎0) and (푋1, 푎1) are in u�0, we

have 푠(푋0, 푎0) ⋅ 푥0 = 푠(푋1, 푎1) ⋅ 푥1. But this is a consequence of the fact every object in u� admits a 휅-small

universally strict-epimorphic sink where the domains are objects in u�0 and the hypothesis that 퐹 is separated. ■

2.1.13 Proposition. Properties of is a composition-closed coverage on . (i) 햩0 u�0 the induced (ii) The restriction functor Psh Psh sends -sheaves on coverage on a (u�) → (u�0) 햪 u� to -sheaves on . site for a pretopos 햩0 u�0 In particular, is a subcanonical coverage on . (iii) 햩0 u�0

85 Charted objects

The restriction functor sends -locally sur- (iv) Psh(u�) → Psh(u�0) 햪 jective morphisms to -locally surjective morphisms. 햩0 The restriction functor is (half of) an equi- (v) Sh(u�, 햪) → Sh(u�0, 햩0) valence of categories.

Proof. (i). By lemma 2.1.3 and proposition 2.1.7, 햩0 is a coverage on u�0, and the fact that the class of effective epimorphisms in u� is closed under

휅-ary coproduct implies that 햩0 is composition-closed.

(ii). Let 퐹 : u� op → Set be a sheaf with respect to the 휅-ary canon- ′ op ical coverage on u� and let 퐹 : (u�0) → Set be the restriction. By ′ lemma a.2.6, with notation as in loc. cit., to show that 퐹 is a 햩0-sheaf, it is enough verify the following:

• For every object 푋 in u�0 and every Φ ∈ 햩0(푋), we have Γ(Φ, 퐹 ) = ′ Γ(Φ, 퐹 ) (as subsets of 퐹 (푋)).

But this is a straightforward consequence of lemma 2.1.12(b).

(iii). Immediate, by (ii).

(iv). Straightforward.

(v). Let 푅 : Psh(u�0) → Psh(u�) be the evident functor defined on objects as follows: Hom h 푅(푃 ) = Psh(u�0)( •, 푃 ) Recalling proposition a.1.4, it is not hard to verify that 푅 is (the func-

tor part of) a right adjoint of the restriction functor Psh(u�) → Psh(u�0).

Since u�0 is a full subcategory of u�, the Yoneda lemma implies that, for

every presheaf 푃 on u�0, the counit 푅(푃 ) → 푃 is an isomorphism. More- over, by adjointness, (iv) implies that 푅(푃 ) is a 햪-separated presheaf on

u� if 푃 is a 햩0-separated presheaf on u�; so, by lemma 2.1.12(b), 푅(푃 ) is

a 햪-sheaf on u� if 푃 is a 햩0-sheaf on u�0. In addition, by the same lemma, for every 햪-sheaf 퐹 on u�, the unit 퐹 → 푅(퐹 ) is an isomorphism. Thus, we indeed have an equivalence of categories. ■

86 2.1. Sites

2.1.14 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in u�. The following are equival- Effective ent: epimorphisms (i) h h h is a -locally surjective morphism in Psh . in pretoposes as ℎ : 퐴 → 퐵 햩0 (u�0) locally surjective (ii) ℎ : 퐴 → 퐵 is an effective epimorphism in u�. morphisms Proof. Omitted. (Compare lemma 2.1.8.) ◊

The Yoneda representation is fully faith- The Yoneda Theorem. h• : u� → Sh(u�0, 햩0) representation ful and preserves 휅-ary coproducts and effective epimorphisms. with respect to a Proof. site for a pretopos Consider the Yoneda embedding h• : u� → Sh(u�, 햪). By the-

orem 1.5.15, h• : u� → Sh(u�, 햪) preserves 휅-ary coproducts. On the

other hand, by theorem 2.1.8, h• : u� → Sh(u�, 햪) preserves effective epimorphisms. Since the Yoneda embedding is known to be fully faithful, the claim follows, by proposition 2.1.13. ■

2.1.15 ¶ Thus, we may identify u� with a certain full subcategory of Sh(u�0, 햩0).

We will give a characterisation of the 햩0-sheaves on u�0 that are representable by an object in u�, but to do so, we require a preliminary result.

Definition.A 햩0-sheaf on u�0 is 휅-generable if it admits a 휅-small 햩0- local generating set of elements.

Let be an object in and let be a -subsheaf of . The Recognition prin- Lemma. 퐵 u� 퐹 햩0 h퐵 ciple for gener- following are equivalent: able subsheaves (i) There is a monomorphism 푚 : 퐴 → 퐵 in u� such that 퐹 is the -closed support of . 햩0 h푚 : h퐴 → h퐵 (ii) 퐹 is 휅-generable.

Proof. (i) ⇒ (ii). It suﬃces to show that h퐴 is 휅-generable. Since u�0 is a

휅-ary site for u�, there is a 휅-small set Φ of objects in u�∕퐴 such that the

induced morphism ∐(푋,푎)∈Φ 푋 → 퐴 is an effective epimorphism in u�

and, for every (푋, 푎) ∈ Φ, 푋 is an object in u�0. Thus, by lemma 2.1.14,

h퐴 is 휅-generable. (ii) ⇒ (i). In view of theorem 2.1.14 and lemma a.3.10, the hypothesis implies there exist an object 푋 in u� and an effective epimorphism 푝 :

87 Charted objects

h푋 ↠ 퐹 in Sh(u�0, 햩0). Let 푏 : 푋 → 퐵 be the morphism in u� correspond-

ing to the composite h푋 ↠ 퐹 ↪ h퐵 and consider im(푏) : Im(푏) → 퐵

in u�. The Yoneda representation u� → Sh(u�0, 햩0) preserves effective

epimorphisms and monomorphisms, so the 햩0-closed support of im(푏) : Im(푏) → 퐵 is 퐹 , as desired. ■

Remark. In particular, h퐵 itself is a 휅-generable 햩0-sheaf on u�0.

2.1.16 Definition.A 햩0-sheaf 퐹 on u�0 is 휅-presentable if there is a set Φ with the following properties:

• Φ is a 휅-small 햩0-local generating set of elements of 퐹 .

• For every ((푋0, 푎0), (푋1, 푎1)) ∈ Φ × Φ, the sheaf Pb(푎0 ⋅ −, 푎1 ⋅ −) is 휅-generable.

Remark. Clearly, every 휅-presentable 햩0-sheaf on u�0 is also 휅-generable.

Let be a -sheaf on . The following are equivalent: Recognition prin- Lemma. 퐹 햩0 u�0 ciple for present- (i) There is an object 퐴 in u� such that h ≅ 퐹 in Sh u� , 햩 . able sheaves 퐴 ( 0 0) is a -presentable sheaf on . (ii) 퐹 휅 햩0 u�0

Proof. (i) ⇒ (ii). It suﬃces to verify that h퐴 itself is 휅-presentable. By

lemma 2.1.15, h퐴 is 휅-generable. Moreover, given any two elements of

h퐴, say (푋0, 푎0) and (푋1, 푎1), the sheaf Pb(푎0 ⋅ −, 푎1 ⋅ −) is represent-

able by an object in u�, so it is also 휅-generable. Thus, h퐴 is indeed 휅- presentable.

(ii) ⇒ (i). In view of theorem 2.1.14 and lemma a.3.10, the hypothesis implies there exist an object 푋 in u� and an effective epimorphism 푝 :

h푋 ↠ 퐹 in Sh(u�0, 햩0). It is straightforward to check that Kr(푝) is a 휅-

generable 햩0-subsheaf of h푋 × h푋, so we have an equivalence relation

(푅, 푑0, 푑1) on 푋 in u� such that the 햩0-closed support of ⟨푑1 ∘ −, 푑0 ∘ −⟩ :

h푅 → h푋 × h푋 is Kr(푝). But u� has exact quotients of equivalence rela-

tions and the Yoneda representation u� → Sh(u�0, 햩0) preserves them, so it follows that 퐹 is representable by an object in u�. ■

88 2.2. Ecumenae

2.2 Ecumenae

Synopsis. We define various notions of categories equipped with structure making it possible to interpret basic notions of topology such as open embeddings and local homeomorphisms.

Prerequisites. §§1.1, 1.2, 1.3, 1.4, 1.5, a.2.

[1] 2.2.1 Definition. An ecumene is a tuple (u�, u�, 햩) where:

• u� is a category.

• u� is a class of fibrations in u�.

• 햩 is a coverage on u�.

• Every morphism in u� of u�-type (u�, 햩)-semilocally on the domain is a member of u�.

We will often abuse notation by referring to the category u� itself as an ecumene, omitting mention of u� and 햩.

Example. Let u� be a category with pullbacks, let u� be the class of quadrable morphisms in u�, and let 햩 be any coverage on u�. Then (u�, u�, 햩) is an ecumene.

2.2.2 ¶ Let u� be a category and let ℬ be a set of morphisms in u�.

Definition. A coverage 햩 on u� is ℬ-adapted if it has the following property:

• For every object 푋 in u� and every 햩-covering sink Φ on 푋, there is ′ ′ ′ ℬ-sink Φ such that Φ ∈ 햩(푋) and ↓(Φ ) ⊆ ↓(Φ).

Example. If ℬ = mor u�, then every coverage on u� is ℬ-adapted.

Remark. Let 햩 be a coverage on u� and let 햩ℬ(푋) be the set of ℬ-sinks on

푋 that are members of 햩(푋). If 햩 is ℬ-adapted, then 햩ℬ is a coverage, and

moreover the 햩-covering sinks coincide with the 햩ℬ-covering sinks.

[1] — from Greek «οἰκουμένη», the inhabited world.

89 Charted objects

Properties Proposition. If 햩 is a ℬ-adapted coverage on u�, then ℬ satisfies the 햩- of adapted local collection axiom. coverages

Proof. Immediate. ■

2.2.3 Definition. The descent axiom for an ecumene (u�, u�, 햩) is the following:

• Every morphism in u� of u�-type 햩-semilocally on the base is a member of u�.

The ecumene Proposition. Let u� be a category, let ℬ be a class of fibrations in u�, let generated by a 햩 be a coverage on u�, and let u� be the class of morphisms in u� that are of class of fibrations ℬ-type (ℬ, 햩)-semilocally on the domain. Assuming every member of u� is a quadrable morphism in u�:

(i) (u�, u�, 햩) is an ecumene.

In addition, assuming 햩 is a ℬ-adapted coverage on u�:

(ii) (u�, u�, 햩) satisfies the descent axiom.

(iii) 햩 is a u�-adapted coverage on u�.

Proof. (i). By proposition 1.2.5 and lemma 1.2.6, u� is a class of fibrations in u� and ℬ ⊆ u�. It remains to be shown that every morphism in u� of u�- type (u�, 햩)-semilocally on the domain is a member of u�. Let 푓 : 푋 → 푌 be a morphism in u� and let Φ be a 햩-covering sink on 푋 such that, for every (푈, 푥) ∈ Φ, both 푥 : 푈 → 푋 and 푓 ∘푥 : 푈 → 푌 are members of u�. Since 푥 : 푈 → 푋 is a member of u�, there is a 햩-covering sink ′ on such that, for every ′ , both Φ(푈,푥) 푈 (푇 , 푢) ∈ Φ(푈,푥) 푢 : 푇 → 푈 and 푥 ∘ 푢 : 푇 → 푋 are members of ℬ. Thus, by proposition a.2.14,

Φ′ = (푇 , 푥 ∘ 푢) (푇 , 푢) ∈ Φ′ ⋃ { | (푈,푥)} (푈,푥)∈Φ

′ ′ is a 햩-covering sink on 푋 such that, for every (푇 , 푥 ) ∈ Φ, both 푥 : 푇 → 푋 and 푓 ∘푥′ : 푇 → 푋 are members of ℬ. Hence, 푓 : 푋 → 푌 is of ℬ-type (ℬ, 햩)-semilocally on the domain. The claim follows.

90 2.2. Ecumenae

(ii). Let 푓 : 푋 → 푌 be a morphism in u� and let Ψ be a 햩-covering sink on 푌 . Suppose, for each (푉 , 푦) ∈ Ψ, there is a pullback square in u� of the form below, 푓 ∗푦 푓 ∗푉 푋

푦∗푓 푓

푉 푦 푌 where 푦∗푓 : 푓 ∗푉 → 푉 is a member of u�. Since u� is a quadrable class of morphisms and 햩 is a ℬ-adapted coverage on u�, we may assume that each 푦 : 푉 → 푌 is a member of ℬ. Then 푦 ∘ 푦∗푓 : 푓 ∗푉 → 푌 is a member of u� and 푓 ∗푦 : 푓 ∗푉 → 푋 is a member of ℬ. Moreover, ∗ ∗ {(푓 푉 , 푓 푦) | (푉 , 푦) ∈ Ψ} is a 햩-covering sink on 푋, so 푓 : 푋 → 푌 is of u�-type (ℬ, 햩)-locally on the domain. Hence, by (i), 푓 : 푋 → 푌 is a member of u�.

(iii). Immediate. ■

Remark. In particular, if (u�, u�, 햩) is an ecumene and 햩 is a u�-adapted coverage on u�, then the descent axiom is satisfied.

2.2.4 Definition.A regulated ecumene is an ecumene (u�, u�, 햩) with the following additional data:

• For each morphism 푓 : 푋 → 푌 in u� that is a member of u�, a mono-

morphism im(푓) : Im(푓) ↣ 푌 in u� such that 푓 = im(푓) ∘ 휂푓 for some

햩-covering morphism 휂푓 : 푋 ↠ Im(푓) in u� and, for every commutative diagram in u� of the form below,

푋′ 푋

푒′ 휂푓 퐼′ Im(푓)

푖′ im(푓) 푌 ′ 푌

if both squares are pullback squares in u�, then 푒′ : 푋′ ↠ 퐼′ is an effective epimorphism in u� and 푖′ : 퐼′ ↣ 푌 ′ is a member of u�.

91 Charted objects

Remark. In the above, 푒′ : 푋′ ↠ 퐼′ is automatically a member of u�, by lemma 1.1.3.

Proposition. Let (u�, u�, 햩) be a regulated ecumene.

(i) Every effective epimorphism in u� that is a member of u� is a 햩- covering morphism in u�.

(ii) (u�, u�) is a regulated category, where u� is the subcategory of u�- perfect morphisms in u�.

Proof. (i). Let 푓 : 푋 → 푌 be an effective epimorphism in u� that is a member of u�. Then im(푓) : Im(푓) → 푌 must be an isomorphism in u�; but 푓 : 푋 → 푌 factors as a 햩-covering morphism in u� followed by im(푓) : Im(푓) → 푌 , so 푓 : 푋 → 푌 itself is also a 햩-covering morphism in u�.

(ii). By proposition 1.1.11, u� is a class of separated fibrations in u�. Every effective epimorphism in u� is a u�-calypsis a fortiori, so every member of u� is quadrably u�-eucalyptic. Thus, every member of u� is u�-agathic, as required. ■

2.2.5 ¶ Let 휅 be a regular cardinal.

Definition.A 휅-ary extensive ecumene is an ecumene (u�, u�, 햩) where: • u� is a 휅-ary extensive category. • 햩 is a 휅-ary superextensive coverage on u�. • Every complemented monomorphism in u� is a member of u�.

Recognition Lemma. Let (u�, u�, 햩) be an ecumene that satisfies the descent axiom. The principle for following are equivalent: extensive (i) is a -ary extensive ecumene. ecumenae that (u�, u�, 햩) 휅 satisfy the (ii) u� is a 휅-ary extensive category and 햩 is a 휅-ary superextensive cov- descent axiom erage on u�.

Proof. (i) ⇒ (ii). Immediate.

92 2.2. Ecumenae

(ii) ⇒ (i). Let 0 be the initial object in u�. Then, for every object 푋 in u�, the unique morphism 0 → 푋 is vacuously of u�-type (u�, 햩)-locally on the

domain, so it is a member of u�. Similarly, given morphisms 푓0 : 푋0 → 푌0

and 푓1 : 푋1 → 푌1 in u� that are members of u�, 푓0⨿푓1 : 푋0⨿푋1 → 푌0⨿푌1 is of u�-type 햩-locally on the base, so it is also a member of u�. Since every isomorphism in u� is a member of u�, it follows that every complemented monomorphism in u� is a member of u�. ■

Properties Proposition. Let (u�, u�, 햩) be a 휅-ary extensive ecumene. of extensive Given a family where is a -small set and each is (i) (푓푖 | 푖 ∈ 퐼) 퐼 휅 푓푖 ecumenae a morphism in , if each is a member of , then 푋푖 → 푌 u� 푓푖 : 푋푖 → 푌 u� the induced morphism is also a member of . 푓 : ∐푖∈퐼 푋푖 → 푌 u� (ii) u� is closed under 휅-ary coproduct in u�. (iii) Assuming 햩 is u�-adapted, a morphism 푔 : 푌 → 푍 in u� is 햩-covering if and only if there is a morphism 푓 : 푋 → 푌 in u� such that 푔 ∘ 푓 : 푋 → 푍 is 햩-covering morphism in u� that is a member of u�.

Proof. (i). By construction, 푓 : ∐푖∈퐼 푋푖 → 푌 is of u�-type (u�, 햩)-semilocally on the domain, so it is a member of u�.

(ii). Since coproduct injections are in u� and u� is closed under composition, the claim is a special case of (i).

(iii). Apply lemma 1.5.16 and (i). ■

2.2.6 ※ For the remainder of this section, (u�, u�, 햩) is an ecumene.

2.2.7 ¶ It is convenient to introduce some terminology for properties of morphisms related to u�.

2.2.7(a) Definition. A morphism in u� is genial if it is a member of u�.

2.2.7(b) Definition. A morphism in u� is étale if it is u�-perfect.

[2] 2.2.7(c) Definition. A morphism in u� is eunoic if it is 햩-semilocally of u�-type.

[2] — from Greek «εὐνοϊκός», favourable.

93 Charted objects

Remark. By lemma 1.2.15, eunoic morphisms are automatically 햩-locally of u�-type.

2.2.8 Definition. An equivalence relation (푅, 푑0, 푑1) on an object 푋 in u� is étale if it has the following property:

• The projections 푑0, 푑1 : 푅 → 푋 are étale morphisms. Let be a kernel pair of a morphism in . 2.2.8(a) Lemma. (푅, 푑0, 푑1) 푓 : 푋 → 푌 u� Recognition The following are equivalent: principle for (i) is a genial morphism in and is an étale kernel pairs of 푓 : 푋 → 푌 u� (푅, 푑0, 푑1) étale morphisms equivalence relation on 푋 in u�. (ii) 푓 : 푋 → 푌 is an étale morphism in u�. Proof. (i) ⇒ (ii). By lemma 1.1.10, the relative diagonal Δ푓 : 푋 → 푅

is étale if either 푑0 : 푅 → 푋 or 푑1 : 푅 → 푋 is étale, in which case 푓 : 푋 → 푌 itself is étale.

(ii) ⇒ (i). The class of étale morphisms in u� is a quadrable class of morphisms in u�, by proposition 1.1.11, so if 푓 : 푋 → 푌 is étale then both

푑0, 푑1 : 푅 → 푋 are also étale. ■ Let be a kernel pair of a -covering morphism 2.2.8(b) Lemma. (푅, 푑0, 푑1) 햩 푓 : Recognition prin- 푋 ↠ 푌 in u�. Assuming (u�, u�, 햩) satisfies the descent axiom, the following ciple for kernel are equivalent: pairs of covering is an étale equivalence relation on in . étale morphisms (i) (푅, 푑0, 푑1) 푋 u� (ii) 푓 : 푋 ↠ 푌 is an étale morphism in u�.

Proof. By definition, the following is a pullback square in u�: 푑 푅 0 푋

푑1 푓 푋 푌 푓 Since 푓 : 푋 ↠ 푌 is a 햩-covering morphism in u�, the descent axiom

implies that 푓 : 푋 ↠ 푌 is a member of u� if and only if either 푑0 : 푅 → 푋

or 푑1 : 푅 → 푋 (or both) is a member of u�. Thus, the claims reduce to lemma 2.2.8(a). ■

94 2.2. Ecumenae

2.2.9 ¶ The following generalises Proposition 1.6 in [JM].

Definition. The ecumene (u�, u�, 햩) is étale if every member of u� is an étale morphism in u�.

The induced Proposition. Let u� be the class of étale morphisms in u�. étale ecumene (i) (u�, u�, 햩) is an étale ecumene. (ii) If (u�, u�, 햩) satisfies the descent axiom, then (u�, u�, 햩) also satisfies the descent axiom.

(iii) (u�, u�, 햩) is 휅-ary extensive if and only if (u�, u�, 햩) is 휅-ary extensive.

Proof. (i). By proposition 1.1.11, u� is a class of separated fibrations in u�. It remains to be shown that every morphism of u�-type (u�, 햩)-semilocally on the domain is a member of u�. Let 푓 : 푋 → 푌 be a morphism in u� and let Φ be a 햩-covering u�-sink on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is a member of u�. Then 푓 : 푋 → 푌 is of u�-type (u�, 햩)-semilocally on the domain, so by lemma 1.2.6, 푓 : 푋 → 푌 is a member of u�. Since 푥 : 푈 → 푋 is

a member of u�, the induced morphism 푈 ×푌 푈 → 푋 ×푌 푋 is also a member of u�. On the other hand, 푓 ∘ 푥 : 푈 → 푌 is a member of u�, so

the relative diagonal Δ푓∘푥 : 푈 → 푈 ×푌 푈 is a member of u�. Thus, the

relative diagonal Δ푓 : 푋 → 푋 ×푌 푋 is of u�-type (u�, 햩)-semilocally on the domain, therefore it is a member of u�. Hence, 푓 : 푋 → 푌 is indeed a member of u�.

(ii). Let 푓 : 푋 → 푌 be a morphism in u� and let Ψ be a 햩-covering sink on 푌 such that, for every (푉 , 푦) ∈ Φ, we have a pullback square in u� of the form below, 푥 푈 푋 푣 푓

푉 푦 푌 where 푣 : 푈 → 푉 is a member of u�. The hypothesis implies 푓 : 푋 → 푌 is a member of u�, and it remains to be shown that 푓 : 푋 → 푌 is u�- separated.

95 Charted objects

By the pullback pasting lemma, every face of the following diagram is a pullback square in u�:

푈 ×푉 푈 푈

푥×푦푥 푈 푉

푋 ×푌 푋 푋

푋 푌 Hence, we have a pullback square in u� of the form below:

푥 푈 푋

Δ푣 Δ푓

푈 ×푉 푈 푋 ×푌 푋 Moreover, by proposition a.2.14,

{(푈 ×푉 푈, 푥 ×푦 푥) | (푉 , 푦) ∈ Ψ}

is a 햩-covering sink on 푋×푌 푋, so the relative diagonal Δ푓 : 푋 → 푋×푌 푋

is of u�-type 햩-semilocally on the base. Thus, Δ푓 : 푋 → 푋 ×푌 푋 is a member of u�, as required. (iii). Immediate, because genial monomorphisms are the same as étale monomorphisms. ■

2.2.10 ¶ Étale morphisms and eunoic morphisms are related as follows.

Recognition Lemma. Let 푓 : 푋 → 푌 be a quadrable morphism in u� such that the principle for relative diagonal is also a quadrable morphism Δ푓 : 푋 → 푋 ×푌 푋 étale morphisms in u�. Assuming (u�, u�, 햩) satisfies the descent axiom, the following are equivalent:

(i) The morphism 푓 : 푋 → 푌 is étale. Both and the relative diagonal are (ii) 푓 : 푋 → 푌 Δ푓 : 푋 → 푋 ×푌 푋 eunoic.

Proof. This is a special case of proposition 1.2.20. ■

96 2.2. Ecumenae

2.2.11 ¶ It is convenient to introduce the following terminology.

2.2.11(a) Definition. An open embedding in u� is a monomorphism in u� that is a member of u�.

2.2.11(b) Definition. The descent axiom for open embeddings in (u�, u�, 햩) is the following:

• Every monomorphism in u� of u�-type 햩-semilocally on the base is an open embedding in u�.

Recognition Lemma. Let 푓 : 푋 → 푌 be a quadrable morphism in u�. Assuming principles for (u�, u�, 햩) satisfies the descent axiom for open embeddings, the following open embeddings are equivalent:

(i) 푓 : 푋 → 푌 is an open embedding in u�. (ii) 푓 : 푋 → 푌 is an étale monomorphism in u�. (iii) 푓 : 푋 → 푌 is a eunoic monomorphism in u�.

Proof. (i) ⇒ (ii). The relative diagonal Δ푓 : 푋 → 푋 ×푌 푋 is an isomorphism in u�.

(ii) ⇒ (iii). Étale morphisms are eunoic a fortiori.

(iii) ⇒ (i). It suﬃces to verify the following:

• Every monomorphism in u� of u�-type 햩-semilocally on the domain is an open embedding in u�. However, by lemma 1.2.19(a), every such monomorphism is automatically of u�-type (u�, 햩)-semilocally on the domain, hence is indeed a member of u�. ■

2.2.12 ¶ Let u�mono be the class of open embeddings in u�. We will see that the following is a specialisation of the notion of étale morphism.

Definition.A local homeomorphism in u� is a morphism in u� of u�mono- type (u�mono, 햩)-semilocally on the domain.

97 Charted objects

Remark. By lemma 1.2.6, local homeomorphisms are automatically of

u�mono-type (u�mono, 햩)-locally on the domain.

Proposition. (i) Every open embedding in u� is a local homeomorphism in u�. (ii) The class of local homeomorphisms in u� is a quadrable class of morphisms in u�. (iii) The class of local homeomorphisms in u� is closed under composition.

(iv) Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u�, if both 푔 : 푌 → 푍 and 푔 ∘ 푓 : 푋 → 푍 are local homeomorphisms, then 푓 : 푋 → 푌 is also a local homeomorphism. (v) Every local homeomorphism in u� is an étale morphism in u�.

Proof. (i). Immediate.

(ii). Clearly, every local homeomorphism in u� is of u�-type (u�, 햩)-semilocally on the domain, hence is a member of u�. In particular, local homeomorphisms in u� are quadrable. The claim follows, by proposition 1.2.4.

(iii). In view of lemma 1.2.6, we may apply proposition 1.2.5.

(iv). Let Ψ be a 햩-covering u�mono-sink on 푌 such that, for every (푉 , 푦) ∈ Ψ, 푔 ∘ 푦 : 푉 → 푍 is an open embedding in u�. By proposition a.2.14

and (ii), there is a 햩-covering u�mono-sink Φ on 푋 such that, for every (푈, 푥) ∈ Φ, 푔 ∘ 푓 ∘ 푥 : 푈 → 푍 is an open embedding in u� and factors

through 푔 ∘ 푦 : 푉 → 푍 for some (푉 , 푦) ∈ Ψ. Recalling that u�mono is a class of fibrations in u�, by lemma 1.1.3, 푓 ∘ 푥 : 푈 → 푌 is also an open embedding in u�. Hence, 푓 : 푋 → 푌 is a local homeomorphism in u�.

(v). If 푓 : 푋 → 푌 is a local homeomorphism in u�, then the relative

diagonal Δ푓 : 푋 → 푋 ×푌 푋 is also a local homeomorphism in u�, by (i), (ii), and (iv). Thus, by lemma 1.1.6, 푓 : 푋 → 푌 is indeed an étale morphism in u�. ■

98 2.2. Ecumenae

2.2.13 Proposition. Let u� be the class of local homeomorphisms in u�. The ecumene of (i) (u�, u�, 햩) is an étale ecumene. local homeomorphisms (ii) (u�, u�, 햩) satisfies the descent axiom for open embeddings if andonly if (u�, u�, 햩) satisfies the descent axiom for open embeddings.

(iii) (u�, u�, 햩) is 휅-ary extensive if and only if (u�, u�, 햩) is 휅-ary extensive.

Proof. (i). By proposition 2.2.12, u� is a class of separated fibrations in u�. It remains to be shown that every morphism of u�-type (u�, 햩)-semilocally on the domain is a member of u�. Let 푓 : 푋 → 푌 be a morphism in u� and let Φ be a 햩-covering u�- sink on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is a local homeomorphism in u�. So, for each (푈, 푥) ∈ Φ, there is a 햩-covering

u�mono sink Θ(푈,푥) on 푈 such that, for every (푇 , 푢) ∈ Θ(푈,푥), 푥∘푢 : 푇 → 푋 is an open embedding in u�. Consider the sink Φ′ defined as follows:

′ ( ) ( ) Φ = ⋃ { 푇 , 푥 ∘ 푢 | 푇 , 푢 ∈ Θ(푈,푥)} (푈,푥)∈Φ

′ By proposition a.2.14, Φ is a 햩-covering u�mono-sink on 푋. Moreover, each 푓 ∘푥∘푢 : 푇 → 푌 is a local homeomorphism in u�, so 푓 : 푋 → 푌 is of

u�-type (u�mono, 햩)-semilocally on the domain. Thus, by proposition 1.2.4, 푓 : 푋 → 푌 itself is indeed a local homeomorphism in u�.

(ii) and (iii). Immediate. ■

2.2.14 ¶ We will now characterise local homeomorphisms as genial morphisms whose kernel pair have a certain property.

Definition. An equivalence relation (푅, 푑0, 푑1) on an object 푋 in u� is tractable if it has the following properties:

• The projections 푑0, 푑1 : 푅 → 푋 are members of u�.

• There is a 햩-covering u�mono-sink Φ on 푋 such that, for every (푈, 푥) ∈

Φ and every object (푇 , 푟) in u�∕푅, if both 푑0 ∘ 푟, 푑1 ∘ 푟 : 푇 → 푋 factor

through 푥 : 푈 → 푋, then 푑0 ∘ 푟 = 푑1 ∘ 푟.

99 Charted objects

Remark. By proposition 2.2.12, if (푅, 푑0, 푑1) is a tractable equivalence relation on 푋, then the relative diagonal Δ : 푋 → 푅 is an open embedding in u�. However, the converse is not true in general, even when we assume

that the projections 푑0, 푑1 : 푅 → 푋 are local homeomorphisms in u�.

Let be an equivalence relation on an object in . 2.2.14(a) Lemma. (푅, 푑0, 푑1) 푋 u� A suﬃcient Assuming the projections are genial morphisms in , if 푑0, 푑1 : 푅 → 푋 u� criterion for there is a local homeomorphism in such that , 푓 : 푋 → 푌 u� 푓 ∘푑0 = 푓 ∘푑1 tractability then is tractable. (푅, 푑0, 푑1)

Proof. Let Φ be a 햩-covering u�mono-sink on 푋 such that, for every (푈, 푥) ∈ Φ, 푓 ∘ 푥 : 푈 → 푌 is an open embedding in u�. For every object (푇 , 푟) in

u�∕푅, we have 푓 ∘ 푑0 ∘ 푟 = 푓 ∘ 푑1 ∘ 푟, so if both 푑0 ∘ 푟, 푑1 ∘ 푟 : 푇 → 푋 factor

through 푥 : 푈 → 푋, then 푑0 ∘ 푟 = 푑1 ∘ 푟. Hence (푅, 푑0, 푑1) is indeed a tractable equivalence relation on 푋 in u�. ■

Let be a kernel pair of a morphism in . 2.2.14(b) Lemma. (푅, 푑0, 푑1) 푓 : 푋 → 푌 u� Recognition prin- The following are equivalent: ciple for kernel (i) is a genial morphism in and is a tractable pairs of local 푓 : 푋 → 푌 u� (푅, 푑0, 푑1) homeomorphisms equivalence relation on 푋 in u�. (ii) 푓 : 푋 → 푌 is a local homeomorphism in u�.

Proof. (i) ⇒ (ii). Let Φ be a 햩-covering u�mono-sink on 푋 such that, for

every (푈, 푥) ∈ Φ and every object (푇 , 푟) in u�∕푅, if both 푑0 ∘ 푟, 푑1 ∘ 푟 :

푇 → 푋 factor through 푥 : 푈 → 푋, then 푑0 ∘ 푟 = 푑1 ∘ 푟. Then 푓 ∘ 푥 :

푈 → 푌 is a monomorphism in u�: indeed, given 푢0, 푢1 : 푇 → 푈 in u�,

if 푓 ∘ 푥 ∘ 푢0 = 푓 ∘ 푥 ∘ 푢1, then we may apply the hypothesis to deduce

that 푢0 = 푢1. Since 푓 : 푋 → 푌 is a member of u� and u� is closed under composition, it follows that 푓 ∘ 푥 : 푈 → 푌 is an open embedding in u�. Hence 푓 : 푋 → 푌 is indeed a local homeomorphism in u�.

(ii) ⇒ (i). This is a special case of lemma 2.2.14(a). ■

100 2.2. Ecumenae

Let be a kernel pair of a -covering morphism 2.2.14(c) Lemma. (푅, 푑0, 푑1) 햩 푓 : Recognition 푋 ↠ 푌 in u�. Assuming (u�, u�, 햩) satisfies the descent axiom for open principle for embeddings, the following are equivalent: kernel pairs of is a tractable equivalence relation on in . covering local (i) (푅, 푑0, 푑1) 푋 u� homeomorphisms (ii) 푓 : 푋 ↠ 푌 is a local homeomorphism in u�.

Proof. (i) ⇒ (ii). By definition, the following is a pullback square in u�:

푑 푅 0 푋

푑1 푓 푋 푌 푓

Suppose 푥 : 푈 → 푋 is an open embedding in u� such that 푓 ∘ 푥 : 푈 → 푌

is a monomorphism in u�. Then the projection 푋 ×푌 푈 → 푋 is also an open embedding in u�, and since 푓 : 푋 ↠ 푌 is a 햩-covering morphism in u�, 푓 ∘ 푥 : 푈 → 푌 is an open embedding in u�. Thus, following the argument of lemma 2.2.14(b), we see that 푓 : 푋 → 푌 itself is a local homeomorphism in u�.

(ii) ⇒ (i). This is a special case of lemma 2.2.14(a). ■

101 Charted objects

2.3 Extents

Synopsis. We examine the properties of categories of objects that are obtained as étale quotients of distinguished objects in pretoposes equipped with a notion of étale morphism.

Prerequisites. §§1.1, 1.2, 1.4, 1.5, 2.1, 2.2.

2.3.1 ¶ Let 휅 be a regular cardinal and let u� be a 휅-ary pretopos. The following definition is due to Joyal [JM, §1].

Definition.A class of étale morphisms in u� is a subset u� ⊆ mor u� with the following properties:

A1. Every isomorphism in u� is a member of u� and u� is closed under composition.

A2. u� is a quadrable class of morphisms in u�. A3. In every pullback square in u� of the form below,

퐴̃ 퐴

ℎ̃ ℎ 퐵̃ 퐵

where 퐵̃ ↠ 퐵 is an effective epimorphism in u�, if ℎ̃ : 퐴̃ → 퐵̃ is a member of u�, then ℎ : 퐴 → 퐵 is also a member of u�.

A4. For every 휅-small set 퐼, the unique morphism ∐푖∈퐼 1 → 1 is a member of u�, where 1 is the terminal object of u�.

A5. For every family (ℎ푖 | 푖 ∈ 퐼) where 퐼 is a 휅-small set, if each ℎ푖 :

퐴푖 → 퐵푖 is a member of u�, then ∐푖∈퐼 ℎ푖 : ∐푖∈퐼 퐴푖 → ∐푖∈퐼 퐵푖 is also a member of u�.

A7. If ℎ : 퐴 → 퐵 is a member of u�, then the relative diagonal Δℎ :

퐴 → 퐴 ×퐵 퐴 is also a member of u�. A8. Given an effective epimorphism ℎ : 퐴 ↠ 퐵 in u� and a morphism 푘 : 퐵 → 퐶 in u�, if both ℎ : 퐴 ↠ 퐵 and 푘 ∘ ℎ : 퐴 → 퐶 are members of u�, then 푘 : 퐵 → 퐶 is also a member of u�.

102 2.3. Extents

Example. The class of all morphisms in u� is a class of étale morphisms in u�.

Recognition Lemma. Let u� be a set of morphisms in u�. The following are equivalent: principle for (i) u� satisfies axioms A1, A2, A4, A5, A7, and A8. classes of étale (ii) , where is the -ary canonical coverage on , is an étale morphisms (u�, u�, 햪) 햪 휅 u� 휅-ary extensive ecumene. Moreover, u� satisfies axiom A3 if and only if (u�, u�, 햪) satisfies the descent axiom.

Proof. (i) ⇒ (ii). By lemma 1.1.10, axioms A1, A2, and A7 imply that u� is a class of separated fibrations in u�. Hence, by axioms A1 and A5,

for every 휅-small set 퐼, every morphism 1 → ∐푖∈퐼 1 is a member of u�, and therefore (by axiom A2 and extensivity) every complemented monomorphism is a member of u�. Axioms A1, A2, A4, and A5 imply

that, for every object 퐵 in u� and every family (ℎ푖 | 푖 ∈ 퐼) where 퐼 is

a 휅-small set and each ℎ푖 : 퐴푖 → 퐵 is a member of u�, the induced

morphism ∐푖∈퐼 퐴푖 → 퐵 is a member of u�. Thus, in view of proposition 1.4.1, lemma 1.5.16, and axiom A8, every morphism in u� of u�-type (u�, 햪)-semilocally on the domain is a member of u�. Hence, (u�, u�, 햪) is an étale 휅-ary extensive ecumene, and it is clear that axiom A3 implies that the descent axiom is satisfied.

(ii) ⇒ (i). Axioms A1, A2, and A7 are immediate. Axiom A8 is also straightforward to verify, given that the 햪-covering morphisms in u� are precisely the effective epimorphisms in u�. The same argument shows that the descent axiom implies axiom A3. Finally, axioms A4 and A5 are special cases of proposition 2.2.5. ■

2.3.2 ¶ Let (u�, u�, 햩) be an étale 휅-ary extensive ecumene that satisfies the descent axiom, let 햤 be the class of 햩-covering morphisms in u�, let u� = Ex(u�, 햤), and let 휄 : u� → u� be the insertion functor. Assume (u�, 햤) satisfies the Shulman condition, so that u� is a 휅-ary pretopos. (Recall proposition 1.5.13 and corollary 1.5.16.) The following is a variation on Theorem 5.2 and Corollary 5.3 in [JM].

103 Charted objects

The exact Proposition. Let u� be the class of morphisms in u� corresponding to completion of an morphisms in Psh(u�) 햩-semilocally of u�-type and let u�̂ be the class of étale extensive u�-perfect morphisms in u�. ecumene (i) A morphism in u� is a member of u� if and only if it corresponds to a morphism in Psh(u�) 햤-semilocally of u�-type. (ii) (u�, u�, 햪) is a 휅-ary extensive ecumene that satisfies the descent axiom, where 햪 is the 휅-ary canonical topology on u�. (iii) The insertion functor 휄 : u� → u� sends members of u� to members of u�. (iv) u�̂ is a class of étale morphisms in u�. (v) For every object 푌 in u� and every morphism ℎ : 퐴 → 휄(푌 ) in u�, if ℎ : 퐴 → 휄(푌 ) is a member of u�̂, then there exist a morphism 푓 : 푋 → 푌 in u� and an effective epimorphism 푝 : 휄(푋) ↠ 퐴 in u� such that 푓 : 푋 → 푌 is a member of u�. (vi) For every quadrable morphism 푓 : 푋 → 푌 in u�, assuming the relative diagonal is a quadrable monomorphism Δ푓 : 푋 → 푋 ×푌 푋 in u�, 푓 : 푋 → 푌 is a member of u� if and only if 휄(푓) : 휄(푋) → 휄(푌 ) is a member of u�̂. (vii) If 햩 is a u�-adapted coverage on u�, then u� and u�̂ both satisfy the 햪-local collection axiom.

Proof. (i). It is clear that every morphism in Psh(u�) 햤-semilocally of u�- type is also 햩-semilocally of u�-type; it remains to be shown that every member of u� corresponds to a morphism in Psh(u�) 햤-semilocally of u�- type. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�) where both 퐴 and 퐵 are 햤-locally 1-presentable 햤-sheaves on u�. Suppose Ψ is a 햩-local generating set of elements of 퐵 such that, for every (푌 , 푏) ∈ Ψ, there is a 햩-

local generating set Φ(푌 ,푏) of elements of Pb(푏 ⋅ −, ℎ) such that, for every (푋, (푓, 푎)) ∈ Φ, 푓 : 푋 → 푌 is a member of u�. By theorem 1.4.16, Pb(푏 ⋅ −, ℎ) is 햤-locally 1-generable, so it is 햩-locally 1-generable a fortiori . Thus, by lemma a.2.16, replacing Φ(푌 ,푏) with a subset if necessary,

104 2.3. Extents

we may assume that Φ(푌 ,푏) is a 휅-small set. Similarly, we may assume that Ψ is a 휅-small set. But proposition 1.5.13, lemmas 1.5.15 and 1.5.16, and corollary 1.5.16 imply that 퐴 and 퐵 are also 햩-sheaves on u�, so by

proposition 2.2.5, we may further assume that Ψ and Φ(푌 ,푏) are singletons. Hence, ℎ : 퐴 → 퐵 is indeed 햤-semilocally of u�-type.

(ii) and (iii). First, note that lemma a.3.10 implies that the effective epimorphisms in u� are precisely the morphisms in u� corresponding to 햤- locally surjective morphisms in Psh(u�), which are 햩-locally surjective a fortiori. We also know that 햪 is a 휅-ary superextensive coverage on u� (lemma 1.5.15 and theorem 1.5.15). Thus, by proposition 1.2.14 and lemma 1.5.16, u� is a class of fibrations in u�, every morphism in u� 햪- locally of u�-type is a member of u�, and 휄 : u� → u� sends members of u� to members of u�. Recalling lemma 2.2.5, we conclude that (u�, u�, 햪) is indeed a 휅-ary extensive ecumene that satisfies the descent axiom.

(iv). Recalling lemma 2.3.1, this is a special case of proposition 2.2.9.

(v). This is a consequence of lemma 1.4.8.

(vi). Apply proposition 1.2.20.

(vii). By construction, for every object 퐴 in u�, there exist an object 푋 in u� and an effective epimorphism 휄(푋) ↠ 퐴 in u�. Since both u� and u�̂ are quadrable classes of morphisms in u�, it suﬃces to verify the following:

• If ℎ : 퐴 → 퐵 is an effective epimorphism in u� and 푘 : 퐵 → 퐶 is a member of u� where 퐵 = 휄(푌 ) and 퐶 = 휄(푍) for some objects 푌 and 푍 in u�, then there is a morphism 푠 : 퐴′ → 퐴 in u� such that ℎ∘푠 : 퐴′ → 퐵 is an effective epimorphism in u� that is a member of u�̂.

It is straightforward to further reduce to the case where 퐴 = 휄(푋), which is an immediate consequence of proposition 2.2.2. ■

2.3.3 ¶ The following is a notion intermediate between pretoposes with a class of étale morphisms (in the sense of Joyal) and étale extensive ecumenae.

105 Charted objects

Definition.A 휅-ary gros pretopos is a pair (u�, u�) where: • u� is a 휅-ary pretopos. • u� is a set of morphisms in u� that satisfies axioms A1, A2, A4, A5, and A7.

• Given an effective epimorphism ℎ : 퐴 ↠ 퐵 in u�, a morphism 푘 :

퐵 → 퐶 in u�, and a kernel pair (푅, 푑0, 푑1) of ℎ : 퐴 ↠ 퐵 in u�, if

푑0, 푑1 : 푅 → 퐴 and 푘 ∘ ℎ : 퐴 → 퐶 are all members of u�, then both ℎ : 퐴 ↠ 퐵 and 푘 : 퐵 → 퐶 are members of u�.

Axiom A8 and Lemma. Let u� be a 휅-ary pretopos and let u� be a set of morphisms in u�. gros pretoposes (i) If (u�, u�) is a 휅-ary gros pretopos, then u� satisfies axiom A8. (ii) If (u�, u�) is a 휅-ary gros pretopos, then (u�, u�, 햪) is an étale 휅-ary extensive regulated ecumene, where 햪 is the 휅-ary canonical coverage on u�. (iii) If u� is a class of étale morphisms in u�, then (u�, u�) is a 휅-ary gros pretopos.

Proof. (i). Let ℎ : 퐴 → 퐵 be an effective epimorphism in u� that is a

member of u� and let (푅, 푑0, 푑1) be a kernel pair of ℎ : 퐴 → 퐵 in u�.

By axiom A2, the projections 푑0, 푑1 : 푅 → 퐴 are members of u�. Let 푘 : 퐵 → 퐶 be a morphism in u� such that 푘 ∘ ℎ : 퐴 → 퐶 is a member of u�. We may then apply the hypothesis to deduce that 푘 : 퐵 → 퐶 is a member of u�, as required.

(ii). By lemma 2.3.1, (u�, u�, 햪) is an étale 휅-ary extensive ecumene. It remains to be shown that (u�, u�, 햪) is regulated. Since u� is a 휅-ary pretopos and 햪 is the 휅-ary canonical coverage on u�, it is enough to verify that the image of a member of u� is also a member of u�.

Let ℎ : 퐴 → 퐵 be a member of u� and let (푅, 푑0, 푑1) be a kernel pair

of ℎ : 퐴 → 퐵 in u�. As before, the projections 푑0, 푑1 : 푅 → 퐴 are members of u�. On the other hand, ℎ = im(ℎ) ∘ 푒 where 푒 : 퐴 ↠ Im(ℎ)

is the coequaliser of 푑0, 푑1 : 푅 → 퐴. Thus, by the hypothesis, im(ℎ) : Im(ℎ) → 퐵 is indeed a member of u�.

106 2.3. Extents

(iii). Let ℎ : 퐴 → 퐵 be an effective epimorphism in u� and let (푅, 푑0, 푑1)

be a kernel pair of ℎ : 퐴 → 퐵 in u�. Suppose 푑0, 푑1 : 푅 → 퐴 are members of u�. Then, by axiom A3, ℎ : 퐴 → 퐵 is also a member of u�. Thus, given a morphism 푘 : 퐵 → 퐶 in u� such that 푘 ∘ ℎ : 퐴 → 퐶 is a member of u�, axiom A8 implies that 푘 : 퐵 → 퐶 is also a member of u�. ■

Gros pretopos Proposition. Let u� be a 휅-ary pretopos, let 햪 be the 휅-ary canonical with local coverage on u�, let u� be a set of morphisms in u� such that (u�, u�, 햪) is a homeomorphisms 휅-ary extensive ecumene that satisfies the descent axiom, and let u� be the class of local homeomorphisms in u�. (i) (u�, u�) is a 휅-ary gros pretopos. (ii) (u�, u�, 햪) satisfies the descent axiom for open embeddings.

Proof. (i). By proposition 2.2.13, (u�, u�, 햪) is an étale 휅-ary extensive ecumene, so by lemmas 2.2.14(c) and 2.3.1, (u�, u�) is a 휅-ary gros pretopos.

(ii). In view of lemma 2.2.11, it is clear that (u�, u�, 햪) satisfies the descent axiom for open embeddings if and only if (u�, u�, 햪) satisfies the descent axiom for open embeddings. ■

2.3.4 ※ For the remainder of this section, (u�, u�) is a 휅-ary gros pretopos.

2.3.5 Definition. The petit 휅-ary pretopos over an object 퐴 in u� is the full

subcategory u�∕퐴 ⊆ u�∕퐴 spanned by the objects (퐹 , 푝) where 푝 : 퐹 → 퐴 is a member of u�.

Properties of the Proposition. petit pretopos For every object in , the inclusion creates limits of (i) 퐴 u� u�∕퐴 ↪ u�∕퐴 over an object finite diagrams, 휅-ary coproducts, and exact quotients. In particular, is a -ary pretopos. (ii) u�∕퐴 휅 (iii) For every morphism ℎ : 퐴 → 퐵 in u�, the pullback functor ℎ∗ : preserves limits of finite diagrams, -ary coproducts, u�∕퐵 → u�∕퐴 휅 and exact forks.

Proof. Straightforward. (Use proposition 2.2.5.) ⧫

107 Charted objects

2.3.6 Definition.A unary basis for (u�, u�) is a full subcategory u� ⊆ u� with the following properties:

• u� is a unary site for u�.

• For every morphism ℎ : 퐴 → 퐵 in u�, if 퐵 is in u� and ℎ : 퐴 → 퐵 is a member of u�, then there is an effective epimorphism 푝 : 푋 ↠ 퐴 in u� such that 푋 is an object in u� and 푝 : 푋 ↠ 퐴 is a member of u�.

Example. Of course, u� is a unary basis for (u�, u�).

2.3.7 Definition.A 휅-ary basis for (u�, u�) is a full subcategory u� ⊆ u� with the following properties:

• u� is a 휅-ary site for u�.

• For every morphism ℎ : 퐴 → 퐵 in u�, if 퐵 is in u� and ℎ : 퐴 → 퐵 is a member of u�, then there is an effective epimorphism 푝 : 푋 ↠ 퐴 in u� such that 푋 is a coproduct of a 휅-small family of objects in u� and 푝 : 푋 ↠ 퐴 is a member of u�.

Example. Every unary basis for (u�, u�) is also a 휅-ary basis for (u�, u�) a fortiori.

Let be a full subcategory of and let be the full subcategory Recognition prin- Lemma. u�0 u� u� ciple for bases of of u� spanned by the objects that are coproducts (in u�) of 휅-small families gros pretoposes of objects in . The following are equivalent: u�0 is a -ary basis for . (i) u�0 휅 (u�, u�)

(ii) u� is a unary basis for (u�, u�).

Proof. Straightforward. (Use lemma 2.1.3 and proposition 2.2.5.) ⧫

2.3.8 ※ For the remainder of this section, u� is a unary basis for (u�, u�).

108 2.3. Extents

2.3.9 Proposition. Let ℬ = u� ∩ mor u�. Properties of (i) Given morphisms 푏 : 푌 → 퐵 and ℎ : 퐴 → 퐵 in u�, if ℎ : 퐴 → 퐵 unary bases of is a member of and is an object in , then there is a commutative gros pretoposes u� 푌 u� square in u� of the form below,

푋 푎 퐴 푓 ℎ 푌 퐵 푏 where is an object in and is an effective 푋 u� ⟨푎, 푓⟩ : 푋 → 퐴 ×퐵 푌 epimorphism in u� that is a member of u�.

(ii) In particular, every member of u� is 햪-locally of ℬ-type.

Proof. (i). By axiom A2, the projection 퐴 ×퐵 푌 → 푌 is a member of u�, so there indeed exist an object 푋 in u� and an effective epimorphism

⟨푎, 푓⟩ : 푋 ↠ 퐴 ×퐵 푌 in u� that is a member of u�.

(ii). Given a morphism ℎ : 퐴 → 퐵 in u� that is a member of u�, there is an effective epimorphism 푏 : 푌 ↠ 퐵 in u� where 푌 is an object in u�. The claim then reduces to (i). ■

2.3.10 Definition.A (u�, u�)-atlas of an object 퐴 in u� is an object (푋, 푎) in u�∕퐴 such that 푋 is an object in u� and 푎 : 푋 → 퐴 is an effective epimorphism in u�.

2.3.10(a) Lemma. Let ℎ : 퐴 → 퐵 be a morphism in u� and let (푌 , 푏) be a (u�, u�)- Atlases for étale atlas of 퐵. If ℎ : 퐴 → 퐵 is a member of u�, then there is a commutative morphisms diagram in u� of the form below,

푋 푎 퐴 푓 ℎ 푌 퐵 푏

where (푋, 푎) is a (u�, u�)-atlas of 퐴 and 푓 : 푋 → 푌 is a member of u�.

Proof. This is a corollary of proposition 2.3.9. ■

109 Charted objects

2.3.10(b) Lemma. Let ℎ : 퐴 → 퐵 be a morphism in u�, let (푋, 푎) be a (u�, u�)-atlas Atlases for of 퐴, and let (푌 , 푏) be a (u�, u�)-atlas of 퐵. morphisms (i) There exist a (u�, u�)-atlas (푈, 푥) of 푋 and a morphism 푦 : 푈 → 푌 in u� such that the following diagram in u� commutes:

푈 푥 푋 푎 퐴 푦 ℎ 푌 퐵 푏

(ii) Moreover, we may choose 푥 : 푈 → 푌 and 푦 : 푈 → 푌 so that the induced morphism is an effective epimorphism ⟨푎 ∘ 푥, 푦⟩ : 푈 → 퐴×퐵푌 in u� that is a member of u�.

Proof. (i). Consider a pullback square in u� of the form below:

푔 푋 ×퐵 푌 푌 푝 푏 푋 퐵 ℎ∘푎

Note that 푝 : 푋 ×퐵 푌 ↠ 푋 is an effective epimorphism in u� that is a member of u�. Since 푋 is an object in u�, we can find a (u�, u�)-atlas

(푈, ⟨푥, 푦⟩) of 푋 ×퐵 푌 such that 푥 = 푝 ∘ ⟨푥, 푦⟩ : 푈 → 푋 is a member of u�. Thus, (푈, 푥) is the required (u�, u�)-atlas of 푋.

(ii). The induced morphism 푎 ×퐵 id푌 : 푋 ×퐵 푌 → 퐴 ×퐵 푌 is an effective epimorphism in u� that is a member of u�, so the same is true of ⟨푎 ∘ 푥, 푦⟩ :

푈 → 퐴 ×퐵 푌 . ■

[1] 2.3.11 Definition.A (u�, u�)-extent in u� is an object 퐴 in u� that admits a (u�, u�)-atlas. We write Xt(u�, u�) for the full subcategory of u� spanned by the (u�, u�)- extents in u�.

[1] — a back-formation from ‘extensive category’.

110 2.3. Extents

Remark. We should justify the omission of u� from the notation Xt(u�, u�). By theorem 2.1.8, the Yoneda representation u� → Sh(u�, 햤) is fully faithful, with essential image given by lemma 2.1.9, so u� is determined up to equivalence by u� and 햤 alone. Moreover, in some cases, u� is determined by ℬ = u� ∩ mor u�. In the case where u� is the category of 휅-ary coproducts of objects in a

휅-ary basis u�0 for (u�, u�), theorem 2.1.14 and lemma 2.1.16 allow us to

substitute Sh(u�0, 햩0) for Sh(u�, 햤) in the above.

Properties of Proposition. the category (i) The class of morphisms in Xt(u�, u�) that are members of u� is a of extents quadrable class of morphisms in Xt(u�, u�).

(ii) Given an effective epimorphism ℎ : 퐴 ↠ 퐵 in u�, if ℎ : 퐴 → 퐵 is a member of u� and 퐴 is a (u�, u�)-extent, then 퐵 is also a (u�, u�)-extent.

(iii) If (u�, u�, 햪) satisfies the descent axiom for open embeddings, then Xt(u�, u�) is closed in u� under exact quotient of tractable equivalence relations.

(iv) If u� satisfies axiom A3, then Xt(u�, u�) is closed in u� under exact quotient of étale equivalence relations.

(v) Xt(u�, u�) is closed in u� under 휅-ary coproduct if and only if the coproduct of every 휅-small family of objects in u� is a (u�, u�)-charted extent.

(vi) If Xt(u�, u�) is closed in u� under 휅-ary coproduct, then Xt(u�, u�) is a 휅-ary extensive category.

(vii) Xt(u�, u�) is closed in u� under finitary product if and only if the product of every finite family of objects in u� is a (u�, u�)-charted extent.

(viii) Xt(u�, u�) is closed in u� under pullback if and only if the pullback of every cospan in u� is a (u�, u�)-charted extent.

Proof. (i). This is an immediate consequence of lemma 2.3.10(a) and the fact that u� is a quadrable class of morphisms in u�.

111 Charted objects

(ii). Straightforward. (Use axiom A2 and the fact that the class of effective epimorphisms in u� is closed under composition.)

(iii). By lemma 2.2.14(b), exact quotients of tractable equivalence relations in u� are local homeomorphisms in u�, and by proposition 2.2.12, local homeomorphisms in u� are members of u�, so the claim reduces to (ii).

(iv). By axiom A3, exact quotients of étale equivalence relations in u� are étale effective epimorphisms in u�, so this claim also reduces to (ii).

(v). Straightforward. (Use axioms A1 and A5.)

(vi). By lemma 2.3.1, every complemented monomorphism in u� is a member of u�, so by (i), the claim reduces to the fact that u� is a 휅-ary extensive category.

(vii). The ‘only if’ direction is clear, so suppose the product of every finite family of objects in u� is a (u�, u�)-charted extent. Since the class of effective epimorphisms in u� that are members of u� is closed under finitary product in u� and composition, it follows that the product of any finite family of objects in Xt(u�, u�) is also a (u�, u�)-charted extent.

(viii). Apply lemma 1.1.5 and lemma 2.3.10(b). ■

2.3.12 Definition. An object 퐴 in u� is u�-localic if u�∕퐴 is a localic 휅-ary pretopos.

2.3.12(a) Lemma. Let 퐴 be an object in u�. The following are equivalent: Recognition (i) is a -localic object in . principle for 퐴 u� u� localic objects For every object in , is a local homeomorph- (ii) (퐹 , 푝) u�∕퐴 푝 : 퐹 → 퐴 ism in u�.

Proof. Straightforward. (Recall lemma 1.5.16.) ⧫

112 2.3. Extents

2.3.12(b) Lemma. Let ℎ : 퐴 → 퐵 be a morphism in u�. If ℎ : 퐴 → 퐵 is a member of u� and 퐵 is u�-localic, then 퐴 is also u�-localic.

Proof. By lemma 2.3.12(a), it is enough to show that, for every object

(퐹 , 푝) in u�∕퐴, 푝 : 퐹 → 퐴 is a local homeomorphism in u�. Since 퐵 is u�-localic, both ℎ : 퐴 → 퐵 and ℎ ∘ 푝 : 퐹 → 퐵 are local homeomorphisms in u�. Thus, by proposition 2.2.12, 푝 : 퐹 → 퐴 is indeed a local homeomorphism in u�. ■

2.3.12(c) Lemma. Let ℎ : 퐴 ↠ 퐵 be an effective epimorphism in u�. If ℎ : 퐴 ↠ 퐵 is a local homeomorphism and 퐴 is u�-localic, then 퐵 is also u�-localic.

Proof. By lemma 2.3.12(a), it is enough to show that, for every object

(퐹 , 푞) in u�∕퐵, 푞 : 퐹 → 퐵 is a local homeomorphism in u�. Consider a pullback square in u� of the form below:

퐴 ×퐵 퐹 퐹 푝 푞 퐴 퐵 ℎ

Since 퐴 is u�-localic, the projection 푝 : 퐴 ×퐵 퐹 → 퐴 is a local homeo-

morphism in u�, so by proposition 2.2.12, ℎ∘푝 : 퐴×퐵 퐹 → 퐵 is also a local

homeomorphism in u�. On the other hand, the projection 퐴 ×퐵 퐹 ↠ 퐴 is both an effective epimorphism and a local homeomorphism in u�, so 푞 : 퐹 → 퐴 is indeed a local homeomorphism in u�. ■

2.3.13 Definition. A morphism ℎ : 퐴 → 퐵 in u� is laminar if there is a 휅-small

set Φ of objects in u�∕퐴 with the following properties: • 퐴 is the disjoint union of Φ.

• For every (푈, 푎) ∈ Φ, ℎ ∘ 푎 : 푈 → 퐵 is an open embedding in u�.

Properties Proposition. of laminar (i) Every isomorphism in u� is a laminar morphism in u�. morphisms (ii) For every object 퐴 in u�, the unique morphism 0 → 퐴 is a laminar morphism in u�.

113 Charted objects

(iii) Every laminar morphism in u� is a local homeomorphism in u�. (iv) For every local homeomorphism ℎ : 퐴 → 퐵 in u�, there is a laminar effective epimorphism 푝 : 퐴̃ → 퐴 in u� such that ℎ ∘ 푝 : 퐴̃ → 퐵 is a laminar morphism in u�. (v) The class of laminar morphisms in u� is a quadrable class of morphisms in u�. (vi) The class of laminar morphisms in u� is closed under composition. (vii) The class of laminar morphisms in u� is closed under 휅-ary coproduct in u�.

Proof. Straightforward. ⧫

2.3.14 ¶ It is clear that every object in u� is a (u�, u�)-extent. Of course, in general, not every (u�, u�)-extent is an object in u�, but we do have the following results.

2.3.14(a) Proposition. Assume the following hypotheses: Extents and exact • u� is a class of étale morphisms in u�. quotients of • is closed in under limit of finite diagrams. étale equival- u� u� ence relations The following are equivalent: For every étale equivalence relation on an object in (i) (푅, 푑0, 푑1) 푋 u�, if 푅 is an object in u�, then there is an exact fork in u� of the form below, 푑 0 푓 푅 푋 푌 푑1 where 푌 is a object in u�. (ii) The inclusion u� ↪ Xt(u�, u�) is (fully faithful and) essentially surjective on objects.

Proof. We may assume without loss of generality that u� is a replete (and full) subcategory of u�.

(i) ⇒ (ii). Let 퐴 be an object in u�, let (푋, 푎) be a (u�, u�)-atlas of 퐴, and

let (푅, 푑0, 푑1) be a kernel pair for 푎 : 푋 ↠ 퐴 in u�. We wish to show that 퐴 is an object in u�.

114 2.3. Extents

By proposition 2.3.11, 푅 is a (u�, u�)-charted extent, so there is a (u�, u�)- ̃ ̃ atlas (푅,̃ 푟) of 푅. Let (푄, 푑0, 푑1) be a kernel pair of 푟 : 푅̃ → 푅. It is not hard to see that 푄 is (the object part of) a limit of the following diagram in u�, 푑 ∘푟 푅̃ 1 푋

푑0∘푟

푑1∘푟

푅̃ 푋 푑0∘푟 and since both 푅̃ and 푋 are objects in u�, 푄 is also an object in u�. But ̃ ̃ (푄, 푑0, 푑1) is an étale equivalence relation on 푅̃ and 푟 : 푅̃ → 푅 is an effective epimorphism in u�, so 푅 is also an object in u�. The same argument (mutatis mutandis) shows that 퐴 is an object in u�.

(ii) ⇒ (i). See proposition 2.3.11. (Note that u� satisfies axiom A3 by hypothesis.) ■

2.3.14(b) Proposition. Assume the following hypotheses: Extents and exact • u� is the class of local homeomorphisms in u�. quotients of tract- • is closed in under -ary coproduct. able equival- u� u� 휅 ence relations • (u�, u�, 햪) satisfies the descent condition for open embeddings. The following are equivalent:

(i) u� has the following properties: • Given morphisms 푓 : 푋 → 푌 and 푦 : 푌 ′ → 푌 in u�, if 푓 : 푋 → 푌 is a member of u�, then there is a pullback square in u� of the form below, 푋′ 푋 푓 ′ 푌 푦 푌 where 푋′ is an object in u�. • For every tractable equivalence relation on an object (푅, 푑0, 푑1) 푋 in u�, if 푅 is an object in u�, then there is an exact fork in u� of the

115 Charted objects

form below, 푑 0 푓 푅 푋 푌 푑1 where 푌 is a object in u�. (ii) The inclusion u� ↪ Xt(u�, u�) is (fully faithful and) essentially surjective on objects.

Proof. We may assume without loss of generality that u� is a replete (and full) subcategory of u�.

(i) ⇒ (ii). First, observe that the hypotheses imply the following:

• For every open embedding ℎ : 퐴 → 퐵 in u�, if 퐵 is an object in u�, then 퐴 is also an object in u�.

Indeed, by lemma 2.3.10(a), there is a (u�, u�)-atlas (푋, 푎) of 퐴, and since ℎ : 퐴 → 퐵 is a monomorphism in u�, the kernel pair of ℎ ∘ 푎 : 푋 → 퐵 is isomorphic to the kernel pair of 푎 : 푋 → 퐴; but by lemma 2.2.14(b), the kernel pair of ℎ ∘ 푎 : 푋 → 퐵 is a tractable equivalence relation, so the hypotheses imply that 퐴 is indeed an object in u�. Consequently, we obtain the following results:

• If ℎ : 퐴 → 퐵 is a laminar morphism in u� and 퐵 is an object in u�, then 퐴 is also an object in u�.

• For every (u�, u�)-extent 퐴 in u�, there is (u�, u�)-atlas (푋, 푎) of 퐴 where 푎 : 푋 ↠ 퐴 is a laminar effective epimorphism in u�. (For the second claim, use proposition 2.3.13.) Now, consider a laminar effective epimorphism 푎 : 푋 ↠ 퐴 in u� where

푋 is an object in u�. Let (푅, 푑0, 푑1) be a kernel pair of 푎 : 푋 ↠ 퐴 in u�.

Then the projections 푑0, 푑1 : 푅 → 푋 are laminar morphisms in u�, so 푅

is an object in u�. But (푅, 푑0, 푑1) is a tractable equivalence relation on 푋, so it follows that 퐴 is also an object in u�. Hence, every (u�, u�)-extent is indeed an object in u�.

(ii) ⇒ (i). See proposition 2.3.11. ■

116 2.4. Functoriality

2.4 Functoriality

Synopsis. We investigate suﬃcient conditions for a functor between gros pretoposes to preserve atlases and extents.

Prerequisites. §§1.1, 1.2, 1.3, 1.4, 1.5, 2.1, 2.2, 2.3, a.2, a.3.

2.4.1 ※ Throughout this section:

• 휅 is a regular cardinal.

• u�0 is a 휅-ary pretopos and u�0 is a unary site for u�0.

• u�1 is a 휅-ary pretopos and u�1 is a unary site for u�1.

2.4.2 ¶ To begin, we need to understand when and how functors u�0 → u�1

extend to functors u�0 → u�1.

Let 햩0 (resp. 햩1) be the restriction of the 휅-ary canonical coverage on

u�0 (resp. u�1) to u�0 (resp. u�1) and let 햤0 (resp. 햤1) be the class of morph-

isms in u�0 (resp. u�1) that are effective epimorphisms in u�0 (resp. u�1).

Note that both (u�0, 햤0) and (u�1, 햤1) satisfy the Shulman condition, by proposition 2.1.7.

Assuming (resp. ) is closed in (resp. ) under -ary co- 2.4.2(a) Lemma. u�0 u�1 u�0 u�1 휅 product, if is a functor that preserves -ary coproducts and 퐹 : u�0 → u�1 휅 is a pre-admissible functor, then 퐹 : (u�0, 햤0) → (u�1, 햤1) 퐹 : (u�0, 햩0) → is a pre-admissible functor. (u�1, 햩1)

Proof. Let 퐴 be a 햩1-sheaf on u�1. Clearly, 퐴 is also an 햤1-sheaf on u�1, so ∗ 퐹 퐴 is an 햤0-sheaf on u�0. Since u�1 is a 휅-ary extensive category and 햩1

is a 휅-ary superextensive coverage on u�1, by lemmas 1.5.15 and 1.5.16, op 퐴 : u�1 → Set sends 휅-ary coproducts in u�1 to 휅-ary products in Set. ∗ op Hence, 퐹 퐴 : u�0 → Set sends 휅-ary coproducts in u�0 to 휅-ary prod-

ucts in Set. But u�0 is a 휅-ary extensive category and 햩0 is a 휅-ary super- ∗ extensive coverage on u�0, so it follows that 퐹 퐴 is a 햩0-sheaf on u�. ■

117 Charted objects

Assuming is an admissible functor and 2.4.2(b) Lemma. 퐹 : (u�0, 햤0) → (u�1, 햤1) is a pre-admissible functor: 퐹 : (u�0, 햩0) → (u�1, 햩1) There exist a functor and an isomorphism (i) 퐹̄ : u�0 → u�1 휂 : 퐹 ⇒ 퐹̄ of functors such that preserves -ary coprod- u�0 → u�1 퐹̄ : u�0 → u�1 휅 ucts and sends right-exact forks in to coequaliser diagrams in . u�0 u�1 (ii) Moreover, any such (퐹̄ , 휂) is a pointwise left Kan extension of 퐹 : along the inclusion . u�0 → u�1 u�0 ↪ u�0

Proof. By lemma 2.1.9, the inclusions u�0 ↪ u�0 and u�1 ↪ u�1 induce

functors Ex(u�0, 햤0) → u�0 and Ex(u�1, 햤1) → u�1 that are fully faithful and essentially surjective on objects, so by lemma 1.4.24, we have a pointwise left Kan extension (퐹̄ , 휂) of the required type. It remains to be

shown that 퐹̄ : u�0 → u�1 preserves 휅-ary coproducts.

Since 퐹 : (u�0, 햤0) → (u�1, 햤1) is an admissible functor, for every

object 퐴 in u�0 and every object 퐵 in u�1, we have the following natural bijection: ̄ Hom h ∗h u�1(퐹 퐴, 퐵) ≅ Sh(u�0,햤0)( 퐴, 퐹 퐵)

On the other hand, since 퐹 : (u�0, 햩0) → (u�1, 햩1) is a pre-admissible func- ∗ tor, 퐹 h퐵 is also a 햩0-sheaf on u�0. Thus, by theorems 1.5.15 and 2.1.13, op u�1(퐹̄ −, 퐵) : u�0 → Set sends 휅-ary coproducts in u�0 to 휅-ary products

in Set. It follows that 퐹̄ : u�0 → u�1 preserves 휅-ary coproducts. ■

Let be a functor, let be a class of fibrations in 2.4.2(c) Lemma. 퐹 : u�0 → u�1 u�0 , and assume the following hypotheses: u�0 • preserves pullbacks of members of . 퐹 : u�0 → u�1 u�0 • sends members of to members of . 퐹 : u�0 → u�1 햤0 ∩ u�0 햤1 • For every morphism in , if is a member of 푓 : 푋 → 푌 u�0 푓 : 푋 → 푌 , then there is a morphism in such that 햤0 푝 : 푋̃ → 푋 u�0 푓 ∘푝 : 푋̃ → 푌 is a member of and also a member of . u�0 햤0 Then is a pre-admissible functor. 퐹 : (u�0, 햤0) → (u�1, 햤1)

Proof. Let ′ . By remark 1.4.5, ′ -covering morphisms are 햤0 = 햤0 ∩ u�0 햤0 the same as -covering morphisms, so by proposition a.3.7, ′ -sheaves 햤0 햤0

118 2.4. Functoriality

are the same as -sheaves. On the other hand, every member of ′ has a 햤0 햤0 kernel pair in , so lemma 1.4.18 implies that ′ u�0 퐹 : (u�0, 햤0) → (u�1, 햤1) is a pre-admissible functor. The claim follows. ■

Let be a functor. Assume the following hypotheses: 2.4.2(d) Lemma. 퐹 : u�0 → u�1 • is a pre-admissible functor. 퐹 : (u�0, 햤0) → (u�1, 햤1) • has -weak pullback squares. u�0 햤0 • sends -weak pullback squares in to -weak pull- 퐹 : u�0 → u�1 햤0 u�0 햤1 back squares in . u�1 Then is an admissible functor. 퐹 : (u�0, 햤0) → (u�1, 햤1)

Proof. By corollary a.3.13, 퐹 : u�0 → u�1 sends 햤0-covering morphisms

in u�0 to 햤1-covering morphisms in u�1, so the hypotheses imply that 퐹 :

u�0 → u�1 sends right-햤0-exact forks in u�0 to right-햤1-exact forks in u�1.

Moreover, by lemma 1.4.21, 퐹 : u�0 → u�1 sends 햤0-local complexes in u�0

to 햤1-local complexes in u�1, so we may apply lemma 1.4.25 to complete the proof. ■

Let be a functor. Assuming , the following 2.4.2(e) Lemma. 퐹 : u�0 → u�1 휅 > ℵ0 are equivalent: is an admissible functor. (i) 퐹 : (u�0, 햤0) → (u�, 햤1) is a pre-admissible functor. (ii) 퐹 : (u�0, 햤0) → (u�, 햤1)

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). Let 햪1 be the 휅-ary canonical coverage on u�1 and let 햩1 be

the restriction of 햪1 to u�1. (Note that u�1 is a 휅-ary site for u�1, so 햩1 is a

coverage on u�1, by proposition 2.1.13.) Since 퐹 : (u�0, 햤0) → (u�, 햤1) is

a pre-admissible functor, 퐹 : (u�0, 햤0) → (u�, 햩1) is also a pre-admissible functor a fortiori. Thus, by proposition a.3.13, the restriction functor ∗ 퐹 : Sh(u�1, 햩1) → Sh(u�0, 햤0) has a left adjoint, say 퐹! : Sh(u�0, 햤0) →

Sh(u�1, 햩1). On the other hand, by proposition 1.5.19, the Yoneda repres-

entation u�1 → Sh(u�1, 햩1) preserves coequalisers, so 퐹! sends 햤0-locally

1-presentable 햤0-sheaves on u�0 to 햩1-locally 휅-presentable 햩1-sheaves on

119 Charted objects

u�1. Moreover, by lemma 2.1.9, 햩1-locally 휅-presentable 햩1-sheaves on u�1

are the same as 햤1-locally 1-presentable 햤1-sheaves on u�1, , so it follows

that 퐹 : (u�0, 햤0) → (u�, 햤1) is an admissible functor. ■

2.4.3 ※ For the remainder of this section:

• (u�0, u�0) is a 휅-ary gros pretopos and u�0 is a unary basis for (u�0, u�0).

• (u�1, u�1) is a 휅-ary gros pretopos and u�1 is a unary basis for (u�1, u�1).

• Given a pullback square in u�0 of the form below,

푋′ 푋 푓 푌 ′ 푌

′ if 푓 : 푋 → 푌 is a member of u�0 and 푋, 푌 , and 푌 are all objects in ′ u�0, then 푋 is also an object in u�0.

• 퐹 : u�0 → u�1 is a functor that preserves 휅-ary coproducts and sends

right-exact forks in u�0 to coequaliser diagrams in u�1.

• 퐹 : u�0 → u�1 sends objects in u�0 to objects in u�1.

• Given a morphism 푓 : 푋 → 푌 in u�0, if 푓 : 푋 → 푌 is a member of

u�0, then 퐹 푓 : 퐹 푋 → 퐹 푌 is a member of u�1.

• Given a pullback square in u�0 of the form below,

푋′ 푋 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of u�0, then 퐹 : u�0 → u�1 preserves this pullback square.

2.4.4 Remark. Under the above assumptions, 퐹 : u�0 → u�1 preserves effective

epimorphisms. However, 퐹 : u�0 → u�1 may fail to preserve kernel pairs.

120 2.4. Functoriality

2.4.5 ¶ If we assume that 퐹 : u�0 → u�1 preserves limits of finite diagrams (or at least limits of finite connected diagrams), things are much simpler;

for instance, it immediately follows that 퐹 : u�0 → u�1 preserves exact

forks. However, in some examples, even 퐹 : u�0 → u�1 fails to preserve pullbacks, so this is not a reasonable assumption to make. We instead

establish the desired preservation properties of 퐹 : u�0 → u�1 in several small steps, starting from the basic assumptions above. We begin with the following results, which will be required later.

Consider a weak pullback square in of the form below: 2.4.5(a) Lemma. u�0 Preservation of special weak 푃 퐴1 pullback squares ℎ1

퐴0 퐵 ℎ0 Assume the following hypotheses: • is an object in . 퐵 u�0 • There is a -atlas of in such that (u�0, u�0) (푋1, 푎1) 퐴1 u�0 ℎ1 ∘푎1 : 푋1 → is a member of . 퐵 u�0 Then preserves the above weak pullback square. 퐹 : u�0 → u�1

Proof. Since u�0 is a unary basis for (u�0, u�0), there exist an object 푋0

in u�0 and an effective epimorphism 푎0 : 푋0 ↠ 퐴0 in u�0. Consider the

following commutative diagram in u�0,

푈 푃 ×퐴1 푋1 푋1

푎1

푋0 ×퐴0 푃 푃 퐴1

ℎ1

푋0 푎 퐴0 퐵 0 ℎ0

where the top left, top right, and bottom left squares are pullback squares

in u�0. By the weak pullback pasting lemma (lemma a.2.19), the outer

square is a weak pullback square in u�0. Since ℎ1 ∘ 푎1 : 푋1 → 퐵 is

a member of u�0 and 푋0, 푋1, and 퐵 are all objects in u�0, 퐹 : u�0 →

121 Charted objects

u�1 sends the outer square to a weak pullback square in u�1. Hence, by

lemma 1.4.19(a), 퐹 : u�0 → u�1 also sends the inner square to a weak

pullback square in u�1. ■

2.4.5(b) Lemma. Preservation If is an open embedding in and is an object in , (i) ℎ : 퐴 → 퐵 u�0 퐵 u�0 of special open then is an open embedding in . 퐹 ℎ : 퐹 퐴 → 퐹 퐵 u�1 embeddings Given a pullback square in of the form below, (ii) u�0

퐴′ 퐴 ℎ′ ℎ 퐵′ 퐵 if is an open embedding in and both ′ and are ℎ : 퐴 → 퐵 u�0 퐵 퐵 objects in , then preserves this pullback square. u�0 퐹 : u�0 → u�1

Proof. (i). By lemma 2.4.5(a), the following is a weak pullback square

in u�1: id 퐹 퐴 퐹 퐴 id 퐹 ℎ 퐹 퐴 퐹 퐵 퐹 ℎ

In other words, the relative diagonal Δ퐹 ℎ : 퐹 퐴 → 퐹 퐴 ×퐹 퐵 퐹 퐴 is an

effective epimorphism in u�1. But Δ퐹 ℎ : 퐹 퐴 → 퐹 퐴 ×퐹 퐵 퐹 퐴 is also a

(split) monomorphism in u�1, so this implies it is an isomorphism. Hence,

퐹 ℎ : 퐹 퐴 → 퐹 퐵 is a monomorphism in u�1. It remains to be shown that 퐹 ℎ : 퐹 퐴 → 퐹 퐵 is an étale morphism in

u�1, and by lemma 1.2.19(a), it is enough to verify that 퐹 ℎ : 퐹 퐴 → 퐹 퐵

is of u�1-type semilocally on the domain. Since u�0 is a unary basis for

(u�0, u�0), there is a (u�0, u�0)-atlas of 퐴 in u�0, say (푋, 푎). Then 퐹 푎 :

퐹 푋 ↠ 퐹 퐴 is an effective epimorphism in u�1 and 퐹ℎ∘퐹푎:퐹푋→퐹퐵

is an étale morphism in u�1, so we are done.

(ii). We know the following is a weak pullback square in u�1:

퐹 퐴′ 퐹 퐴 퐹 ℎ′ 퐹 ℎ 퐹 퐵′ 퐹 퐵

122 2.4. Functoriality

′ ′ ′ Since 퐹 ℎ : 퐹 퐴 → 퐹 퐵 is a monomorphism in u�1, the induced morph- ′ ′ ism 퐹 퐴 → 퐹 퐵 ×퐹 퐵 퐹 퐴 is also a monomorphism in u�1. But any monomorphism that is an effective epimorphism is an isomorphism, so

the above is indeed a pullback square in u�1. ■

Let be an object in and let be an étale equival- 2.4.5(c) Lemma. 푋 u�0 (푅, 푑0, 푑1) Preservation ence relation on in . 푋 u�0 of special (i) If is a monomorphism in , then étale equival- ⟨퐹 푑1, 퐹 푑0⟩ : 퐹 푅 → 퐹 푋 × 퐹 푋 u�1 is an equivalence relation on in . ence relations (퐹 푅, 퐹 푑0, 퐹 푑1) 퐹 푋 u�1 Furthermore, if there is an étale morphism in such that 푓 : 푋 → 푌 u�0 , then: 푓 ∘ 푑0 = 푓 ∘ 푑1 is an étale equivalence relation on in . (ii) (퐹 푅, 퐹 푑0, 퐹 푑1) 퐹 푋 u�1 If is a kernel pair of an (étale) effective epimorphism (iii) (푅, 푑0, 푑1) 푎 : in , then is an étale effective epimorphism 푋 ↠ 퐴 u�0 퐹 푎 : 퐹 푋 ↠ 퐹 퐴 in and is a kernel pair of in . u�1 (퐹 푅, 퐹 푑0, 퐹 푑1) 퐹 푎 : 퐹 푋 ↠ 퐹 퐴 u�1

Proof. (i). (퐹 푅, 퐹 푑0, 퐹 푑1) is clearly reflexive and symmetric as a relation on 퐹 푋, so it suﬃces to verify transitivity. Consider a pullback square

in u�0 of the form below:

푑 푅2 0 푅

푑2 푑1 푅 푋 푑0

By lemma 2.4.5(a), its image is a weak pullback square in u�1. On the

other hand, since (푅, 푑0, 푑1) is an equivalence relation on 푋 in u�1, there 2 is a (necessarily unique) morphism 푑1 : 푅 → 푅 in u�1 such that 푑0 ∘ 푑1 =

푑0 ∘ 푑0 and 푑1 ∘ 푑1 = 푑1 ∘ 푑2. It follows that (퐹 푅, 퐹 푑0, 퐹 푑1) is transitive,

so we indeed have an equivalence relation on 퐹 푋 in u�1.

(ii). Since ⟨푑1, 푑0⟩ : 푅 → 푋 ×푌 푋 is an open embedding in u�0 (by

lemma 1.1.10) and 푋 ×푌 푋 is an object in u�0, lemma 2.4.5(b) implies

that ⟨퐹 푑1, 퐹 푑0⟩ : 퐹 푅 → 퐹 푋 ×퐹 푌 퐹 푋 is an open embedding in u�1. In

particular, the projections 퐹 푑0, 퐹 푑1 : 퐹 푅 → 퐹 푋 are étale morphisms in

123 Charted objects

u�1 and ⟨퐹 푑1, 퐹 푑0⟩ : 퐹 푅 → 퐹 푋 × 퐹 푋 is a monomorphism in u�1. Thus,

by (i), (퐹 푅, 퐹 푑1, 퐹 푑0) is an étale equivalence relation on 퐹 푋 in u�1.

(iii). The following is a coequaliser diagram in u�1,

퐹 푑 0 퐹 푎 퐹 푅 퐹 푋 퐹 퐴 퐹 푑1

but (퐹 푅, 퐹 푑0, 퐹 푑1) is an étale equivalence relation on 퐹 푋 in u�1, so it is indeed the kernel pair of 퐹 푎 : 퐹 푋 ↠ 퐹 퐴. Furthermore, there is a unique

morphism 푦 : 퐴 → 푌 in u�0 such that 푦 ∘ 푎 = 푓, and 퐹 푓 : 퐹 푋 → 퐹 푌 is

an étale morphism in u�1, so both 퐹 푎 : 퐹 푋 ↠ 퐹 퐴 and 퐹 푦 : 퐹 퐴 → 퐹 푌

are étale morphisms in u�1. ■

If is a member of and is an object in , then 2.4.6 Lemma. ℎ : 퐴 → 퐵 u�0 퐵 u�0 Preservation is a member of . 퐹 ℎ : 퐹 퐴 → 퐹 퐵 u�1 of special étale morphisms Proof. Since u�0 is a unary basis for (u�0, u�0), there is a (u�0, u�0)-atlas

of 퐴, say (푋, 푎). Let (푅, 푑0, 푑1) be a kernel pair of 푎 : 푋 ↠ in u�0. By

lemma 2.4.5(c), (퐹 푅, 퐹 푑0, 퐹 푑1) is an étale equivalence relation and also

a kernel pair of 퐹 푎 : 퐹 푋 ↠ 퐹 퐴 in u�1. We know that 퐹 ℎ ∘ 퐹 푎 : 퐹 푋 →

퐹 퐵 is an étale morphism in u�1, so it follows that both 퐹 푎 : 퐹 푋 ↠ 퐹 퐴

and 퐹 ℎ : 퐹 퐴 → 퐹 퐵 are also étale morphisms in u�1. ■

For every object in , the functor Properties of the Proposition. 푍 u�0 (u�0)∕푍 → (u�1)∕퐹 푍 induced functor induced by preserves: 퐹 : u�0 → u�1 between special -ary coproducts, petit pretoposes (i) 휅 (ii) effective epimorphisms,

(iii) finitary products,

(iv) pullbacks, and

(v) exact forks.

Proof. (i) and (ii). Since 퐹 : u�0 → u�1 preserves 휅-ary coproducts

and effective epimorphisms, the same is true of the functor (u�0)∕푍 →

(u�1)∕퐹 푍, by proposition 2.3.5.

124 2.4. Functoriality

(iii). It is clear that the induced functor (u�0)∕푍 → (u�1)∕퐹 푍 preserves terminal objects, and it is enough to verify that the functor preserves binary products.

Let ℎ0 : 퐴0 → 푍 and ℎ1 : 퐴1 → 푍 be étale morphisms in u�0,

let (푋0, 푎0) be a (u�0, u�0)-atlas of 퐴0, let (푋1, 푎1) be a (u�0, u�0)-atlas

of 퐴1, let (푅0, 푑0,0, 푑0,1) be a kernel pair of 푎0 : 푋0 ↠ 퐴0, and let

(푅1, 푑1,0, 푑1,1) be a kernel pair of 푎1 : 푋1 ↠ 퐴1. It is straightforward to

verify that (푅0 ×푍 푅1, 푑0,0 ×푍 푑1,0, 푑0,1 ×푍 푑1,1) is a kernel pair of the

effective epimorphism 푎0 ×푍 푎1 : 푋0 ×푍 푋1 ↠ 퐴0 ×푍 퐴1. On the other

hand, by lemma 2.4.5(b), we see that 퐹 : u�0 → u�1 preserves the pullback

squares in the following commutative diagram in u�0,

푅0 ×푍 푅1 (푋0 ×푍 푋0) ×푍 푅1 푅1

푅0 ×푍 (푋1 ×푍 푋1) (푋0 ×푍 푋0) ×푍 (푋1 ×푍 푋1) 푋1 ×푍 푋1

푅0 푋0 ×푍 푋0 푍

so by lemma 2.4.5(c), we have the following coequaliser diagram in u�1,

퐹 푑0,0×퐹 푍퐹 푑1,0 퐹 (푎0×푍푎1) 퐹 푅0 ×퐹 푍 퐹 푅1 퐹 푋0 ×퐹 푍 퐹 푋1 퐹 (퐴0 ×푍 퐴1) 퐹 푑0,1×퐹 푍퐹 푑1,1

where (by abuse of notation) we have elided the canonical isomorphism

퐹 푋0 ×퐹 푍 퐹 푋1 ≅ 퐹 (푋0 ×푍 푋1). In particular, 퐹 : u�0 → u�1 preserves

퐴0 ×푍 퐴1.

(iv). First, we will show that (u�0)∕푍 → (u�1)∕퐹 푍 sends pullback squares

in (u�0)∕푍 to weak pullback squares in (u�1)∕퐹 푍.

Let ℎ0 : 퐴0 → 퐵 be a morphism in u�0, let ℎ1 : 퐴1 → 퐵 and 푧 :

퐵 → 푍 be étale morphisms in u�0, let (푌 , 푏) be a (u�0, u�0)-atlas of 퐵

in u�0, let (푋1, ⟨푎1, 푓1⟩) be a (u�0, u�0)-atlas of 퐴1 ×퐵 푌 in u�0, and let

⟨푎0, 푓0⟩ : 푋0 ↠ 퐴0 ×퐵 푌 be an effective epimorphism in u�0 where 푋0 is

125 Charted objects

an object in u�0. We have the following pullback square in u�0,

푓0×퐵푓1 푋0 ×퐵 푋1 푌 ×퐵 푌 id id id id 푋0×푧 푋1 푌 ×푧 푌

푋0 ×푍 푋1 푌 ×푍 푌 푓0×푍푓1

and since 푋0×푍 푋1 and 푌 ×푍 푌 are both objects in u�0 and id푌 ×푧id푌 : 푌 ×퐵

푌 → 푌 ×푍 푌 is an open embedding, 퐹 : u�0 → u�1 preserves this pullback square, by lemma 2.4.5(b). On the other hand, by lemma 2.4.5(c), 퐹 :

u�0 → u�1 preserves kernel pairs of 푏 : 푌 ↠ 퐵 and 푧 ∘ 푏 : 푌 → 푍. Thus,

in the following commutative diagram in u�1,

퐹 (푋0 ×퐵 푋1) 퐹 (푌 ×퐵 푌 ) 퐹 퐵 퐹 푧 Δ퐹 푧

퐹 (푋0 ×푍 푋1) 퐹 (푌 ×푍 푌 ) 퐹 퐵 ×퐹 푍 퐹 퐵 퐹 푍

Δ퐹 푍

퐹 푋0 × 퐹 푋1 퐹 푌 × 퐹 푌 퐹 퐵 × 퐹 퐵 퐹 푍 × 퐹 푍 퐹 푓0×퐹 푓1 퐹 푏×퐹 푏 퐹 푧×퐹 푧

each square is a pullback square in u�1. Hence, in the commutative dia-

gram in u�1 shown below,

퐹 (푋0 ×퐵 푋1) 퐹 푋1

퐹 푎1

퐹 (퐴0 ×퐵 퐴1) 퐹 퐴1

퐹 ℎ1

퐹 푋0 퐹 퐴0 퐹 퐵 퐹 푎0 퐹 ℎ

the outer square is a pullback square in u�1, so the inner square is indeed

a weak pullback square in u�1, by lemma 1.4.19(a).

It follows from the above that the functor (u�0)∕푍 → (u�1)∕퐹 푍 preserves monomorphisms and pullbacks of monomorphisms. Thus, we have

the following commutative square in u�1,

퐹 (퐴0 ×퐵 퐴1) 퐹 퐴0 ×퐹 퐵 퐹 퐴1

퐹 (퐴0 ×푍 퐴1) ≅ 퐹 퐴0 ×퐹 푍 퐹 퐴1

126 2.4. Functoriality

where the vertical arrows are monomorphisms and the horizontal arrows are the canonical comparison morphisms. It follows that the comparison

퐹 (퐴0 ×퐵 퐴1) → 퐹 퐴0 ×퐹 퐵 퐹 퐴1 is both an effective epimorphism and a monomorphism, so it is indeed an isomorphism.

(v). Combine (ii) and (iv). ■

Remark. The arguments used above can also be used to establish the universal property of the exact completion of a category with limits of finite diagrams and a subcanonical unary topology.

Given a pullback square in of the form below, 2.4.7 Proposition. u�0 Preservation of ℎ base changes 퐴 퐵 of special étale 푝 푞 morphisms 푋 푌 푓 if is a morphism in and is a member of , 푓 : 푋 → 푌 u�0 푞 : 퐵 → 푌 u�0 then preserves this pullback square. 퐹 : u�0 → u�1

Proof. There is a exact fork in u�0 of the form below,

푑 0 푏 푅 퐵̃ 퐵 푑1

where 푑0, 푑1 : 푅 → 퐵̃ and 푏 : 퐵̃ → 퐵 are all étale morphisms in u�0 and

퐵̃ is an object in u�0. Consider the following commutative diagram in u�0,

푋 ×푌 푅 푅

푑0 푑1

푋 ×푌 퐵̃ 퐵̃ 푏 ℎ 퐴 퐵 푝 푞 푋 푌 푓

where the squares are pullback squares in u�0. Note that ⟨푑1, 푑0⟩ : 푅 →

퐵̃ ×푌 퐵̃ is an open embedding (by lemma 1.1.10) and that 퐵̃ ×푌 퐵̃ is

127 Charted objects

an object in u�0. Since exact forks are preserved by pullback in u�0, by lemma 2.4.5(b) and proposition 2.4.6, both squares in the following dia-

gram in u�1 are pullback squares in u�1,

퐹 (푋 ×푌 푅) 퐹 푅

퐹 ⟨푑1,푑0⟩

퐹 (푋 ×푌 (퐵̃ ×푌 퐵̃)) 퐹 (퐵̃ ×푌 퐵̃)

퐹 푋 퐹 푌 퐹 푓

and we have a commutative diagram in u�1 of the form below,

퐹 (푋 ×푌 푅) 퐹 푅

퐹 (푋 ×푌 퐵̃) 퐹 퐵̃ 퐹 푏 퐹 ℎ 퐹 퐴 퐹 퐵 퐹 푝 퐹 푞 퐹 푋 퐹 푌 퐹 푓

where the columns are exact forks in u�1 and every rectangle with base

퐹 푓 : 퐹 푋 → 퐹 푌 is a pullback diagram in u�1, except possibly the bottom square. But lemma 1.4.19(b) is applicable, so we have the following

pullback square in u�1:

퐹 (푋 ×푌 퐵̃) 퐹 퐵̃ 퐹 푏 퐹 퐴 퐹 퐵 퐹 ℎ

We can then apply lemma 1.4.19(c) to complete the proof. ■

2.4.8 ¶ Under some additional assumptions, we can show that 퐹 : u�0 → u�1

sends (u�0, u�0)-extents in u�0 to (u�1, u�1)-extents in u�1.

128 2.4. Functoriality

Assuming sends étale equivalence relations 2.4.8(a) Proposition. 퐹 : u�0 → u�1 Preservation on objects in to equivalence relations in and is a class of étale u�0 u�1 u�1 of special morphisms in : u�1 étale effective If is an étale effective epimorphism in and is an epimorphisms (i) ℎ : 퐴 ↠ 퐵 u�0 퐴 object in , then is a étale effective epimorphism in u�0 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 , and preserves kernel pairs of . u�1 퐹 : u�0 → u�1 ℎ : 퐴 ↠ 퐵 In particular, sends -extents in to - (ii) 퐹 : u�0 → u�1 (u�0, u�0) u�0 (u�1, u�1) extents in . u�1

Proof. (i). Let (푅, 푑0, 푑1) be a kernel pair of ℎ : 퐴 → 퐵 in u�0. Then

(푅, 푑0, 푑1) is an étale equivalence relation on 퐴, so lemma 2.4.6 and the

hypothesis implies that (퐹 푅, 퐹 푑0, 퐹 푑1) is an étale equivalence relation

on 퐹 퐴. On the other hand, the following is a coequaliser diagram in u�1,

퐹 푑 0 퐹 ℎ 퐹 푅 퐹 퐴 퐹 퐵 퐹 푑1

so (퐹 푅, 퐹 푑0, 퐹 푑1) is indeed a kernel pair of 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 in u�1. Hence, by lemma 2.2.8(b), 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 is indeed an étale morphism

in u�1.

(ii). This is an immediate consequence of (i). ■

Assuming is the class of local homeomorphisms in 2.4.8(b) Proposition. u�0 u�0 Preservation of and satisfies the descent axiom for open embeddings: (u�1, u�1) special effective- (i) If is a laminar effective epimorphism in and there epimorphic local ℎ : 퐴 ↠ 퐵 u�0 is a local homeomorphism in where is an object in , homeomorphisms 퐴 → 푋 u�0 푋 u�0 then is also a laminar effective epimorphism in , 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 u�1 and preserves kernel pairs of . 퐹 : u�0 → u�1 ℎ : 퐴 ↠ 퐵 (ii) If ℎ : 퐴 ↠ 퐵 is both an effective epimorphism and a local homeomorphism in and is an object in , then is an u�0 퐴 u�0 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 effective epimorphism and a local homeomorphism in . u�1 In particular, sends -extents in to - (iii) 퐹 : u�0 → u�1 (u�0, u�0) u�0 (u�1, u�1) extents in . u�1

129 Charted objects

Proof. (i). First, note that the existence of an object 푋 in u�0 and a local

homeomorphism 퐴 → 푋 in u�0 ensures that (u�0)∕퐴 → (u�1)∕퐹 퐴 preserves limits of finite diagrams, by proposition 2.4.6.

Let (푅, 푑0, 푑1) be a kernel pair of ℎ : 퐴 ↠ 퐵 in u�0. We will now

show that (퐹 푅, 퐹 푑0, 퐹 푑1) is a tractable equivalence relation in u�1. By hypothesis, 퐴 is the disjoint union of a 휅-small set Φ of subobjects of 퐴 such that, for every (푈, 푎) ∈ Φ, ℎ ∘ 푎 : 푈 → 퐵 is an open embedding in

u�0. Thus,

푅 ≅ ∐ 푈0 ×퐵 푈1 (푈0,푎0)∈Φ (푈1,푎1)∈Φ

as objects in (u�0)∕퐴×퐴, and since the projections 푈0 ×퐵 푈1 → 푈0 and

푈0 ×퐵 푈1 → 푈1 are open embeddings in u�0, the projections 푑0, 푑1 : 푅 →

퐴 are laminar morphisms in u�0. Thus,

퐹 푅 ≅ ∐ 퐹 (푈0 ×퐵 푈1) (푈0,푎0)∈Φ (푈1,푎1)∈Φ

as objects in (u�1)∕퐹 퐴×퐹 퐴 and, by proposition 2.4.6, 퐹 sends the projec-

tions 푈0 ×퐵 푈1 → 푈0 and 푈0 ×퐵 푈1 → 푈1 to open embeddings in u�1, so

the induced morphism 퐹 (푈0 ×퐵 푈1) → 퐹 푈0 × 퐹 푈1 is a monomorphism

in u�1. Since the class of monomorphisms in u�1 is closed under 휅-ary co-

product, it follows that ⟨푑1, 푑0⟩ : 퐹 푅 → 퐹 퐴×퐹 퐴 is a monomorphism in

u�1. Moreover, 퐹 푑0, 퐹 푑1 : 퐹 푅 → 퐹 퐴 are laminar morphisms in u�1, so

by (proposition 2.3.13 and) lemma 2.4.5(c), (퐹 푅, 퐹 푑0, 퐹 푑1) is an étale

equivalence relation on 퐹 퐴 in u�1. In addition, for every (푈, 푎) ∈ Φ, we

have the following commutative diagram in u�0,

⟨⟨푎,푎⟩,id푈 ⟩ 푈 푅 ×퐴 푈 푈

⟨id푈 ,⟨푎,푎⟩⟩ 푎

푈 ×퐴 푅 푅 퐴 푑0 푑1

푈 푎 퐴

where every square is a pullback square in u�0. Using the pullback pasting lemma, it is not hard to see that these pullback squares are preserved by

130 2.4. Functoriality

퐹 : u�0 → u�1, and it follows that (퐹 푅, 퐹 푑0, 퐹 푑1) is a tractable equi-

valence relation on 퐹 퐴 in u�1. The above also implies that, for every

(푈, 푎) ∈ Φ, 퐹ℎ∘퐹푎:퐹푈→퐹퐵 is a monomorphism in u�1.

On the other hand, the following is a coequaliser diagram in u�1, 퐹 푑 0 퐹 ℎ 퐹 푅 퐹 퐴 퐹 퐵 퐹 푑1

so (퐹 푅, 퐹 푑0, 퐹 푑1) is indeed a kernel pair of 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 in u�1. Hence, by lemma 2.2.14(c), 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 is a local homeomorphism

in u�1. In particular, for every (푈, 푎) ∈ Φ, 퐹ℎ∘퐹푎:퐹푈 →퐹퐵 is an

open embedding in u�1, so 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 is indeed a laminar effective

epimorphism in u�1.

(ii). Let ℎ : 퐴 → 퐵 be a local homeomorphism in u�0 where 퐴 is

an object in u�0. By proposition 2.3.13, there is a laminar effective epi-

morphism 푝 : 퐴̃ ↠ 퐴 in u�0 such that ℎ ∘ 푝 : 퐴̃ → 퐵 is a laminar

morphism in u�0. Since the induced functor (u�0)∕퐴 → (u�1)∕퐹 퐴 preserves limits of finite diagrams, 휅-ary coproducts, and effective epimorph-

isms, 퐹 푝 : 퐹 퐴̃ → 퐹 퐴 is a laminar effective epimorphism in u�1. More-

over, by (i), if ℎ : 퐴 → 퐵 is an effective epimorphism in u�1, then 퐹 ℎ ∘ 퐹 푝 : 퐹 퐴̃ ↠ 퐹 퐵 is also a laminar effective epimorphism in u�. Thus, by proposition 2.2.12, 퐹 ℎ : 퐹 퐴 → 퐹 퐵 is indeed (an effective

epimorphism and) a local homeomorphism in u�1. (iii). This is an immediate consequence of (ii). ■

2.4.9 ※ For the remainder of this section, we make the following additional assumptions:

• Given an effective epimorphism ℎ : 퐴 ↠ 퐵 in u�0, if ℎ : 퐴 ↠ 퐵 is

a member of u�0 and 퐴 is an object in u�0, then 퐹 ℎ : 퐹 퐴 ↠ 퐹 퐵 is a

member of u�1.

• For every (u�0, u�0)-extent 퐴 in u�0, there exist an effective epimorph-

ism 푝 : 퐴̃ → 퐴 in u�0 and a morphism 푥 : 퐴̃ → 푋 in u�0 such that 푋 is

an object in u�0, both 푝 : 퐴̃ → 퐴 and 푥 : 퐴̃ → 푋 are members of u�0,

and 퐹 : u�0 → u�1 preserves kernel pairs of 푝 : 퐴̃ → 퐴.

131 Charted objects

2.4.10 ¶ Though the above assumptions seem weak, we will see that it has some important consequences, which eventually lead to a more elegant formu- lation. For instance, the first assumption can be replaced with the following:

If is a member of and is a -extent in Lemma. ℎ : 퐴 → 퐵 u�0 퐵 (u�0, u�0) , then is a member of . u�0 퐹 ℎ : 퐹 퐴 → 퐹 퐵 u�1

Proof. By lemma 2.3.10(a), there is a commutative square in u�0 of the form below, 푎 푋 퐴 푓 ℎ 푌 퐵 푏

where 푓 : 푋 → 푌 is an étale morphism in u�0 and both 푎 : 푋 ↠ 퐴 and

푏 : 푌 ↠ 퐵 are étale effective epimorphisms in u�0. Then, by assumption,

퐹 푓 : 퐹 푋 → 퐹 푌 is an étale morphism in u�1 and both 퐹 푎 : 퐹 푋 ↠ 퐹 퐴

and 퐹 푏 : 퐹 푌 ↠ 퐹 퐵 are étale effective epimorphisms in u�1, so 퐹 ℎ :

퐹 퐴 → 퐹 퐵 is indeed an étale morphism in u�1. ■

Consider a pullback square in of the form below: 2.4.11 Lemma. u�0 Preservation 푝 of pullbacks 퐴̃ 퐴 of special ℎ̃ ℎ étale effective ̃ 퐵 푞 퐵 epimorphisms Assuming the following hypotheses:

• is an étale effective epimorphism in . 푝 : 퐴̃ ↠ 퐴 u�0 • is an étale morphism in . 푞 : 퐵̃ → 퐵 u�0 • preserves kernel pairs of and . 퐹 : u�0 → u�1 푝 : 퐴̃ ↠ 퐴 푞 : 퐵̃ → 퐵 • preserves pullbacks of étale morphisms along 퐹 : u�0 → u�1 ℎ̃ : 퐴̃ → 퐵̃. Then also preserves the above pullback square. 퐹 : u�0 → u�1

Proof. Apply lemma 1.4.19(b). ■

132 2.4. Functoriality

For every -extent in , the functor Properties of Proposition. (u�0, u�0) 퐸 u�0 (u�0)∕퐸 → the induced induced by preserves: (u�1)∕퐹 퐸 퐹 : u�0 → u�1 functor between (i) limits of finite diagrams, petit pretoposes over extents (ii) 휅-ary coproducts, and (iii) exact forks.

Proof. (i). It suﬃces to prove the following:

• Given a pullback square in u�0 of the form below,

푃 퐴1

ℎ1

퐴0 퐵 ℎ0

if 퐵 is a (u�0, u�0)-extent in u�0 and both ℎ0 : 퐴0 → 퐵 and ℎ1 : 퐴1 → 퐵

are members of u�0, then 퐹 : u�0 → u�1 preserves this pullback square.

Let 푞 : 퐵̃ → 퐵 be an étale effective epimorphism in u�0 such that there

exist an object 푌 in u�0 and a étale morphism 퐵̃ → 푌 in u�0 and 퐹 : u�0 →

u�1 preserves kernel pairs of 푞 : 퐵̃ → 퐵. By the pullback pasting lemma,

we have the following pullback square in u�0,

퐵̃ ×퐵 푃 퐵̃ ×퐵 퐴1

퐵̃ ×퐵 퐴0 퐵̃

and by proposition 2.4.6, 퐹 : u�0 → u�1 preserves this pullback square.

On the other hand, by (lemma 2.3.10(a) and) lemma 2.4.10, 퐹 : u�0 → u�1

sends the projections 퐵̃ ×퐵 푃 ↠ 푃 , 퐵̃ ×퐵 퐴0 ↠ 퐴0, and 퐵̃ ×퐵 퐴1 ↠ 퐴1

to étale effective epimorphisms in u�1. Let (푅, 푑0, 푑1) be a kernel pair of

푞 : 퐵̃ ↠ 퐵 in u�0. Then 퐹 : u�0 → u�1 preserves each of the following parallel pairs of pullback squares,

푑0×퐵id푃 푅 ×퐵 푃 퐵̃ ×퐵 푃 푑1×퐵id푃

푑0 푅 퐵̃ 푑1

133 Charted objects

id id 푑0×퐵 퐴0 푑0×퐵 퐴1 푅 ×퐵 퐴0 퐵̃ ×퐵 퐴0 푅 ×퐵 퐴1 퐵̃ ×퐵 퐴1 id id 푑1×퐵 퐴0 푑1×퐵 퐴1

푑0 푑0 푅 퐵̃ 푅 퐵̃ 푑1 푑1

and it follows that 퐹 : u�0 → u�1 preserves the following exact forks:

푑0×퐵id푃 푅 ×퐵 푃 퐵̃ ×퐵 푃 푃 푑1×퐵id푃

id 푑0×퐵 퐴0 푅 ×퐵 퐴0 퐵̃ ×퐵 퐴0 퐴0 id 푑1×퐵 퐴0

id 푑0×퐵 퐴1 푅 ×퐵 퐴1 퐵̃ ×퐵 퐴1 퐴1 id 푑1×퐵 퐴1

Moreover, 퐹 : u�0 → u�1 preserves pullbacks of étale morphisms along

the projections 퐵̃ ×퐵 푃 → 퐵̃, 퐵̃ ×퐵 퐴0 → 퐵̃, and 퐵̃ ×퐵 퐴1 → 퐵̃, so by

lemma 2.4.11, 퐹 : u�0 → u�1 also preserves these pullback squares:

퐵̃ ×퐵 푃 푃 퐵̃ ×퐵 퐴0 퐴0 퐵̃ ×퐵 퐴1 퐴1

̃ ̃ ̃ 퐵 푞 퐵 퐵 푞 퐵 퐵 푞 퐵

Thus, in the following commutative diagrams in u�1,

퐹 (퐵̃ ×퐵 푃 ) 퐹 (퐵̃ ×퐵 퐴1) 퐹 퐴1

퐹 퐵̃ × 퐴 퐹 퐵̃ 퐹 퐵 ( 퐵 0) 퐹 푞

퐹 (퐵̃ ×퐵 푃 ) 퐹 푃

퐹 (퐵̃ ×퐵 퐴0) 퐹 퐴0

퐹 퐵̃ 퐹 퐵 퐹 푞

134 2.4. Functoriality

all the squares are pullback squares in u�1, and by the pullback pasting

lemma, in the following commutative diagram in u�1,

퐹 (퐵̃ ×퐵 푃 ) 퐹 푃 퐹 퐴1

퐹 ℎ1

퐹 (퐵̃ ×퐵 퐴0) 퐹 퐴0 퐹 퐵 퐹 ℎ0

the left square and outer rectangle are pullback diagrams in u�1. Hence,

by lemma 1.4.19(c), the right square is indeed a pullback square in u�1.

(ii) and (iii). Since 퐹 : u�0 → u�1 preserves 휅-ary coproducts and exact

forks, the same is true of the functor (u�0)∕퐸 → (u�1)∕퐹 퐸, by proposition 2.3.5 and (i). ■

Given a pullback square in of the form below, 2.4.12 Proposition. u�0 Preservation of 푘 base changes 퐶 퐷 of étale morph- 푝 푞 isms of extents 퐴 퐵 ℎ

if is a morphism in and is a member ℎ : 퐴 → 퐵 Xt(u�0, u�0) 푞 : 퐷 → 퐵 of , then preserves this pullback square. u�0 퐹 : u�0 → u�1

Proof. Let (푌 , 푏) be a (u�0, u�0)-atlas of 퐵 and let (푋, ⟨푎, 푓⟩) be a (u�0, u�0)-

atlas of 퐴 ×퐵 푌 . By the pullback pasting lemma, we have the following

commutative diagram in u�0,

푘×퐵id푌 퐷 ×퐵 푋 퐶 ×퐵 푌 퐷 ×퐵 푌

푝×퐵id푌 푋 퐴 × 푌 푌 ⟨푎,푓⟩ 퐵

where the squares are pullback squares in u�0, the vertical arrows are étale

morphisms in u�0, the composite 퐷 ×퐵 푋 → 퐶 ×퐵 푌 → 퐷 ×퐵 푌 is

id퐷 ×퐵 푓 : 퐷 ×퐵 푋 → 퐷 ×퐵 푌 , and 퐷 ×퐵 푋 ↠ 퐶 ×퐵 푌 is an étale

effective epimorphism in u�0. Thus, by proposition 2.4.7, the following is

135 Charted objects

a pullback square in u�1:

퐹 (id퐷×퐵푓) 퐹 (퐷 ×퐵 푋) 퐹 (퐷 ×퐵 푌 )

퐹 푋 퐹 푌 퐹 푓 On the other hand, by lemma 2.3.10(a) and proposition 2.4.11, we also

have the following pullback square in u�1,

퐹 (퐷 ×퐵 푋) 퐹 (퐶 ×퐵 푌 )

퐹 (푝×퐵id푌 ) 퐹 푋 퐹 퐴 × 푌 퐹 ⟨푎,푓⟩ ( 퐵 )

where the horizontal arrows are étale effective epimorphisms in u�1, so we may apply lemma 1.4.19(c) to deduce that the following is a pullback

square in u�1:

퐹 (푘×퐵id푌 ) 퐹 (퐶 ×퐵 푌 ) 퐹 (퐷 ×퐵 푌 )

퐹 (푝×퐵id푌 )

퐹 (퐴 ×퐵 푌 ) 퐹 푌

On the other hand, we have the following pullback square in u�1,

퐹 (퐷 ×퐵 푌 ) 퐹 퐷 퐹 푞

퐹 푌 퐹 퐵 퐹 푏

so the outer rectangle of the following commutative diagram in u�1 is a

pullback diagram in u�1:

퐹 푘 퐹 (퐶 ×퐵 푌 ) 퐹 퐶 퐹 퐷

퐹 (푝×퐵id푌 ) 퐹 푝 퐹 푞 퐹 퐴 × 푌 퐹 퐴 퐹 퐵 ( 퐵 ) 퐹 ℎ

Moreover, the left square is a pullback square in u�1 wherein the horizontal

arrows are étale effective epimorphisms in u�1, so the right square is indeed

a pullback square in u�1. ■

136 2.4. Functoriality

2.4.13 ¶ To summarise:

Properties of Theorem. Under the standing assumptions (2.4.3 and 2.4.9): the induced sends -extents in to -extents in (i) 퐹 : u�0 → u�1 (u�0, u�0) u�0 (u�1, u�1) functor between u� . categories 1 The induced functor preserves étale of extents (ii) 퐹 : Xt(u�0, u�0) → Xt(u�1, u�1) morphisms and pullbacks of étale morphisms. Given an exact fork in of the form below, (iii) u�0

푑0 푅 퐴 ℎ 퐵 푑1

if is a -extent in and is an étale effective 퐴 (u�0, u�0) u�0 ℎ : 퐴 → 퐵 epimorphism in , then preserves this exact fork. u�0 퐹 : u�0 → u�1 In particular, is a regulated 퐹 : (Xt(u�0, u�0), u�0) → (Xt(u�1, u�1), u�1) functor.

Proof. (i). This is an immediate consequence of lemma 2.4.10 and the

assumption that 퐹 : u�0 → u�1 preserves effective epimorphisms.

(ii). We know Xt(u�0, u�0) → Xt(u�1, u�1) preserves étale morphisms. For the preservation of pullbacks of étale morphisms, see proposition 2.4.12.

(iii). Apply lemma 2.3.10(a) and (ii). ■

137 Charted objects

2.5 Universality

Synopsis. We characterise the category of extents by a universal property in a special case.

Prerequisites. §§1.1, 1.4, 1.5, 2.1, 2.2, 2.3, 2.4, a.1, a.2, a.3.

2.5.1 ※ Throughout this section, 휅 is a regular cardinal.

2.5.2 ¶ For convenience, we introduce the following terminology.

Definition.A 휅-ary admissible ecumene is a tuple (u�, u�, 햩) where: • (u�, u�, 햩) is a 휅-ary extensive ecumene. • u� is the class of local homeomorphisms in u�. • 햩 is a subcanonical u�-adapted coverage on u�. • (u�, 햤) satisfies the Shulman condition, where 햤 is the class of 햩-covering morphisms in u�.

Remark. By remark 2.2.3 and proposition 2.2.12, any 휅-ary admissible ecumene is an étale ecumene that satisfies the descent axiom.

2.5.3 ¶ Let (u�, u�, 햩) be a 휅-ary admissible ecumene, let 햤 be the class of 햩- covering morphisms in u�, and let u� = Ex(u�, 햤). By corollary 1.5.16, 햤 is a 휅-summable saturated unary coverage on u�, so by proposition 1.5.13, u� is a 휅-ary pretopos. Let u� be the class of morphisms in u� corresponding to morphisms in Psh(u�) that are 햩-semilocally of u�-type and let 햪 be the 휅-ary canonical coverage on u�. By proposition 2.3.2, (u�, u�, 햪) is a 휅-ary extensive ecumene that satisfies the descent axiom. Let u�̄ be the class of local homeomorphisms in u�. Then (u�, u�̄) is a gros 휅-ary pretopos that satisfies the descent axiom for open embeddings, by proposition 2.3.3.

Definition. The gros 휅-ary pretopos associated with a 휅-ary admissible ecumene (u�, u�, 햩) is (u�, u�̄) as defined above.

138 2.5. Universality

2.5.4 ¶ With notation as in paragraph 2.5.3, since 햩 is a subcanonical coverage on u�, 햤 is a subcanonical unary coverage on u�, and therefore the insertion u� → u� is fully faithful. By abuse of notation, we will pretend that the insertion is the inclusion of a full subcategory. By construction, u� is a unary site for u�. Moreover, by proposition 2.3.2, u� is a unary basis for (u�, u�̄). Recalling lemma a.3.10, a morphism in u� is an effective epimorphism in u� if and only if it is a 햩-covering morphism in u�. Since 햩 is a u�-adapted coverage on u�, for every effective epimorphism ℎ : 퐴 ↠ 퐵 in u�, if 퐵 is an object in u�, then there is a morphism 푎 : 푋 → 퐴 in u� such that ℎ ∘ 푝 : 푋 → 퐵 is both a 햩-covering morphism in u� and a member of u�. A similar argument shows that u�̄ ∩ mor u� = u�; furthermore, u� is a unary basis for (u�, u�̄). Thus, in a further abuse of notation, we may simply write u� instead of u�̄.

Definition.A (u�, u�, 햩)-charted object is a (u�, u�)-extent in u�.

Properties of Proposition. the category of (i) (Xt(u�, u�), u�, 햪) is a 휅-ary admissible ecumene. charted objects (ii) Xt(u�, u�) has exact quotients of tractable equivalence relations, and every effective epimorphism in Xt(u�, u�) that is a member of u� is 햪- covering.

(iii) If u� has exact quotients of tractable equivalence relations, and every quotient of every tractable equivalence relation in u� is 햩-covering, then the inclusion u� ↪ Xt(u�, u�) is (fully faithful and) essentially surjective on objects.

Proof. (i). First, note that u� ⊆ Xt(u�, u�) ⊆ u�, so Xt(u�, u�) is also a unary site for u�. Thus, by proposition 2.1.7, the Shulman condition is satisfied. Next, by proposition 2.3.11, (Xt(u�, u�), u�, 햪) is a 휅-ary extensive ecumene, and clearly, the local homeomorphisms therein are the morphisms in Xt(u�, u�) that are local homeomorphisms in u�. It remains to be shown that 햪 is a u�-adapted coverage on Xt(u�, u�). Since 햪 is a 휅-ary superextensive coverage on Xt(u�, u�), it is enough to

139 Charted objects

verify the following:

• For every 햪-covering morphism ℎ : 퐴 ↠ 퐵 in Xt(u�, u�), there is a morphism 푝 : 퐴̃ → 퐴 in Xt(u�, u�) such that ℎ ∘ 푝 : 퐴̃ → 퐵 is a 햪-covering local homeomorphism in Xt(u�, u�).

Let (푌 , 푏) be a (u�, u�)-atlas of 퐵. Then the projection 퐴 ×퐵 푌 ↠ 푌 is an

effective epimorphism in u�, so there is a morphism ⟨푎, 푓⟩ : 푋 → 퐴 ×퐵 푌 in u� such that 푓 : 푋 → 푌 is a 햩-covering local homeomorphism in u�. Thus, 푏 ∘ 푓 : 푋 → 퐵 is a 햪-covering local homeomorphism in Xt(u�, u�); but ℎ ∘ 푎 = 푏 ∘ 푓, so we are done.

(ii). Since (u�, u�, 햪) satisfies the descent axiom for open embeddings, by lemma 2.2.14(c), Xt(u�, u�) is closed in u� under exact quotient of tractable equivalence relations. On the other hand, u� is a quadrable class of morphisms in Xt(u�, u�) and the inclusion Xt(u�, u�) ↪ u� preserves (these) pullbacks, so any effective epimorphism in Xt(u�, u�) that is a member of u� is an exact quotient in u� of its kernel pair, i.e. is an effective epimorphism in u�.

(iii). See proposition 2.3.14(b). ■

2.5.5 ¶ Let (u�0, u�0, 햩0) and (u�1, u�1, 햩1) be 휅-ary admissible ecumenae.

Definition.A 휅-ary admissible functor 퐹 : (u�0, u�0, 햩0) → (u�1, u�1, 햩1)

is a functor 퐹 : u�0 → u�1 with the following properties:

• 퐹 : u�0 → u�1 preserves 휅-ary coproducts.

• 퐹 : (u�0, 햤0) → (u�1, 햤1) is an admissible functor, where 햤0 (resp. 햤1)

is the class of 햩0-covering (resp. 햩1-covering) morphisms in u�0 (resp.

u�1).

• For every pullback square in u�0 of the form below,

푋′ 푋 푓 푌 ′ 푌

if 푓 : 푋 → 푌 is a member of u�0, then 퐹 푓 : 퐹 푋 → 퐹 푌 is a member

of u�1 and 퐹 : u�0 → u�1 preserves this pullback square.

140 2.5. Universality

Let and be the gros -ary pretoposes Properties of Proposition. (u�0, u�0) (u�1, u�1) 휅 admissible func- associated with and , respectively. (u�0, u�0, 햩0) (u�1, u�1, 햩1) tors between (i) 퐹 : u�, 햩 → u�, 햩 is a pre-admissible functor. admissible ( 0) ( 1) There exist a functor and an isomorphism ecumenae (ii) 퐹̄ : u�0 → u�1 휂 : 퐹 ⇒ 퐹̄ of functors such that preserves -ary coprod- u�0 → u�1 퐹̄ : u�0 → u�1 휅 ucts and sends right-exact forks in to coequaliser diagrams in . u�0 u�1 (iii) Moreover, any such (퐹̄ , 휂) is a pointwise left Kan extension of 퐹 : along the inclusion . u�0 → u�1 u�0 ↪ u�0 sends -extents in to -extents in (iv) 퐹̄ : u�0 → u�1 (u�0, u�0) u�0 (u�1, u�1) . u�1 is a -ary ad- (v) 퐹̄ : (Xt(u�0, u�0), u�0, 햪0) → (Xt(u�1, u�1), u�1, 햪1) 휅 missible functor, where (resp. ) is the restriction of the -ary 햪0 햪1 휅 canonical coverage on (resp. ). u�0 u�1

Proof. (i). This is a special case of lemma 2.4.2(a).

(ii) and (iii). See lemma 2.4.2(b).

(iv). In view of propositions 2.3.13 and 2.4.8(b), we may apply theorem 2.4.13.

(v). We have seen that 퐹̄ : (Xt(u�0, u�0), u�0, 햪0) → (Xt(u�1, u�1), u�1, 햪1) preserves 휅-ary coproducts, local homeomorphisms, and pullbacks of

local homeomorphisms, and sends right-햤0-exact forks to coequaliser diagrams. Moreover, by lemma 2.1.10,

퐹̄ : (Xt(u�0, u�0), 햤0) → (Xt(u�1, u�1), 햤1)

is a admissible functor, where 햤0 (resp. 햤1) is the class of 햪0-covering

(resp. 햪1-covering) morphisms in u�0 (resp. u�1). Thus,

퐹̄ : (Xt(u�0, u�0), u�0, 햪0) → (Xt(u�1, u�1), u�1, 햪1)

is indeed a 휅-ary admissible functor. ■

141 Charted objects

2.5.6 Definition. An effective 휅-ary admissible ecumene is a 휅-ary admissible ecumene (u�, u�, 햩) with the following additional data:

• For each tractable equivalence relation (푅, 푑0, 푑1) on each object 푋 in u�, an exact quotient 푞 : 푋 → 푋̄ in u� such that 푞 : 푋 → 푋̄ is a 햩-covering morphism.

Properties of Proposition. Let (u�, u�, 햩) be an effective 휅-ary admissible ecumene and effective admiss- let (u�, u�) be the associated gros 휅-ary pretopos. ible ecumenae (i) (u�, u�, 햩) is a regulated ecumene.

(ii) Effective epimorphisms in u� that are members of u� are 햩-covering morphisms in u�.

(iii) Every 햩-covering morphism in u� that is a member of u� is an effective epimorphism in both u� and u�.

Proof. (i). Let 푓 : 푋 → 푌 be a local homeomorphism in u� and let

(푅, 푑0, 푑1) be a kernel pair of 푓 : 푋 → 푌 in u�. By lemma 2.2.14(b),

(푅, 푑0, 푑1) is a tractable equivalence relation on 푋, so it has a 햩-covering exact quotient 푞 : 푋 → 푋̄ in u�. Let 푚 : 푋̄ → 푌 be the unique morphism in u� such that 푚 ∘ 푞 = 푓. Then, by proposition 1.4.23, the following is an exact fork in u�: 푑 0 푞 푅 푋 푋̄ 푑1 Since u� is a regular category, it follows that 푚 : 푋̄ → 푌 is a monomorphism in u�; hence, 푚 : 푋̄ → 푌 is an open embedding in u�. Thus, we have the required factorisation of 푓 : 푋 → 푌 .

(ii). With notation as above, suppose 푓 : 푋 → 푌 is also an effective epimorphism in u�. Then 푚 : 푋̄ → 푌 is an isomorphism in u�, so 푓 : 푋 → 푌 is also a 햩-covering morphism in u�.

(iii). Every 햩-covering morphism in u� is an effective epimorphism in u�, and local homeomorphisms in u� have kernel pairs that are preserved by the inclusion u� ↪ u�, so the claim follows. ■

142 2.5. Universality

Remark. In view of propositions 1.5.17 and 2.5.6, the coverage associated with an effective 휅-ary admissible ecumene is equivalent to the 휅-ary superextensive coverage generated by the class of effective epimorphisms that are local homeomorphisms.

Let be a -ary admissible 2.5.7 Theorem. 퐹 : (u�0, u�0, 햩0) → (u�1, u�1, 햩1) 휅 The universal functor. Assuming is an effective -ary admissible ecumene: (u�1, u�1, 햩1) 휅 property of (i) The inclusion Xt is a -ary ad- the category of (u�0, u�0, 햩0) ↪ ( (u�0, u�0), u�0, 햪0) 휅 charted objects missible functor. There exist a -ary admissible functor (ii) 휅 퐹̄ : (Xt(u�0, u�0), u�0, 햪0) → and an isomorphism of functors . (u�1, u�1, 햩1) 휂 : 퐹 ⇒ 퐹̄ u�0 → u�1

(iii) Moreover, any such (퐹̄ , 휂) is a pointwise left Kan extension of 퐹 : along the inclusion . u�0 → u�1 u�0 ↪ Xt(u�0, u�0)

Proof. (i). Apply lemma 2.1.10.

(ii) and (iii). By proposition 2.5.4, the inclusion u�1 ↪ Xt(u�1, u�1) is fully faithful and essentially surjective on objects, so the claims reduce to proposition 2.5.5. ■

Remark. The above theorem does most of the hard work of showing that the 2-submetacategory of effective 휅-ary admissible ecumenae is bireflect- ive in the 2-metacategory of 휅-ary admissible ecumenae.

143

Chapter III Specificities

3.1 Compactness

Synopsis. We examine three classes of continuous maps of topological spaces that arise by relativising the notion of compactness.

Prerequisites. §§1.1, 1.2, a.2.

3.1.1 Definition.A universal topological quotient is a continuous map 푓 : 푋 ↠ 푌 with the following property: For every pullback square in Top of the form below, 푋′ 푋 푓 ′ 푓 푌 ′ 푌 the map 푓 ′ : 푋′ → 푌 ′ is a topological quotient.

Remark. Since the effective epimorphisms in Top are precisely the topological quotients, by proposition 1.4.1, the universally strict epimorphisms in Top are precisely the universal topological quotients.

Example. Every surjective open map of topological spaces is a universal topological quotient. Indeed, every surjective open map is a topological quotient, and the class of surjective open maps is a quadrable class of morphisms in Top.

3.1.2 ※ Throughout this section:

• u� is a full subcategory of Top.

• u� is closed under finitary disjoint union.

145 Specificities

• u� is closed under pullback.

• For each object 푋 in u�:

– 햩햿 (푋) is the set of all finite and jointly surjective sinks on 푋.

– 햩햿헊(푋) is the set of all finite sinks Φ on 푋 such that the induced

map ∐(푈,푥)∈Φ 푈 → 푋 is a universal topological quotient.

3.1.3 ¶ The following terminology is non-standard.

Definition. A continuous map 푓 : 푋 → 푌 is semiproper if it has the following property: • For every pullback square in Top of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

if 푌 ′ is compact, then 푋′ is also compact.

Remark. Thus, from the relative point of view, a semiproper map of topological spaces is a continuous family of compact spaces.

Example. Every continuous map from a compact space to a Hausdorff space is semiproper. Indeed, given a pullback square in Top as in the definition, if 푋 is compact and 푌 is Hausdorff, then the comparison map 푋′ → 푌 ′ × 푋 is a closed embedding, so 푋′ is compact when 푌 ′ is.

Properties of Proposition. semiproper maps (i) Every closed embedding of topological spaces is semiproper. For every topological space , the codiagonal (ii) 푋 ∇푋 : 푋 ⨿ 푋 → 푋 is semiproper. (iii) The class of semiproper maps of topological spaces is a quadrable class of morphisms in Top. (iv) The class of semiproper maps of topological spaces is closed under composition. (v) The class of semiproper maps of topological spaces is closed under (possibly infinitary) coproduct in Top.

146 3.1. Compactness

(vi) Given a surjective continuous map 푓 : 푋 ↠ 푌 and a continuous map 푔 : 푌 → 푍, if 푔 ∘ 푓 : 푋 → 푍 is semiproper, then 푔 : 푌 → 푍 is also semiproper.

Proof. Straightforward. ⧫

Let be the class of semiproper maps in . Then every Corollary. ℱ헌헉 u� morphism in that is of -type -semilocally on the domain is semi- u� ℱ헌헉 햩햿 proper.

Proof. Apply proposition 3.1.3. ■

3.1.4 ¶ We will see that the following is a specialisation of the notion of semiproper map.

Definition. A continuous map 푓 : 푋 → 푌 is proper if it has the following property: • For every pullback square in Top of the form below,

푋′ 푋 푓 ′ 푓 푌 ′ 푌

the map 푓 ′ : 푋′ → 푌 ′ is closed, i.e. the image of every closed subspace of 푋′ is a closed subspace of 푌 ′.

Example. If 푋 is a compact topological space, then the unique map 푋 → 1 is proper: this is the precisely the statement of the tube lemma.

Properties of Proposition. proper maps (i) An injective continuous map is proper if and only if it is a closed embedding. For every topological space , the codiagonal (ii) 푋 ∇푋 : 푋 ⨿ 푋 → 푋 is proper. (iii) The class of proper maps of topological spaces is a quadrable class of morphisms in Top.

147 Specificities

(iv) The class of proper maps of topological spaces is closed under composition. (v) The class of proper maps of topological spaces is closed under (possibly infinitary) coproduct in Top.

(vi) Given a surjective continuous map 푓 : 푋 ↠ 푌 and a continuous map 푔 : 푌 → 푍, if 푔 ∘ 푓 : 푋 → 푍 is proper, then 푔 : 푌 → 푍 is also proper. (vii) Given a pullback square in Top of the form below,

푋̃ 푋 푓 ̃ 푓 푌̃ 푌

where 푌̃ ↠ 푌 is a universal topological quotient, if 푓̃ : 푋̃ → 푌̃ is proper, then 푓 : 푋 → 푌 is also proper.

Proof. Straightforward. ⧫

Let be the class of proper maps in . Then every morph- Corollary. ℱ헉 u� ism in that is -semilocally of -type is proper. u� 햩햿헊 ℱ헉

Proof. Apply proposition 3.1.3. ■

Remark. In the language of §2.2, what we have shown is that (u�, ℱ헉, 햩햿헊)

is a finitary (i.e. ℵ0-ary) extensive regulated ecumene that satisfies the descent axiom and in which every eunoic morphism is genial.

3.1.5 ¶ Properness is closely related to compactness. For instance, suppose 푋 is a topological space such that the unique map 푋 → 1 is proper. Let 푆 = 1 − 1 푛 ∈ ℕ ∪ {1} ⊆ ℝ and let 푥 푛 ∈ ℕ be a sequence { 푛+1 | } ( 푛 | ) of points of 푋. Consider 푇 = 1 − 1 , 푥 푛 ∈ ℕ ⊆ 푆 × 푋. The {( 푛+1 푛) | } closure of 푇 is 푇̄ = 푇 ∪{1}×퐴, where 퐴 is the set of accumulation points

of (푥푛 | 푛 ∈ ℕ). Since 푇̄ is a closed subspace of 푆 × 푋, its image is a closed subspace of 푆. In particular, 1 is in the image of 푇̄, i.e. 퐴 contains a point. Thus, every sequence in 푋 contains a convergent subsequence,

148 3.1. Compactness

i.e. 푋 is sequentially compact. A similar argument using nets instead of sequences can be used to show that 푋 is compact. Much more generally, we have the following result.

Recognition Theorem. Let 푓 : 푋 → 푌 be a continuous map. The following are principle for equivalent: proper maps (i) The map 푓 : 푋 → 푌 is proper. For every topological space , the map (ii) 푇 id푇 × 푓 : 푇 × 푋 → 푇 × 푌 is closed.

(iii) The map 푓 : 푋 → 푌 is closed and, for every 푦 ∈ 푌 , 푓 −1{푦} is compact.

(iv) The map 푓 : 푋 → 푌 is closed and, for every subspace 푌 ′ ⊆ 푌 , if 푌 ′ is compact, then 푓 −1푌 ′ is also compact. (v) The map 푓 : 푋 → 푌 is closed and semiproper.

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (iii), (iii) ⇒ (i). See tag 005R in [Stacks].

(i) ⇒ (v). Consider a pullback square in Top of the form below:

푋′ 푋 푓 ′ 푓 푌 ′ 푌

Suppose 푓 : 푋 → 푌 is proper and 푌 ′ is compact. We must show that 푋′ is compact. Then, by proposition 3.1.4, 푓 ′ : 푋′ → 푌 ′ is also proper. Since the unique map 푌 ′ → 1 is proper (by the tube lemma), it follows that 푋′ → 1 is also proper. But we know (i) ⇒ (iii), so 푋′ is indeed compact.

(v) ⇒ (iv), (iv) ⇒ (iii). Immediate. □

Example. If 푋 is a compact topological space and 푌 is a Hausdorff space, then every continuous map 푋 → 푌 is proper: in view of proposition 3.1.4 and theorem 3.1.5, this is a special case of lemma 1.1.9.

149 Specificities

3.1.6 Lemma. Let 푓 : 푋 → 푌 be a continuous map. Assuming 푌 is a compactly When semiproper generated Hausdorff space, the following are equivalent: implies proper (i) The map 푓 : 푋 → 푌 is proper.

(ii) The map 푓 : 푋 → 푌 is semiproper.

(iii) For every subspace 푌 ′ ⊆ 푌 , if 푌 ′ is compact, then 푓 −1푌 ′ is also compact.

Proof. (i) ⇒ (ii). See theorem 3.1.5.

(ii) ⇒ (iii). Immediate.

(iii) ⇒ (i). In view of the theorem, it is enough to check that 푓 : 푋 → 푌 is a closed map. Let 푋′ be a closed subspace of 푋 and let 푌 ′ be its image in 푌 . We wish to show that 푌 ′ is a closed subspace of 푌 . Since 푌 is compactly generated, it is enough to show that 푌 ′ ∩ 푉 is a closed subspace of 푉 for all compact subspaces 푉 ⊆ 푌 . Let 푉 be a compact subspace of 푌 . Then 푓 −1푉 is a compact subspace of 푋. Since 푋′ ∩ 푓 −1푉 is a closed subspace of 푓 −1푉 , it is compact. The image of 푋′ ∩ 푓 −1푉 in 푉 is 푌 ′ ∩ 푉 , and since 푉 is Hausdorff, it follows that 푌 ′ ∩ 푉 is indeed a closed subspace of 푉 . ■

3.1.7 Definition. A continuous map 푓 : 푋 → 푌 is perfect if it has the following properties:

• 푓 : 푋 → 푌 is proper.

• 푓 : 푋 → 푌 is separated, i.e. the relative diagonal Δ푓 : 푋 → 푋 ×푌 푋 is a closed embedding.

Example. For a topological space 푋, the unique map 푋 → 1 is perfect if and only if 푋 is a compact Hausdorff space.

Properties of Proposition. perfect maps (i) An injective continuous map is perfect if and only if it is a closed embedding.

150 3.1. Compactness

For every topological space , the codiagonal (ii) 푋 ∇푋 : 푋 ⨿ 푋 → 푋 is perfect.

(iii) The class of perfect maps of topological spaces is a quadrable class of morphisms in Top.

(iv) The class of perfect maps of topological spaces is closed under composition.

(v) The class of perfect maps of topological spaces is closed under (possibly infinitary) coproduct in Top.

(vi) Given continuous maps 푓 : 푋 → 푌 and 푔 : 푌 → 푍, if both 푔 : 푌 → 푍 and 푔 ∘ 푓 : 푋 → 푍 are perfect, then 푓 : 푋 → 푌 is also perfect.

(vii) Given a surjective continuous map 푓 : 푋 ↠ 푌 and a continuous map 푔 : 푌 → 푍, if 푓 : 푋 ↠ 푌 is proper and 푔 ∘푓 : 푋 → 푍 is perfect, then 푔 : 푌 → 푍 is also perfect. (viii) Given a pullback square in Top of the form below,

푋̃ 푋 푓 ̃ 푓 푌̃ 푌

where 푌̃ ↠ 푌 is a universal topological quotient, if 푓̃ : 푋̃ → 푌̃ is perfect, then 푓 : 푋 → 푌 is also perfect.

Proof. (i)–(vi). Straightforward. (Recall propositions 1.1.11 and 3.1.4.)

(vii). Under the hypotheses, 푔 : 푌 → 푍 is proper. It remains to be shown that 푔 : 푌 → 푍 is separated. Consider the following commutative square in Top:

푓 푋 푌

Δ푔∘푓 Δ푔

푋 ×푍 푋 푌 ×푍 푌 푓×푍푓

Since 푓 : 푋 ↠ 푌 is proper, so too is 푓 ×푍 푓 : 푋 ×푍 푋 → 푌 ×푍 푌 . On the other hand, since 푔 ∘ 푓 : 푋 → 푍 is separated, the relative diagonal

151 Specificities

Δ푔∘푓 : 푋 → 푋 ×푍 푋 is a closed embedding. Hence, Δ푔 : 푌 → 푌 ×푍 푌 is indeed a closed embedding.

(viii). Under the hypotheses, the following is a pullback square in Top,

푋̃ 푋

Δ푓 ̃ Δ푓

푋̃ ×푌̃ 푋̃ 푋 ×푌 푋

and the claim follows. ■

Let be the class of perfect maps in . Then every morph- Corollary. u�헉 u� ism in that is -semilocally of -type is perfect. u� (u�헉, 햩햿헊) u�헉

Remark. In the language of §2.2, what we have shown is that (u�, u�헉, 햩햿헊)

is an étale finitary (i.e. ℵ0-ary) extensive regulated ecumene that satisfies the descent axiom.

152 3.2. Discrete fibrations

3.2 Discrete fibrations

Synopsis. We construct a combinatorial example of a gros pretopos based on discrete fibrations of simplicial sets.

Prerequisites. §§1.1, 2.2, 2.3, a.1.

3.2.1 ¶ Roughly speaking, a discrete fibration of simplicial sets is a morphism with a unique path lifting property, similar to covering maps in algebraic topology. However, because the edges of a simplicial set are oriented, it is perhaps better to think of discrete fibrations of simplicial sets (in the sense below) as generalisations of discrete fibrations of categories. We will see a more precise statement later.

Definition.A discrete fibration of simplicial sets is a morphism 푓 : 푋 → 푌 in sSet with the following property: • For every natural number 푛, the following is a pullback square in Set:

푓푛+1 푋푛+1 푌푛+1

푑0 푑0

푋푛 푌푛 푓푛

Recognition Lemma. Let 푓 : 푋 → 푌 be a morphism of simplicial sets. The following principle for are equivalent: discrete fibrations (i) 푓 : 푋 → 푌 is a discrete fibration of simplicial sets. of simplicial sets (ii) 푓 : 푋 → 푌 is right orthogonal to 훿0 :Δ푛 → Δ푛+1 for every natural number 푛.

Proof. Straightforward. ⧫

Properties of Proposition. discrete fibrations (i) Every isomorphism of simplicial sets is a discrete fibration of sim- of simplicial sets plicial sets. (ii) The class of discrete fibrations of simplicial sets is a quadrable class of morphisms in sSet.

153 Specificities

(iii) The class of discrete fibrations of simplicial sets is closed under composition.

(iv) Given morphisms 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in sSet, if both 푔 : 푌 → 푍 and 푔 ∘ 푓 : 푋 → 푍 are discrete fibrations of simplicial sets, then 푓 : 푋 → 푌 is also a discrete fibration of simplicial sets. (v) For every simplicial set 푋 and every set 퐼, the codiagonal ∇ : is a discrete fibration of simplicial sets. ∐푖∈퐼 푋 → 푋 (vi) The class of discrete fibrations of simplicial sets is closed under (possibly infinitary) coproduct in sSet. (vii) Given a pullback square in sSet of the form below,

푝 푋̃ 푋 푓 ̃ 푓 ̃ 푌 푞 푌

where 푞 : 푌̃ ↠ 푌 is degreewise surjective, if 푓̃ : 푋̃ → 푌̃ is a discrete fibration of simplicial sets, then 푓 : 푋 → 푌 is also a discrete fibration of simplicial sets.

(viii) Given a degreewise surjective morphism 푓 : 푋 ↠ 푌 in sSet and a morphism 푔 : 푌 → 푍 in sSet, if both 푓 : 푋 ↠ 푌 and 푔 ∘ 푓 : 푋 → 푍 are discrete fibrations of simplicial sets, then 푔 : 푌 → 푍 is also a discrete fibration of simplicial sets.

Proof. (i)–(iv). In view of lemma 3.2.1, this is a special case of proposition 1.1.14.

(v) and (vi). Straightforward.

(vii). In the given situation, we have the following commutative diagram in sSet, 푑0 푝푛 푋̃푛+1 푋̃푛 푋푛 ̃ ̃ 푓푛+1 푓푛 푓푛

푌푛+1̃ 푌푛̃ 푌푛 푑0 푞푛 in which both squares are pullback squares in sSet. Thus, by the pullback pasting lemma, the outer rectangle in the commutative diagram in sSet

154 3.2. Discrete fibrations

shown below is a pullback diagram in sSet:

푝푛+1 푑0 푋̃푛+1 푋푛+1 푋푛 ̃ 푓푛+1 푓푛+1 푓푛

푌푛+1̃ 푌푛+1 푌푛 푞푛+1 푑0

But the left square is also a pullback square in sSet, so we may apply lemma 1.4.19(c) to deduce that the right square is indeed a pullback square in sSet.

(viii). In the given situation, we have the following commutative diagram in sSet, 푓푛+1 푔푛+1 푋푛+1 푌푛+1 푍푛+1

푑0 푑0 푑0

푋푛 푌푛 푍푛 푓푛 푔푛 where the outer rectangle and the left square are both pullback diagrams

in sSet. Since 푓푛 : 푋푛 ↠ 푌푛 is surjective, it follows that the right square is a pullback square in sSet. ■

3.2.2 ¶ Let 푋 be a simplicial set. Suppose we have the following data:

• For each 푥 ∈ 푋0, a set 퐴(푥). ∗ • For each 푒 ∈ 푋1, a map 푒 : 퐴(푑0(푒)) → 퐴(푑1(푒)). ∗ • For every 푥 ∈ 푋0, 푠0(푥) = id퐴(푥).

• For every positive integer 푛 and every 휎 ∈ 푋푛+1:

∗ ∗ 푑0(⋯ (푑푛−2(푑푛+1(휎))) ⋯) ∘ 푑0(⋯ (푑푛−2(푑푛−1(휎))) ⋯) ∗ = 푑0(⋯ (푑푛−2(푑푛(휎))) ⋯)

∗ ∗ ∗ (So, for instance, for every 휎 ∈ 푋2, 푑1(휎) = 푑2(휎) ∘ 푑0(휎) .) We may then construct a simplicial set 퐴 as follows:

• The elements of 퐴0 are pairs (푥, 푎) where 푥 ∈ 푋0 and 푎 ∈ 퐴(푥).

• For each positive integer 푛, the elements of 퐴푛 are pairs (휎, 푎) where

휎 ∈ 푋푛 and 푎 ∈ 퐴(푑0(⋯ (푑푛−1(휎)))).

155 Specificities

• The face operators 푑0, 푑1 : 퐴1 → 퐴0 and the degeneracy operator

푠0 : 퐴0 → 퐴1 are defined as follows:

푑0(푒, 푎) = (푑0(푒), 푎) ∗ 푑1(푒, 푎) = (푑1(푒), 푒 (푎))

푠0(푥, 푎) = (푠0(푥), 푎)

• For every positive integer 푛, the face operators 푑0, … , 푑푛+1 : 퐴푛+1 →

퐴푛 and the degeneracy operators 푠0, … , 푠푛 : 퐴푛 → 퐴푛+1 are defined as follows:

푑푖(휎, 푎) = (푑푖(휎), 푎) for 0 ≤ 푖 ≤ 푛 ∗ 푑푛+1(휎, 푎) = (푑푛+1(휎), 푑0(⋯ (푑푛−1(휎))) (푎))

푠푖(휎, 푎) = (푠푖(휎), 푎) for 0 ≤ 푖 ≤ 푛

(It is straightforward to verify the simplicial identities.) There is an evident projection 푝 : 퐴 → 푋, and by construction, it is a discrete fibration of simplicial sets. In fact, every discrete fibration with codomain 푋 arises in this fashion up to isomorphism. On the other hand, consider the category u� defined inductively as follows:

• For each 푥 ∈ 푋0, 푥 is an object in u�.

• For each 푒 ∈ 푋1, 푒 : 푑1(푥) → 푑0(푥) is a morphism in u�.

• For each 푥 ∈ 푋0, 푠0(푥) = id푥 as morphisms in u�.

• For each 휎 ∈ 푋2, we have 푑0(휎) ∘ 푑2(휎) = 푑1(휎) as morphisms in u�. It is clear from the description above that 퐴 defines a presheaf on u�. Thus:

There is a functor that is fully faithful and Theorem. Psh(u�) → sSet∕푋 essentially surjective onto the full subcategory spanned by the discrete fibrations of simplicial sets with codomain 푋.

Proof. Straightforward, given the discussion above. ⧫

156 3.2. Discrete fibrations

3.2.3 ※ For the remainder of this section:

• 휅 is an uncountable regular cardinal.

• u� is the category of simplicial sets 푋 such that each 푋푛 is hereditarily 휅-small. • u� is the class of discrete fibrations of simplicial sets in u�.

3.2.4 ¶ We will now use the above results to construct a combinatorial example of a gros pretopos.

Proposition. (i) u� is a 휅-ary pretopos. (ii) u� is a class of étale morphisms in u�. [1] In particular, (u�, u�) is a gros 휅-ary pretopos.

Proof. (i). Straightforward. (The inclusion u� ↪ sSet creates limits of finite diagrams and colimits of 휅-small diagrams.)

(ii). See proposition 3.2.1. ■

3.2.5 Remark. (u�, u�) is an example of a gros 휅-ary pretopos in which the class of étale morphisms strictly contains the class of local homeomorphisms. Indeed, given lemma 2.3.12(a) and theorem 3.2.2, this is essentially just the observation that Psh(u�) is not always a localic topos.

3.2.6 ¶ Unlike the gros pretoposes we will see later, there is no obvious candid- ate for a unary basis for (u�, u�). Nonetheless, for the sake of illustration, we may consider the following.

Definition. The strict Segal condition on a simplicial set 푋 is the following: • For every positive integer 푛, the following is a pullback square in Set:

푑푛+1 푋푛+1 푋푛

푑0∘⋯∘푑푛−1 푑0∘⋯∘푑푛−1

푋1 푋0 푑1

[1] Recall lemma 2.3.3.

157 Specificities

Example. Let u� be a category. The nerve of u� is the simplicial set N(u�) defined as follows:

• N(u�)0 is the set of objects in u�.

• N(u�)1 is the set of morphisms in u�.

• For 푛 ≥ 2, N(u�)푛 is the set of 푛-tuples (푓푛, … , 푓1) of morphisms in

u� such that 푓푛 ∘ ⋯ ∘ 푓1 is defined in u�.

• The face operators 푑0, 푑1 : N(u�)1 → N(u�)0 and the degeneracy oper-

ator 푠0 : N(u�)0 → N(u�)1 are defined as follows:

푑0(푓) = codom 푓

푑1(푓) = dom 푓

푠0(푥) = id푥

• The face operators 푑0, 푑1, 푑2 : N(u�)2 → N(u�)1 and the degeneracy

operators 푠0, 푠1 : N(u�)1 → N(u�)2 are defined as follows:

푑0(푓1, 푓0) = 푓1

푑1(푓1, 푓0) = 푓1 ∘ 푓0

푑2(푓1, 푓0) = 푓0

푠0(푓) = (푓, iddom 푓 )

푠1(푓) = (idcodom 푓 , 푓)

• For 푛 > 2, the face operators 푑0, … , 푑푛 : N(u�)푛 → N(u�)푛−1 and

degeneracy operators 푠0, … , 푠푛−1 : N(u�)푛−1 → N(u�)푛 are defined analogously. Then N(u�) satisfies the strict Segal condition. In fact, a simplicial set satisfies the strict Segal condition if and only if it is isomorphic to N(u�) for some category u�.

Lemma. Let 푝 : 푋 → 푌 be a discrete fibration of simplicial sets. (i) If 푌 satisfies the strict Segal condition, then 푋 also satisfies the strict Segal condition.

(ii) If 푝 : 푋 → 푌 is degreewise surjective and 푋 satisfies the strict Segal condition, then 푌 also satisfies the strict Segal condition.

158 3.2. Discrete fibrations

Proof. The pullback pasting lemma implies that the following are pullback squares in Set:

푓푛 푓푛+1 푋푛 푌푛 푋푛+1 푌푛+1

푑0∘⋯∘푑푛−1 푑0∘…∘푑푛−1 푑0∘⋯∘푑푛−1 푑0∘…∘푑푛−1

푋0 푌0 푋1 푌1 푓0 푓1 Then (i) is another application of the pullback pasting lemma, and (ii) is consequence of lemma 1.4.19(c). ■

3.2.7 ¶ Let u� be the full subcategory of u� spanned by the simplicial sets 푋 with the following properties:

• 푋 satisfies the strict Segal condition.

• The map ⟨푑1, 푑0⟩ : 푋1 → 푋0 × 푋0 is injective. Proposition. (i) u� is equivalent to the metacategory of 휅-small preordered sets. (ii) u� is a unary basis for (u�, u�). (iii) Xt(u�, u�) is equivalent to the metacategory of 휅-small categories in which every morphism is a monomorphism.

(iv) A (u�, u�)-extent in u� is u�-localic if and only if it is an object in u�.

Proof. (i). A simplicial set 푋 is isomorphic to an object in u� if and only if 푋 is isomorphic to the nerve of some preordered set (considered as a category). Since N : Cat → sSet is fully faithful, the claim follows.

(ii). Since 휅 > ℵ0, every object in u� is a simplicial set with < 휅 elements. Thus, the Yoneda lemma implies that u� is a unary site for u�. In addition,

given a discrete fibration 푝 : 푋 → 푌 in u�, if ⟨푑1, 푑0⟩ : 푌1 → 푌0 ×

푌0 is injective, then ⟨푑1, 푑0⟩ : 푋1 → 푋0 × 푋0 is also injective; so, by lemma 3.2.6, if 푌 is an object in u�, then 푋 is also an object in u�. Hence, u� is indeed a unary basis for (u�, u�). (iii). It suﬃces to identify necessary and suﬃcient conditions for a category u� to admit a surjective discrete fibration u�̃ → u� where u�̃ is a preorder category.

159 Specificities

Clearly, if every morphism in u� is a monomorphism, then every slice ̃ category u�∕푥 is a preorder category, so we may take u� = ∐푥∈ob u� u�∕푥. Conversely, suppose we have a surjective discrete fibration 푝 : u�̃ → u� where u�̃ is a preorder category. Let 푔 : 푦 → 푧 be a morphism in u� and let

푓0, 푓1 : 푥 → 푦 be a parallel pair of morphisms in u� such that 푔∘푓0 = 푔∘푓1. By hypothesis, there is an object ̃푧 in u�̃ such that 푝( )̃푧 = 푧, and there exist a unique object ̃푦 and a unique morphism ̃푔: ̃푦→ ̃푧 in u�̃ such that 푝( )̃푦 = 푦 and 푝( )̃푔 = 푔. Similarly, there exist a unique object ̃푥 and a ̃ ̃ ̃ unique morphism 푓 : ̃푥→ ̃푦 in u� such that 푝( ̃푥) = 푥 and 푝(푓) = 푓0 = 푓1. This shows 푔 : 푦 → 푧 is indeed a monomorphism in u�.

(iv). The claim reduces to the fact that Psh(u�) is localic if and only if u� is a preorder category. ■

160 3.3. Manifolds

3.3 Manifolds

Synopsis. We construct an admissible ecumene for which the charted objects are the smooth manifolds of fixed dimension and cardinality.

Prerequisites. §§1.1, 1.4, 2.2, 2.3.

3.3.1 ¶ Manifolds are probably the best-known notion of space built from local models. Unfortunately, from a category-theoretic point of view, manifolds are somewhat awkward: for instance, the category of manifolds does not have all pullbacks. On the other hand, this can still be accommodated within the theory of charted objects with respect to an admissible ecumene —after all, an ecumene is not required to have pullbacks. In fact, even finitary products are unnecessary—we will see how to define manifolds of a fixed dimension as charted objects.

3.3.2 ※ Throughout this section:

• 푛 is a natural number. connected 푛 • u�0 is the category of open subspaces of ℝ and smooth maps.

• 휅 is a regular cardinal such that u�0 is essentially 휅-small.

• u� = Fam휅(u�0).

• u�0 is the class of local diffeomorphisms in u�0.

• u� is the class of morphisms in u� that are familially of u�0-type. • 햤 is the class of morphisms 푓 : 푋 → 푌 in u� such that the corresponding continuous map is a surjective local homeomorphism.[1]

• For each object 푋 in u�0, 햩0(푋) is the set of open covers of 푋. • For each object 푋 in u�, 햩(푋) is the set of 휅-small sinks Φ on 푋 such

that the induced morphism ∐(푈,푥)∈Φ 푈 → 푋 in u� is a member of 햤.

[1] Here, we are using the functor u� → Top induced by the forgetful functor u�0 → Top.

161 Specificities

3.3.3 Proposition. (u�, u�, 햩) is a 휅-ary admissible ecumene, i.e.:

(i) u� is a 휅-ary extensive category.

(ii) u� is a class of separated fibrations in u�.

(iii) 햩 is a subcanonical u�-adapted 휅-ary superextensive coverage on u�.

(iv) Every morphism in u� of u�-type (u�, 햩)-semilocally on the domain is a member of u�.

(v) Every complemented monomorphism in u� is a member of u�. [2] (vi) (u�, 햤) satisfies the Shulman condition.

Proof. (i). This is a special case of proposition 1.5.7.

(ii)–(v). Straightforward.

(vi). By extensivity, it suﬃces to verify the following:

• Given objects 푋 , … , 푋 in u� , El h × ⋯ × h is an essen- 0 푘−1 0 ( 푋0 푋푘−1) tially 휅-small category.

But this is an immediate consequence of the assumption that u�0 itself is an essentially 휅-small category. ■

3.3.4 ¶ Let (u�, u�) be the gros 휅-ary pretopos associated with (u�, u�, 햩). By

abuse of notation, we will consider u�0 to be a full subcategory of u� and u� to be a full subcategory of u�. It should come as no surprise that (u�, u�)- extents are 푛-dimensional manifolds, but we should be more precise about what that means.

3.3.4(a) Definition.A smooth 푛-dimensional atlas of a topological space 푋 is a set Φ with the following properties: • Every element of Φ is a pair (푈, 푥) where 푈 is a connected open subspace of 푋 and 푥 : 푈 → ℝ푛 is an open embedding of topological spaces.

• 푋 = ⋃(푈,푥)∈Φ 푈. [2] Note that a morphism in u� is 햩-covering if and only if it is 햤-covering.

162 3.3. Manifolds

• For every , −1 ′ ′ is a ((푈0, 푥0), (푈1, 푥1)) ∈ Φ × Φ 푥1 ∘ (푥0) : 푈0 → 푈1 diffeomorphism, where ′ is the image of 푛 and ′ 푈0 푥0 : 푈0 ∩ 푈1 → ℝ 푈1 푛 is the image of 푥1 : 푈0 ∩ 푈1 → ℝ .

3.3.4(b) Definition.A 푛-dimensional manifold is a pair (푋, Φ) where 푋 is a topological space and Φ is a smooth 푛-dimensional atlas of 푋.

Remark. In particular, we do not require manifolds to be Hausdorff spaces, nor do we require manifolds to be second-countable.

3.3.5 ¶ Though it is more usual to first define ‘smooth map’, we will instead define the presheaf represented by a manifold directly and take for granted that the Yoneda representation is fully faithful.

Definition. The presheaf represented by an 푛-dimensional manifold

(푋, Φ) is the presheaf h(푋,Φ) on u�0 defined as follows: 푛 • For each connected open subspace 푇 ⊆ ℝ , h(푋,Φ)(푇 ) is the set of all continuous maps 푓 : 푇 → 푋 with the following property: – For every (푈, 푥) ∈ Φ, the map 푥 ∘ 푓 : 푇 ∩ 푓 −1푈 → ℝ푛 is smooth.

• The action of u�0 is composition (of continuous maps). Let be an -dimensional manifold. Then is a - Lemma. (푋, Φ) 푛 h(푋,Φ) 햩0 sheaf on . u�0

Proof. Straightforward. (This is essentially the fact that smoothness of maps is a local property.) ⧫

3.3.6 Theorem. The essential image of the Yoneda representation Xt(u�, u�) → is the full and replete subcategory spanned by the -sheaves Sh(u�0, 햩0) 햩0 represented by 푛-dimensional manifolds (푋, Φ) where Φ is 휅-small.

Proof. For ease of notation, we will identify u� with its essential image in

Sh(u�0, 햩0). First, let (푋, Φ) be an 푛-dimensional manifold where Φ is 휅-small.

We will show that h(푋,Φ) is in Xt(u�, u�). It is not hard to see that Φ is

햩0-local generating set of elements of h(푋,Φ). In other words, we have a

햩0-locally surjective morphism 푝 : 퐴 ↠ h(푋,Φ) where 퐴 = ∐(푈,푥)∈Φ h푥푈

163 Specificities

푛 in Sh(u�0, 햩0), where 푥푈 is the image of 푥 : 푈 → ℝ . Since u� is closed

under 휅-ary coproduct, 퐴 is an object in u�. Let (푅, 푑0, 푑1) be a kernel

pair of 푝 : 퐴 ↠ h(푋,Φ) in Sh(u�0, 햩0). Then 푅 is a 휅-ary disjoint union of

representable 햩0-sheaves on u�0, so (푅, 푑0, 푑1) is an equivalence relation

on 퐴 in u�. Moreover, by lemma a.3.10,(h(푋,Φ) is an object in u� and) 푝 :

퐴 ↠ h(푋,Φ) is an exact quotient of (푅, 푑0, 푑1) in u�. It is straightforward

to verify that (푅, 푑0, 푑1) is a tractable equivalence relation in u�. Hence,

by lemma 2.2.14(c), 푝 : 퐴 ↠ h(푋,Φ) is a local homeomorphism in u�, and

therefore h(푋,Φ) is indeed a (u�, u�)-extent in u�. Now, let 퐵 be a (u�, u�)-extent in u�. By definition, there is a (u�, u�)- atlas of 퐵 in u�, say (퐴, 푝), and by proposition 2.3.13, we may assume that 푝 : 퐴 ↠ 퐵 is a laminar morphism. (Note that, for every laminar morphism ℎ : 퐶′ → 퐶 in u�, if 퐶 is isomorphic to an object in ′ u�, then 퐶 is also isomorphic to an object in u�.) Suppose 퐴 = ∐푖∈퐼 푇푖

for some family (푇푖 | 푖 ∈ 퐼) of objects in u�0 where 퐼 is a 휅-small set,

such that each composite 푇푖 → 퐴 ↠ 퐵 is an open embedding in u�. Let 푅 = ∐ ∐ 푇 × 푇 . Then 푅, 푑 , 푑 is a kernel pair of 푖0∈퐼 푖1∈퐼 0 퐵 1 ( 0 1) 푝 : 퐴 ↠ 퐵, where 푑0, 푑1 : 푅 → 퐴 are the two evident projections. Let 푋 be the quotient of the corresponding equivalence relation in Top. Since

the projections 푑0, 푑1 : 푅 → 퐴 and the relative diagonal Δ푝 : 퐴 → 푅 correspond to open maps of topological spaces, the quotient map is a local homeomorphism of topological spaces. In particular, we obtain open

embeddings 푇푖 → 푋, and it is straightforward to verify that their inverses comprise a smooth 푛-dimensional atlas Φ of 푋. Moreover, by the argu-

ment of the previous paragraph, we obtain h(푋,Φ) ≅ 퐵, as desired. ■

3.3.7 ¶ Finally, we should remark that the material covered in this section does not depend very strongly on the meaning of ‘smooth’. Indeed, everything still works if we replace ‘smooth’ with ‘푚-times continuously differentiable’—then we get the category of 푛-dimensional manifolds and 푚-times continuously differentiable maps. We could even replace ℝ with ℂ and ‘smooth’ with ‘analytic’ to obtain the category of 푛-dimensional complex analytic manifolds and analytic maps. The dimension restriction can also

164 3.3. Manifolds be lifted; then, by proposition 2.3.11, Xt(u�, u�) will have finitary products (as is well known).

165 Specificities

3.4 Topological spaces

Synopsis. We construct admissible ecumenae from categories of topological spaces and investigate when a topological space is representable by a charted object.

Prerequisites. §§1.1, 2.2, 2.3, 2.5, a.1, a.2, a.3.

3.4.1 ※ Throughout this section:

• 휅 is a regular cardinal.

• u� is a (small) full subcategory of the metacategory of topological spaces (and continuous maps) that is closed under finitary products, equalisers, and 휅-ary disjoint unions.

• u� is the class of local homeomorphisms between objects in u�.

• For every object 푋 in u� and every open subspace 푈 ⊆ 푋, there is 휅-small set Φ of open subspaces of 푈 with the following properties:

– For every 푉 ∈ Φ, 푉 is homeomorphic to an object in u�.

– 푈 = ⋃푉 ∈Φ 푉 . • For each object 푋 in u�, 햩(푋) is the set of 휅-small sinks Φ on 푋 in u� such that Φ is a jointly surjective family of open embeddings.

• 햤 is the class of surjective local homeomorphisms between objects in u�.

3.4.2 Proposition. (u�, u�, 햩) is a 휅-ary admissible ecumene, i.e.:

(i) u� is a 휅-ary extensive category.

(ii) u� is a class of separated fibrations in u�.

(iii) 햩 is a subcanonical u�-adapted 휅-ary superextensive coverage on u�.

(iv) Every morphism in u� of u�-type (u�, 햩)-semilocally on the domain is a member of u�.

(v) Every complemented monomorphism in u� is a member of u�.

166 3.4. Topological spaces

[1] (vi) (u�, 햤) satisfies the Shulman condition.

Proof. Straightforward. ⧫

Remark. In particular, every local homeomorphism in u� in the sense of definition 2.2.12 is a member of u�, so there is no danger of confusion in using the phrase ‘local homeomorphism in u�’.

3.4.3 ※ For the remainder of this section:

• u� = Ex(u�, 햤).

• u� is the class of morphisms in u� corresponding to morphisms in Psh(u�) that are 햩-locally of u�-type.

• 햪 is the 휅-ary canonical coverage on u�.

• u�̂ is the class of u�-perfect morphisms in u�.

Furthermore, by abuse of notation, we will identify u� with the image of the insertion u� → u�.

3.4.4 Proposition. (i) (u�, u�̂) is a 휅-ary gros pretopos.

(ii) Moreover, (u�, u�̂, 햪) satisfies the descent axiom. (iii) A morphism in u� is a member of u� if and only if it is a member of u�̂.

Proof. This is a special case of proposition 2.3.2. ■

3.4.5 ¶ Consider the Yoneda representation h• : Top → Psh(u�). Since u� has pullbacks and the inclusion u� ↪ Top preserves them, lemma a.2.6

implies that, for every topological space 푋, h푋 is a 햩-sheaf on u�. Thus, by proposition a.1.4, for every 햩-sheaf 퐴 on u�, there is a topological space

|퐴| and a morphism 휂퐴 : 퐴 → h|퐴| in Sh(u�, 햩) such that the following map is a bijection for every topological space 푌 :

Top(|퐴|, 푌 ) → HomSh(u�,햩)(퐴, h푌 ) [1] Note that a morphism in u� is 햩-covering if and only if it is 햤-covering.

167 Specificities

푓 ↦ h푓 ∘ 휂퐴

Indeed, we may take |퐴| = lim 푋. This yields an adjunction: −−→(푋,푎):El(퐴)

|−| Top ⊥ Sh(u�, 햩) h•

It is clear (by construction) that the counit 휀푋 : |h푋| → 푋 is a homeomorphism for every object 푋 in u�. We would like to know if this happens for topological spaces that are not necessarily in u�.

3.4.5(a) Lemma. Let 푋 be a topological space. The following are equivalent: The counit is a homeomorphism. (i) 휀푋 : |h푋| → 푋 (ii) For every topological space 푌 , the following is a bijection:

h• : Top(푋, 푌 ) → HomSh(u�,햩)(h푋, h푌 )

Proof. Straightforward. ⧫

3.4.5(b) Lemma. The functor |−| : Sh(u�, 햩) → Top preserves monomorphisms.

Proof. Let Γ: Sh(u�, 햩) → Set be the evident functor defined on objects by 퐴 ↦ 퐴(1). It is clear that 1 is a 햩-local object in u�, so by lemma a.3.12, Γ: Sh(u�, 햩) → Set preserves colimits. On the other hand, proposition a.1.4 implies that Γ: Sh(u�, 햩) → Set is isomorphic to the composite of |−| : Sh(u�, 햩) → Top and the forgetful functor Top → Set. Since the forgetful functor Top → Set is faithful, it follows that |−| : Sh(u�, 햩) → Top preserves monomorphisms. ■

3.4.5(c) Lemma. Let 푓 : 푋 ↠ 푌 be a surjective local homeomorphism of topological spaces and let be the kernel pair of in (푅, 푑0, 푑1) 푓 : 푋 ↠ 푌 Top.

(i) The following is an exact fork in Sh(u�, 햩):

푑 ∘− 0 푓∘− h푅 h푋 h푌 푑1∘−

168 3.4. Topological spaces

If both and are homeomorphisms, (ii) 휀푅 : |h푅| → 푅 휀푋 : |h푋| → 푋 then is also a homeomorphism. 휀푌 : |h푌 | → 푌

Proof. (i). It is not hard to verify that h푓 : h푋 → h푌 is a 햩-locally surjective morphism in Psh(u�). Thus, by lemma a.3.10, we have the desired exact fork.

(ii). |−| : Sh(u�, 햩) → Top preserves coequalisers, and 푓 : 푋 ↠ 푌 is

an effective epimorphism in Top, so it follows that 휀푌 : |h푌 | → 푌 is

a homeomorphism if both 휀푅 : |h푅| → 푅 and 휀푋 : |h푋| → |h푌 | are homeomorphisms. ■

Let be a family of topological spaces where is a 3.4.5(d) Lemma. (푋푖 | 푖 ∈ 퐼) 퐼 -small set and let . 휅 푋 = ∐푖∈퐼 푋푖 (i) h is a coproduct of h 푖 ∈ 퐼 in Sh(u�, 햩) (with the evident co- 푋 ( 푋푖 | ) product injections).

(ii) If each 휀 : h → 푋 is a homeomorphism, then 휀 : h → 푋 푋푖 | 푋푖| 푖 푋 | 푋| is also a homeomorphism.

Proof. (i). Using lemma 1.5.4, it is not hard to see that the Yoneda rep-

resentation h• : Top → Sh(u�, 햩) preserves 휅-ary coproducts.

(ii). On the other hand, |−| : Sh(u�, 햩) also preserves (휅-ary) coproducts. The claim follows. ■

3.4.5(e) Lemma. Let 푋 be an object in u� and let 푈 be an open subspace of 푋. is a monomorphism in that is -semilocally of (i) h푈 → h푋 Psh(u�) 햩 u�-type. is a homeomorphism in . (ii) 휀푈 : |h푈 | → 푈 u�

Proof. (i). By hypothesis, there is a 휅-small set Φ of open subspaces of 푉 such that:

• For each 푉 ∈ Φ, 푉 is homeomorphic to an object in u�.

• 푈 = ⋃푉 ∈Φ 푉 .

169 Specificities

It is clear that h푈 → h푋 is a monomorphism in Psh(u�), and it follows ̄ that h푈 → h푋 is 햩-semilocally of u�-type. Let 푉 = ∐푉 ∈Φ 푉 and let 푝 : 푉̄ ↠ 푈 be the evident projection. Clearly, 푝 : 푉̄ ↠ 푈 is a surjective

local homeomorphism. Let (푅, 푑0, 푑1) be the kernel pair of 푝 : 푉̄ ↠ 푉 . Then 푅 ≅ ∐ ∐ 푉 ∩푉 , and each 푉 ∩푉 is homeomorphic to an 푉0∈Φ 푉1∈Φ 0 1 0 1 object in u�, so by lemma 3.4.5(d), both 휀푅 : |h푅| → 푅 and 휀푉̄ : |h푉̄ | → 푉̄

are homeomorphisms. Hence, by lemma 3.4.5(c), 휀푈 : |h푈 | → 푈 is also a homeomorphism. ■

3.4.6 ¶ In view of the discussion above, we make the following definition.

Definition. A topological space 푋 is of u�-type if there is a 휅-small set Φ of open subspaces of 푋 with the following properties: • For every 푈 ∈ Φ, 푈 is homeomorphic to an object in u�.

• 푋 = ⋃푈∈Φ 푈. We write ℳ for the metacategory of topological spaces of u�-type (and continuous maps).

Proposition. (i) ℳ is closed in Top under 휅-ary disjoint union. (ii) Given an object 푋 in ℳ, if 푈 is an open subspace of 푋, then 푈 is also an object in ℳ. (iii) For every object 푌 in ℳ, there exist an object 푋 in u� and a surjective local homeomorphism such that is 푓 : 푋 ↠ 푌 푋 ×푌 푋 homeomorphic to a 휅-ary disjoint union of open subspaces of 푋 and is a morphism in that is -semilocally of -type. h푓 : h푋 → h푌 Psh(u�) 햩 u� For every object in , the counit is a homeo- (iv) 푌 ℳ 휀푌 : |h푌 | → 푌 morphism.

(v) The Yoneda representation ℳ → Sh(u�, 햩) is fully faithful, preserves 휅-ary coproducts, and sends surjective local homeomorphisms in ℳ to effective epimorphisms in Sh(u�, 햩).

Proof. (i) and (ii). Straightforward.

170 3.4. Topological spaces

(iii). Let 푌 be an object in ℳ. By definition, there is a 휅-small set Ψ of open subspaces of 푌 with the following properties:

• For every 푉 ∈ Ψ, 푉 is homeomorphic to an object in u�.

• 푌 = ⋃푉 ∈Ψ 푉 .

Since u� is closed under 휅-ary disjoint union, ∐푉 ∈Ψ 푉 is also homeomorphic to an object in u�, say 푋. There is an evident surjective local

homeomorphism 푓 : 푋 ↠ 푌 , and it is clear that 푋 ×푌 푋 is homeomorphic to a 휅-ary disjoint union of open subspaces of 푋. Moreover, by

proposition 1.2.13 and lemma 3.4.5(e), h푓 : h푋 → h푌 is 햩-semilocally of u�-type, as claimed. (iv). Apply lemmas 3.4.5(c) and 3.4.5(d) to (ii) and (iii).

(v). By lemma 3.4.5(a) and (iv), the Yonedarepresentation ℳ → Sh(u�, 햩) is fully faithful. We already know that the Yoneda representation Top → Sh(u�, 햩) preserves 휅-ary coproducts and sends surjective local homeomorphisms in Top to effective epimorphisms in Sh(u�, 햩), so we are done. ■

3.4.7 ¶ By theorem 2.1.14, the Yoneda representation u� → Sh(u�, 햩) is fully faithful and preserves limits of finite diagrams, 휅-ary coproducts, and exact quotients. Moreover, by lemma 2.1.16, a 햩-sheaf on u� is in the essential image of the Yoneda representation if and only if it is 햩-locally 휅-presentable.

Lemma. If 푥 : 푈 → 푋 is an open embedding in u� and 푋 is an object in , then is an open embedding of topological spaces. u� |h푥| : |h푈 | → |h푋|

Proof. Since h푥 : h푈 → h푋 is a monomorphism in Psh(u�) that is 햩-

semilocally of u�-type, there is a 휅-small set Φ of objects in u�∕푈 with the following properties:

• For every (푉 , 푢) ∈ Φ, 푉 is an object in u� and 푥 ∘ 푢 : 푉 → 푋 is an open embedding of topological spaces.

• The induced morphism 푝 : ∐(푉 ,푢)∈Φ 푉 → 푈 in u� is an effective epimorphism.

171 Specificities

Thus, h : h ↠ |푈| is an effective epimorphism in Top and | 푝| | ∐(푉 ,푢)∈Φ 푉 | the composite h ∘ h : h → |푋| is a local homeomorph- | 푥| | 푝| | ∐(푉 ,푢)∈Φ 푉 | ism of topological spaces. On the other hand, by lemma 3.4.5(b), |h푥| :

|h푈 | → |h푋| is an injective continuous map. Thus, |h푥| : |h푈 | → |h푋| is indeed an open embedding of topological spaces. ■

Theorem. Let u�̄ be the class of local homeomorphisms in u�.

(i) If 푋 is an object in ℳ, then there is a (u�, u�̄)-extent 퐴 in u� such that in . h푋 ≅ h퐴 Sh(u�, 햩) If is a ̄ -extent in , then is a topological space of (ii) 퐴 (u�, u�) u� |h퐴| u�-type. The functor ̄ is fully faithful and essentially (iii) |h•| : Xt(u�, u�) → ℳ surjective on objects.

Proof. (i). First, consider a subspace 푈 of an object 푋 in u�. Recalling the proof of lemma 3.4.5(e), we see that there is an open subobject 퐴 of

푋 in u� such that h푈 ≅ h퐴. Thus, by proposition 3.4.6, for every object

푋 in ℳ, there is an object 퐴 in u� such that h푋 ≅ h퐴. Moreover, by tracing the proof of that proposition, it is straightforward to verify that 퐴 is a (u�, u�̄)-extent in u�.

(ii). In view of lemma 3.4.7, a similar argument shows that |h퐴| is a topological space of u�-type if 퐴 is a (u�, u�̄)-extent in u�. ̄ (iii). Hence, by lemma 3.4.5(a) and proposition 3.4.6, |h•| : Xt(u�, u�) → ℳ is fully faithful and essentially surjective on objects. ■

3.4.8 ¶ By proposition 2.2.12, we have u�̄ ⊆ u�̂, so Xt(u�, u�̄) ⊆ Xt(u�, u�̂). We will now see an explicit example where these inclusions are strict.

Example. Let u� be the category of topological spaces 푋 such that the set of points of 푋 is hereditarily 휅-small. It is straightforward to check that the hypotheses of proposition 2.3.14(b) are satisfied, so each (u�, u�̄)- extent in u� is isomorphic to an object in u�. On the other hand, consider the unit circle in the complex plane:

푆1 = {푧 ∈ ℂ | |푧| = 1}

172 3.4. Topological spaces

1 1 Let 푅 = ℤ × 푆 and let 푑0, 푑1 : 푅 → 푆 be defined as follows:

푑0(푛, 푧) = exp(푖푛)푧

푑1(푛, 푧) = 푧

1 Then 푑0, 푑1 : 푅 → 푆 are local homeomorphisms of topological spaces. 1 Moreover, (푅, 푑0, 푑1) is an equivalence relation on 푆 , but there does not 1 exist a local homeomorphism 푓 : 푆 → 푌 such that 푓 ∘ 푑0 = 푓 ∘ not 푑1: indeed, (푅, 푑0, 푑1) is a tractable equivalence relation. (Recall lemma 2.2.14(a).) Nonetheless, assuming 푅 and 푆1 are objects in u�, an

exact quotient of (푅, 푑0, 푑1) exists in u�, and lemma 2.2.8(b) says that it is a (u�, u�̂)-extent; but by the preceding discussion, it is not a (u�, u�̄)-extent. In this context, it is also worth noting that a (u�, u�̄)-extent is the same thing as a u�̂-localic (u�, u�̂)-extent. In other words, a (u�, u�̂)-extent is a (u�, u�̄)-extent precisely when it has enough open subobjects.

3.4.9 Example. Let u� be the category of Hausdorff spaces 푋 such that the set of points of 푋 is hereditarily 휅-small. Then, by theorem 3.4.7, the essential image of |−| : Xt(u�, u�̄) → Top is spanned by the locally Hausdorff spaces 푋 such that the set of points of 푋 is 휅-small. In particular, since there are locally Hausdorff spaces that are not Hausdorff spaces, assuming ̄ 휅 > ℵ0, we have a (u�, u�)-extent that is not isomorphic to any object in u�.

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3.5 Schemes

Synopsis. We see two prima facie different ways of defining schemes as extents in a gros pretopos and show that they are the same.

Prerequisites. §§1.1, 1.2, 2.2, 2.3, 2.5, a.1, a.2, a.3.

3.5.1 ※ Throughout this section, rings and algebras are associative, unital, and commutative.

3.5.2 ¶ Modern algebraic geometry begins with the observation that the opposite of the category of rings is like a category of spaces. Indeed, we will see

how to equip it with the structure of an étale finitary (i.e. ℵ0-ary) extensive ecumene in two different ways—one that starts from the notion that the formal dual of a principal localisation of a ring is analogous to a basic open subspace, and another that starts from the notion that the formal dual of a flat ring homomorphism of finite presentation is analogous toan open map. The associated gros pretoposes are also different, but we will see that schemes can be defined as objects obtained by gluing together the formal duals of rings along local homeomorphisms in either pretopos.

3.5.3 ¶ To begin, we briefly recall some commutative algebra.

Proposition. CRingop is a finitary extensive category.

Proof. Omitted. (Use the Chinese remainder theorem.) ◊

3.5.4 Definition. A ring homomorphism 푓 : 퐴 → 퐵 is flat if 퐵 is flat as an

퐴-module, i.e. 퐵 ⊗퐴 (−) preserves injective homomorphisms of modules.

Properties of Proposition. flat ring homo- (i) Every ring isomorphism is flat. morphisms (ii) Every principal localisation is flat.

(iii) For every ring 퐴, the unique homomorphism 퐴 → {0} is flat. For every ring , the diagonal is flat. (iv) 퐴 Δ퐴 : 퐴 → 퐴 × 퐴

174 3.5. Schemes

(v) The class of flat ring homomorphisms is a coquadrable class of morphisms in CRing.

(vi) The class of flat ring homomorphisms is closed under composition.

(vii) The class of flat ring homomorphisms is closed under finitary product in CRing.

Proof. Straightforward. ⧫

3.5.5 Definition. A ring homomorphism 푓 : 퐴 → 퐵 is faithfully flat if it has the following properties:

• 푓 : 퐴 → 퐵 is flat. • For every pushout square in CRing of the form below,

퐴 퐴′ 푓 푓 ′ 퐵 퐵′

푓 ′ : 퐴′ → 퐵′ is an injective homomorphism of rings.

Recognition Lemma. Let 푓 : 퐴 → 퐵 be a ring homomorphism. The following are principles for equivalent: faithfully flat ring (i) is faithfully flat. homomorphisms 푓 : 퐴 → 퐵 (ii) 푓 : 퐴 → 퐵 is flat and, for every prime ideal 픭 of 퐴, there is a prime ideal 픮 of 퐵 such that 푓 −1픮 = 픭. is flat and reflects isomorphisms of modules. (iii) 푓 : 퐴 → 퐵 퐵 ⊗퐴 (−)

Proof. (i) ⇒ (ii). Let 퐹 be the fraction field of 퐴/픭 and consider the following pushout square in CRing:

퐴 퐹 푓 푓 ′

퐵 퐹 ⊗퐴 퐵

′ By hypothesis, 푓 : 퐹 → 퐹 ⊗퐴 퐵 is injective, so 퐹 ⊗퐴 퐵 is non-zero. In

particular, 퐹 ⊗퐴 퐵 has a prime ideal. Let 픮 be its preimage in 퐵. Then 푓 −1픮 = 픭, as desired.

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(ii) ⇒ (iii). This is well known: see e.g. tag 00HQ in [Stacks].

(iii) ⇒ (i). Consider a pushout square in CRing of the form below:

퐴 퐴′ 푓 푓 ′ 퐵 퐵′

′ ′ ′ We wish to show that 푓 : 퐴 → 퐵 is injective. Since 퐵⊗퐴 (−) preserves ′ kernels and reflects isomorphisms, it suﬃces to verify that id퐵 ⊗퐴 푓 : ′ ′ 퐵 ⊗퐴 퐴 → 퐵 ⊗퐴 퐵 is injective. But the following is also a pushout square in CRing,

′ 퐵 퐵 ⊗퐴 퐴 ′ id퐵⊗퐴푓 ′ 퐵 ⊗퐴 퐵 퐵 ⊗퐴 퐵

′ and 퐵 → 퐵 ⊗퐴 퐵 is a split monomorphism in CRing, so id퐵 ⊗퐴 푓 : ′ ′ 퐵 ⊗퐴 퐴 → 퐵 ⊗퐴 퐵 is indeed injective. ■

Remark. In the lemma above, one can also prove (i) ⇒ (iii) directly, thereby avoiding the use of the prime ideal theorem.

Properties of Proposition. faithfully flat ring (i) Every flat split monomorphism in CRing is faithfully flat. homomorphisms (ii) The class of faithfully flat ring homomorphisms is a coquadrable class of morphisms in CRing.

(iii) The class of faithfully flat ring homomorphisms is closed under composition.

(iv) The class of faithfully flat ring homomorphisms is closed under finitary product in CRing.

Proof. Straightforward. (Recall proposition 3.5.4.) ⧫

3.5.6 Definition. A ring homomorphism 푓 : 퐴 → 퐵 is of finite presentation if 퐵 is finitely presentable as an 퐴-algebra.

176 3.5. Schemes

Properties of ring Proposition. homomorph- (i) Every ring isomorphism is of finite presentation. isms of finite (ii) Every principal localisation is of finite presentation. presentation (iii) For every ring 퐴, the unique homomorphism 퐴 → {0} is of finite presentation. For every ring , the diagonal is of finite present- (iv) 퐴 Δ퐴 : 퐴 → 퐴 × 퐴 ation.

(v) The class of ring homomorphisms of finite presentation is a coquadrable class of morphisms in CRing. (vi) The class of ring homomorphisms of finite presentation is closed under composition.

(vii) The class of ring homomorphisms of finite presentation is closed under finitary product in CRing.

(viii) Given ring homomorphisms 푓 : 퐴 → 퐵 and 푔 : 퐵 → 퐶, if both 푓 : 퐴 → 퐵 and 푔 ∘ 푓 : 퐴 → 퐶 are of finite presentation, then 푔 : 퐵 → 퐶 is also of finite presentation.

Proof. Straightforward. ⧫

3.5.7 Definition. A ring homomorphism 푓 : 퐴 → 퐵 is étale if it has the following properties:

• 푓 : 퐴 → 퐵 is a flat ring homomorphism of finite presentation.

• The relative codiagonal ∇푓 : 퐵 ⊗퐴 퐵 → 퐵 is flat.

Recognition Lemma. Let 푓 : 퐴 → 퐵 be a ring homomorphism. The following are principle for equivalent: étale ring homo- (i) is an étale ring homomorphism. morphisms 푓 : 퐴 → 퐵 (ii) 푓 : 퐴 → 퐵 is a flat and unramified ring homomorphism of finite presentation.

Proof. (i) ⇒ (ii). By tag 092M in [Stacks], étale ring homomorphisms (in our sense) are formally unramified, and by tag 00UU in op. cit., a formally unramified ring homomorphism of finite presentation is unramified.

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(ii) ⇒ (i). By tag 08WD in [Stacks], a flat unramified ring homomorphism of finite presentation is étale in their sense, and tag 00U7 in op. cit. implies that an étale ring homomorphism in their sense is étale in our sense. ■

Properties of Proposition. étale ring homo- (i) Every flat epimorphism of finite presentation in CRing is étale. morphisms (ii) In particular, every principal localisation is étale.

(iii) For every ring 퐴, the unique homomorphism 퐴 → {0} is étale. For every ring , the diagonal is étale. (iv) 퐴 Δ퐴 : 퐴 → 퐴 × 퐴 (v) The class of étale ring homomorphisms is a coquadrable class of morphisms in CRing.

(vi) The class of étale ring homomorphisms is closed under composition.

(vii) The class of étale ring homomorphisms is closed under finitary product in CRing.

(viii) Given ring homomorphisms 푓 : 퐴 → 퐵 and 푔 : 퐵 → 퐶, if both 푓 : 퐴 → 퐵 and 푔 ∘ 푓 : 퐴 → 퐶 are étale, then 푔 : 퐵 → 퐶 is also étale.

Proof. Apply lemma 1.1.10 and propositions 3.5.4 and 3.5.6. ■

3.5.8 ¶ We will also need some results from descent theory.

Definition. A ring homomorphism is fppf[1] if it is both faithfully flat and of finite presentation.

3.5.8(a) Proposition. Every faithfully flat ring homomorphism is an effective monomorphism in CRing.

Proof. Let 푓 : 퐴 → 퐵 be a faithfully flat ring homomorphism and let 0 1 푑 , 푑 : 퐵 → 퐵 ⊗퐴 퐵 be defined as follows:

푑0(푏) = 1 ⊗ 푏 푑1(푏) = 푏 ⊗ 1

[1] — from French « fidèlement plat de présentation finie ».

178 3.5. Schemes

We must show that the following is an equaliser diagram in CRing:

0 푓 푑 퐴 퐵 퐵 ⊗퐴 퐵 푑1 By lemma 3.5.5, it is enough to verify that the following is an equaliser diagram in CRing,

1 푑1 푑 퐵 퐵 ⊗퐴 퐵 퐵 ⊗퐴 퐵 ⊗퐴 퐵 푑2

1 2 where 푑 , 푑 : 퐵 ⊗퐴 퐵 → 퐵 ⊗퐴 퐵 ⊗퐴 퐵 are defined as follows:

1 2 푑 (푏0 ⊗ 푏1) = 푏0 ⊗ 1 ⊗ 푏1 푑 (푏0 ⊗ 푏1) = 푏0 ⊗ 푏1 ⊗ 1

In fact, this is a split equaliser diagram, so we are done. ■

3.5.8(b) Proposition. Let 푓 : 퐴 → 퐵 and 푔 : 퐵 → 퐶 be ring homomorphisms. (i) Assuming 푔 : 퐵 → 퐶 is faithfully flat, 푔 ∘ 푓 : 퐴 → 퐶 is flat if and only if 푓 : 퐴 → 퐵 is flat. (ii) Assuming 푔 : 퐵 → 퐶 is fppf, 푔 ∘ 푓 : 퐴 → 퐶 is of finite presentation if and only if 푓 : 퐴 → 퐵 is of finite presentation. (iii) Assuming 푔 : 퐵 → 퐶 is faithfully flat and étale, 푔 ∘ 푓 : 퐴 → 퐶 is étale if and only if 푓 : 퐴 → 퐵 is étale.

Proof. (i). Straightforward. (Use proposition 3.5.4 and lemma 3.5.5.)

(ii). For the ‘if’ direction, see proposition 3.5.6; for the ‘only if’ direction, see tag 02KK in [Stacks].

(iii). For the ‘if’ direction, see proposition 3.5.7; for the ‘only if’ direction, see tag 02K6 in [Stacks]. □

3.5.8(c) Proposition. Consider a pushout square in CRing:

퐴 퐴′ 푓 푓 ′ 퐵 퐵′

Assuming 퐴 → 퐴′ is faithfully flat: (i) 푓 ′ : 퐴′ → 퐵′ is flat if and only if 푓 : 퐴 → 퐵 is flat.

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(ii) 푓 ′ : 퐴′ → 퐵′ is of finite presentation if and only if 푓 : 퐴 → 퐵 is of finite presentation.

(iii) 푓 ′ : 퐴′ → 퐵′ is étale if and only if 푓 : 퐴 → 퐵 is étale.

Proof. (i). By proposition 3.5.4, if 푓 : 퐴 → 퐵 is flat, then 푓 ′ : 퐴′ → 퐵′ is also flat. Conversely, by proposition 3.5.5, 퐵 → 퐵′ is faithfully flat, and if 푓 ′ : 퐴′ → 퐵′ is flat, then the composite 퐴 → 퐵 → 퐵′ is also flat, so by proposition 3.5.8(b), 푓 : 퐴 → 퐵 is indeed flat. (ii). For the ‘if’ direction, see proposition 3.5.6; for the ‘only if’ direction, see tag 00QQ in [Stacks].

(iii). By proposition 3.5.7, if 푓 : 퐴 → 퐵 is étale, then 푓 ′ : 퐴′ → 퐵′ is étale. For the converse, suppose 푓 ′ : 퐴′ → 퐵′ is étale. By (i) and (ii), 푓 : 퐴 → 퐵 is flat and of finite presentation. Moreover, we have the following commutative diagram in CRing,

퐵 퐵′

′ ′ 퐵 ⊗퐴 퐵 퐵 ⊗퐴′ 퐵

∇푓 ∇푓′ 퐵 퐵′ where every square is a pushout square in CRing and every horizontal arrow is a faithfully flat ring homomorphism, so the relative codiagonal

∇푓 : 퐵 ⊗퐴 퐵 → 퐵 is flat. Thus, 푓 : 퐴 → 퐵 is indeed étale. □

3.5.9 Lemma. Let 퐴 be a ring, let 퐼 be a finite subset of 퐴, and let 퐴̃ = −1 . The following are equivalent: ∏푎∈퐼 퐴[푎 ] (i) 퐼 generates the unit ideal of 퐴.

(ii) The induced homomorphism 퐴 → 퐴̃ is faithfully flat.

(iii) There is a ring homomorphism 퐴̃ → 퐵 such that the composite 퐴 → 퐴̃ → 퐵 is faithfully flat.

Proof. (i) ⇒ (ii). By proposition 3.5.4, 퐴 → 퐴̃ is flat. We will now show that 퐴 → 퐴̃ is injective. Since 퐼 generates the unit ideal of 퐴, there

180 3.5. Schemes

exist elements 푟0, … , 푟푛−1 of 퐴 and elements 푎0, … , 푎푛−1 of 퐼 such that

푟0푎0 + ⋯ + 푟푛−1푎푛−1 = 1. Suppose 푠 is an element of 퐴 such that 푠 = 0 −1 in each 퐴[푎푖 ], i.e. for 0 ≤ 푖 < 푛, there is a natural number 푘푖 such that 푘푖 푎푖 푠 = 0 in 퐴; hence, for 푘 = 푘0 + ⋯ + 푘푛−1:

푘 푠 = (푟0푎0 + ⋯ + 푟푛−1푎푛−1) 푠 = 0

Thus 퐴 → 퐴̃ is indeed injective. The same argument shows that 퐵 → −1 ∏푎∈퐴 퐵[푓(푎) ] is injective for every ring homomorphism 푓 : 퐴 → 퐵, so 퐴 → 퐴̃ is indeed faithfully flat.

(ii) ⇒ (iii). Immediate.

(iii) ⇒ (i). Let 픞 be the ideal of 퐴 generated by 퐼 and let 픟 be the ideal of 퐵 generated by the image of 퐼. We have the following pushout square in CRing: 퐴 퐴/픞

퐵 퐵/픟

By hypothesis, 퐴/픞 → 퐵/픟 is injective. On the other hand, 픟 is the unit ideal of 퐵, so 퐴/픞 ≅ {0}. Hence, 픞 is the unit ideal of 퐴. ■

3.5.10 ¶ The following is a deep result in algebraic geometry.

Flat morphisms Proposition. Let 푓 : 퐴 → 퐵 be a flat ring homomorphism of finite have open image presentation and let 퐼 be the subset of 퐴 defined as follows: • 푎 ∈ 퐼 if and only if the induced homomorphism 퐴[푎−1] → 퐵[푓(푎)−1] is fppf.

Then {푓(푎) | 푎 ∈ 퐼} generates the unit ideal of 퐵.

Proof. In the language of algebraic geometry, this is the statement that a flat morphism (of schemes) of finite presentation has an open image.See tag 01UA in [Stacks]. □

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3.5.11 Definition. An open quasilocalisation is an epimorphism 푓 : 퐴 → 퐵 in CRing for which there is a (finite) subset 퐼 ⊆ 퐴 with the following properties:

• For every 푎 ∈ 퐼, the induced homomorphism 퐴[푎−1] → 퐵[푓(푎)−1] is an isomorphism.

• {푓(푎) | 푎 ∈ 퐼} generates the unit ideal of 퐵.

Recognition prin- Lemma. Let 푓 : 퐴 → 퐵 be a ring homomorphism. The following are ciples for open equivalent: quasilocalisations (i) 푓 : 퐴 → 퐵 is an open quasilocalisation.

(ii) 푓 : 퐴 → 퐵 is an epimorphism in CRing, flat, and of finite presentation.

(iii) 푓 : 퐴 → 퐵 is an epimorphism in CRing and étale.

Proof. (i) ⇒ (ii). By propositions 3.5.4 and 3.5.6 and lemma 3.5.9, there is a finite subset 퐼 ⊆ 퐴 with the following properties:

−1 • The induced homomorphism 퐵 → ∏푎∈퐼 퐵[푎 ] is fppf.

−1 • The composite 퐴 → 퐵 → ∏푎∈퐼 퐵[푎 ] is flat and of finite presentation.

Thus, by proposition 3.5.8(b), 푓 : 퐴 → 퐵 is flat and of finite presentation. But open quasilocalisations are epimorphisms in CRing by definition, so we are done.

(ii) ⇒ (iii). Apply proposition 3.5.7.

(iii) ⇒ (i). By proposition 3.5.10, there is a finite subset 퐼 ⊆ 퐴 with the following properties:

• For every 푎 ∈ 퐼, the induced homomorphism 퐴[푎−1] → 퐵[푓(푎)−1] is faithfully flat.

• {푓(푎) | 푎 ∈ 퐼} generates the unit ideal of 퐵.

182 3.5. Schemes

But proposition 3.5.8(a) implies that every faithfully flat epimorphism in CRing is an isomorphism, so 푓 : 퐴 → 퐵 is indeed an open quasilocalisation. ■

Properties of Proposition. open quasi- (i) Every ring isomorphism is an open quasilocalisation. localisations (ii) Every principal localisation is an open quasilocalisation.

(iii) The class of open quasilocalisations is a coquadrable class of morphisms in CRing.

(iv) The class of open quasilocalisations is closed under composition.

(v) The class of open quasilocalisations is closed under finitary product in CRing.

(vi) Given ring homomorphisms 푓 : 퐴 → 퐵 and 푔 : 퐵 → 퐶, if both 푓 : 퐴 → 퐵 and 푔 ∘ 푓 : 퐴 → 퐶 are open quasilocalisations, then 푔 : 퐵 → 퐶 is also an open quasilocalisation.

Proof. By lemma 3.5.11, these reduce to propositions 3.5.3 and 3.5.7. ■

3.5.12 ※ For the remainder of this section:

• 퐾 is a ring.

• u� is a (small) full subcategory of the metacategory of 퐾-algebras that is closed under finitary coproducts, coequalisers, and finitary products.

• For every object 퐴 in u� and every element 푎 ∈ 퐴, the principal localisation 퐴[푎−1] is also an object in u�.

3.5.13 ¶ To avoid confusion, we write Spec 퐴 for the object in u�op corresponding to an object 퐴 in u�, and we write Spec 푓 : Spec 퐵 → Spec 퐴 for the morphism in u�op corresponding to a morphism 푓 : 퐴 → 퐵 in u�. The four classical Grothendieck topologies are defined as follows:

183 Specificities

3.5.13(a) Definition. The Zariski coverage on u�op consists of all sinks of the form below up to isomorphism,

−1 {Spec 푓푎 : Spec 퐴[푎 ] → Spec 퐴 | 푎 ∈ 퐼}

where 퐴 is an object in u�, 퐼 is a finite subset of 퐴 that generates the unit −1 ideal, and 푓푎 : 퐴 → 퐴[푎 ] is the principal localisation.

3.5.13(b) Definition. The étale coverage on u�op is the smallest composition- closed coverage on u�op containing the Zariski coverage as well as the singleton {Spec 푓 : Spec 퐵 → Spec 퐴} for every faithfully flat étale ring homomorphism 푓 : 퐴 → 퐵 in u�.

3.5.13(c) Definition. The fppf coverage on u�op is the smallest composition- closed coverage on u�op containing the Zariski coverage as well as the singleton {Spec 푓 : Spec 퐵 → Spec 퐴} for every fppf ring homomorphism 푓 : 퐴 → 퐵 in u�.

[2] 3.5.13(d) Definition. The fpqc coverage on u�op is the smallest composition- closed coverage on u�op containing the Zariski coverage as well as the singleton {Spec 푓 : Spec 퐵 → Spec 퐴} for every faithfully flat ring homomorphism 푓 : 퐴 → 퐵 in u�.

Properties of Proposition. the classical (i) The Zariski (resp. étale, fppf, fpqc) coverage on u�op is finitary Grothendieck superextensive. topologies (ii) The Zariski (resp. étale, fppf, fpqc) coverage on u�op is subcanonical.

Proof. (i). It suﬃces to verify that the Zariski coverage on u�op is finitary superextensive. Let 퐴 and 퐵 be rings and consider the projections 퐴 × 퐵 → 퐴 and 퐴 × 퐵 → 퐵. It is straightforward to verify that 퐴 × 퐵 → 퐴 is isomorphic to the principal localisation 퐴 × 퐵 → (퐴 × 퐵)[(1, 0)−1]. Similarly, 퐴×퐵 → 퐵 is isomorphic to the principal localisation 퐴×퐵 → (퐴 × 퐵)[(0, 1)−1]. Hence, every finite coproduct cocone in u�op is in the Zariski coverage. [2] — from French « fidèlement plat et quasi-compacte »

184 3.5. Schemes

(ii). It suﬃces to verify that the fpqc coverage on u�op is subcanonical. Since the fpqc coverage is finitary superextensive, by lemmas 1.4.18, 1.5.15 and 1.5.16, it is enough to show that every faithfully flat ring homomorphism in u� is an effective monomorphism in u�. Since the forgetful functor u� → CRing creates cokernel pairs, this reduces to proposition 3.5.8(a). ■

3.5.14 Definition. A morphism ℎ : 푋 → 푌 in Psh(u�op) is étale if both ℎ :

푋 → 푌 and the relative diagonal Δℎ : 푋 → 푋 ×푌 푋 are fppf-semilocally of u�-type, where u� is the opposite of the class of flat ring homomorphisms of finite presentation in u�.

Remark. Since u� is a quadrable class of morphisms in u�op (by proposition 3.5.6), by lemma 1.2.15, every morphism in Psh(u�op) fppf- semilocally of u�-type is also fppf-locally of u�-type.

Properties of Proposition. étale morphisms (i) Every fppf-locally bijective morphism in Psh(u�op) is étale. of presheaves (ii) Forevery presheaf 푋 on u�op and every set 퐼, the codiagonal morphism is étale. ∐푖∈퐼 푋 → 푋 (iii) The class of étale morphisms in Psh(u�op) is a quadrable class of morphisms in Psh(u�op).

(iv) The class of étale morphisms in Psh(u�op) is closed under composition.

(v) The class of étale morphisms in Psh(u�op) is closed under (possibly infinitary) coproduct in Psh(u�op).

(vi) Given a pullback square in Psh(u�op) of the form below,

푋̃ 푋

ℎ̃ ℎ 푌̃ 푌

where 푌̃ ↠ 푌 is fppf-locally surjective, if ℎ̃ : 푋̃ → 푌̃ is étale, then ℎ : 푋 → 푌 is also étale.

185 Specificities

(vii) Given morphisms ℎ : 푋 → 푌 and 푘 : 푌 → 푍 in Psh(u�op), if both 푘 : 푌 → 푍 and 푘 ∘ ℎ : 푋 → 푍 are étale, then ℎ : 푋 → 푌 is also étale.

(viii) Given a fppf-locally surjective morphism ℎ : 푋 ↠ 푌 in Psh(u�op) and a morphism 푘 : 푌 → 푍 in Psh(u�op), if both ℎ : 푋 ↠ 푌 and 푘 ∘ ℎ : 푋 → 푍 are étale, then 푘 : 푌 → 푍 is also étale.

(ix) For every morphism 푓 : 퐴 → 퐵 in u�, h푓 : h퐴 → h퐵 is étale if and only if 푓 : 퐴 → 퐵 is étale.

Proof. (i)–(viii). Combine proposition 1.1.11 and remark 1.2.16 with propositions 1.2.14 and 1.2.17.

(ix). Apply proposition 1.2.20 to propositions 3.5.8(b) and 3.5.8(c). ■

3.5.15 Definition. An open immersion of presheaves on u�op is a monomorphism ℎ : 푋 → 푌 in Psh(u�op) that is Zariski-semilocally of ℬ-type, where ℬ is the opposite of the class of morphisms in u� that are principal localisations up to isomorphism.

Remark. It is clear that ℬ is a quadrable class of morphisms in u�op. Thus, by lemma 1.2.15, any morphism that is Zariski-semilocally of ℬ- type is also Zariski-locally of ℬ-type.

Recognition Lemma. Let 퐴 be an object in u� and let 푈 be an fppf-closed subpresheaf principles for of h퐴. The following are equivalent: open fppf-closed 퐴 subpresheaves (i) The inclusion 푈 ↪ h is an open immersion.

(ii) The inclusion 푈 ↪ h퐴 is an étale monomorphism.

(iii) There is a (possibly infinite) set Φ of elements of 푈 with the following properties:

• Φ is a Zariski-local generating set of elements of 푈. • For every (Spec 퐵, Spec 푓) ∈ Φ, (퐵, 푓) is isomorphic (in 퐴∕u�) to a principal localisation of 퐴.

186 3.5. Schemes

Proof. (i) ⇒ (ii). Since every principal localisation is an étale ring homomorphism (proposition 3.5.7) and Zariski-locally surjective morphisms are fppf-locally surjective, open immersions are étale.

−1 (ii) ⇒ (iii). Let 퐼 be the set of all 푎 ∈ 퐴 such that (Spec 퐴[푎 ], Spec 푓푎) −1 is an element of 푈, where 푓푎 : 퐴 → 퐴[푎 ] is the principal localisation, −1 and let Φ = {(Spec 퐴[푎 ], Spec 푓푎) | 푎 ∈ 퐼}. Clearly, it is enough to prove that Φ is a Zariski-local generating set of elements of 푈. By lemma 1.5.16 and propositions 3.5.4, 3.5.6, and 3.5.13, there exist an fppf ring homomorphism 푗 : 퐴 → 퐴̃ in u� and a set Φ̃ of objects in ̃ 퐴∕u� with the following properties:

̃ • Φ̃ is an fppf-local generating set of elements of the preimage 푈̃ ⊆ h퐴 of 푈 ⊆ h퐴.

• For every (퐵,̃ 푓)̃ ∈ Φ̃ , 푓̃ : 퐴̃ → 퐵̃ is flat and of finite presentation.

Note that 푓̃ ∘ 푗 : 퐴 → 퐵̃ is flat and of finite presentation, soby proposition 3.5.10, there is a finite subset 퐼′ ⊆ 퐴 with the following properties:

′ −1 −1 • For every 푎 ∈ 퐼 , the induced homomorphism 퐴[푎 ] → 퐵̃[푓(̃ 푗(푎)) ] is fppf.

′ • {푓(̃ 푗(푎)) | 푎 ∈ 퐼 } generates the unit ideal of 퐵̃.

Since (Spec 퐵,̃ Spec(푓̃ ∘ 푗)) is an element of 푈 and 푈 is an fppf-closed subpresheaf of h퐴, it follows that 퐼′ ⊆ 퐼. On the other hand, by proposition a.2.14, the morphism 푈̃ → 푈 is fppf-locally surjective, so it follows that Φ is an fppf-local generating set of elements of 푈. Hence, by lemma 3.5.9, Φ is also a Zariski-local generating set of elements of 푈.

(iii) ⇒ (i). Immediate. ■

Properties of Proposition. open immersions (i) Every isomorphism in Psh(u�op) is an open immersion. of presheaves (ii) Every open immersion in Psh(u�op) is étale. (iii) The class of open immersions in Psh(u�op) is a quadrable class of morphisms in Psh(u�op).

187 Specificities

(iv) The class of open immersions in Psh(u�op) is closed under composition.

(v) The class of open immersions in Psh(u�op) is closed under (possibly infinitary) coproduct in Psh(u�op).

(vi) Given a pullback square in Psh(u�op) of the form below,

푋̃ 푋

ℎ̃ ℎ 푌̃ 푌

where 푌̃ ↠ 푌 is Zariski-locally surjective, if ℎ̃ : 푋̃ → 푌̃ is an open immersion and ℎ : 푋 → 푌 is a monomorphism, then ℎ : 푋 → 푌 is also an open immersion.

(vii) Given morphisms ℎ : 푋 → 푌 and 푘 : 푌 → 푍 in Psh(u�op), if both 푘 : 푌 → 푍 and 푘 ∘ ℎ : 푋 → 푍 are open immersions, then ℎ : 푋 → 푌 is also an open immersion.

(viii) For every morphism 푓 : 퐴 → 퐵 in u�, h푓 : h퐴 → h퐵 is an open immersion if and only if 푓 : 퐴 → 퐵 is an open quasilocalisation.

Proof. (i)–(vi). Apply proposition 1.2.14.

(vii). This is a special case of lemma 1.1.3.

(viii). Suppose 푓 : 퐴 → 퐵 is an open quasilocalisation. Then, by definition, Spec 푓 : Spec 퐵 → Spec 퐴 is monomorphism in u�op that is Zariski- locally of ℬ-type, where ℬ is the opposite of the class of principal localisations, so h푓 : h퐵 → h퐴 is indeed an open immersion. For the converse, suppose h푓 : h퐵 → h퐴 is an open immersion. Since the fppf coverage on u�op is subcanonical (proposition 3.5.13), the presheaf image is an fppf-closed subpresheaf of h퐴. Thus, we may apply lemma 3.5.15 to deduce that 푓 : 퐴 → 퐵 is an open quasilocalisation. ■

188 3.5. Schemes

3.5.16 Definition.A local isomorphism of presheaves on u�op is a morphism ℎ : 푋 → 푌 in Psh(u�op) for which there is a set Φ of subpresheaves of 푋 with the following properties:

• ⋃푈∈Φ 푈 is a Zariski-dense subpresheaf of 푋. • For every 푈 ∈ Φ, both the inclusion 푈 ↪ 푋 and the composite 푈 ↪ 푋 → 푌 are open immersions in Psh(u�op).

Properties of Proposition. local isomorph- (i) Every open immersion in Psh(u�op) is a local isomorphism. isms of pre- Every local isomorphism in op is étale. sheaves (ii) Psh(u� ) (iii) A monomorphism in Psh(u�op) is an open immersion if and only if it is a local isomorphism.

(iv) Forevery presheaf 푋 on u�op and every set 퐼, the codiagonal morphism is a local isomorphism. ∐푖∈퐼 푋 → 푋 (v) The class of local isomorphisms in Psh(u�op) is a quadrable class of morphisms in Psh(u�op).

(vi) The class of local isomorphisms in Psh(u�op) is closed under composition.

(vii) The class of local isomorphisms in Psh(u�op) is closed under (possibly infinitary) coproduct in Psh(u�op).

(viii) Given morphisms ℎ : 푋 → 푌 and 푘 : 푌 → 푍 in Psh(u�op), if both 푘 : 푌 → 푍 and 푘 ∘ ℎ : 푋 → 푍 are local isomorphisms, then ℎ : 푋 → 푌 is also a local isomorphism.

Proof. Straightforward given proposition 3.5.15. (Compare the proof of proposition 2.2.12.) ⧫

3.5.17 ¶ Every Zariski-locally surjective morphism in Psh(u�op) is also fpqc- locally surjective. The converse is not true in general, but we do have the following result.

189 Specificities

Recognition prin- Lemma. Let ℎ : 푋 → 푌 be a local isomorphism in Psh(u�op). The ciple for locally following are equivalent: surjective local (i) ℎ : 푋 → 푌 is Zariski-locally surjective. isomorphisms of presheaves (ii) ℎ : 푋 → 푌 is fpqc-locally surjective.

Proof. (i) ⇒ (ii). Immediate.

(ii) ⇒ (i). In view of propositions 3.5.16 and a.2.14, it is enough to verify the case where 푌 = h퐴 for some object 퐴 in u�. Let Φ be a set of subpresheaves of 푋 with the following properties:

• ⋃푈∈Φ 푈 is a Zariski-dense subpresheaf of 푋. • For every 푈 ∈ Φ, both the inclusion 푈 ↪ 푋 and the composite 푈 ↪ 푋 → 푌 are open immersions in Psh(u�op).

Moreover, recalling proposition 3.5.15, we may assume that each 푈 ∈ Φ can be represented by some principal localisation of 퐴. This yields an fpqc-covering sink on Spec 퐴. Since every fpqc-covering sink in u�op contains a finite fpqc-covering sink, 푋 → 푌 is indeed Zariski-locally surjective, by lemma 3.5.9. ■

3.5.18 ※ For the remainder of this section:

• 휅 is a regular cardinal.

ℵ0 op • u� = Fam휅 (u� ). ′ • For each object 푋 in u�, 햩 (푋) (resp. 햩햿 (푋)) is the set of 휅-small sinks

Φ on 푋 in u� such that the induced morphism ∐(푈,푥)∈Φ 푈 → 푋 corresponds to a Zariski-locally surjective local isomorphism (resp. fppf- locally surjective morphism) in Psh(u�op).

′ • 햤 , (resp. 햤햿 ) is the class of morphisms in u� that correspond to Zariski- locally (resp. fppf-locally) surjective morphisms in Psh(u�op).

′ ′ • u� = Ex(u�, 햤 ) and u�햿 = Ex(u�, 햤햿 ). • u� (resp. u�′) is the class of morphisms that correspond to étale morphisms (resp. local isomorphisms) in Psh(u�op).

190 3.5. Schemes

′ 3.5.19 ¶ We will consider both u� and u�햿 as settings for algebraic geometry. First, we need to establish some basic properties.

3.5.19(a) Proposition. (i) u� is a 휅-ary extensive category with limits of finite diagrams. In particular, both ′ and satisfy the Shulman condi- (ii) (u�, 햤 ) (u�, 햤햿 ) tion. Both ′ and are -ary pretoposes. (iii) u� u�햿 휅

Proof. (i). This is a special case of proposition 1.5.21.

(ii). Immediate.

(iii). Apply proposition 1.5.13. ■

′ ′ 3.5.19(b) Proposition. (u�, u� , 햩 ) is a 휅-ary admissible ecumene, i.e.: (i) u�′ is a class of separated fibrations in u�. (ii) 햩′ is a subcanonical u�′-adapted 휅-ary superextensive coverage on u�. ′ ′ ′ (iii) Every morphism in u� of u� -type (u� , 햩 )-semilocally on the domain is a member of u�′. (iv) Every complemented monomorphism in u� is a member of u�′. Moreover:

′ ′ (v) (u� , u� ) is the associated gros 휅-ary pretopos.

Proof. (i) and (iv). See proposition 3.5.16.

(ii). The Zariski coverage on u�op is subcanonical (proposition 3.5.13), so 햩′ is a subcanonical coverage on u�, which is 휅-ary superextensive by proposition 1.5.17 and u�′-adapted by construction.

(iii). It suﬃces to verify the following:

• Given an Zariski-locally surjective morphism ℎ : 푋 → 푌 in Psh(u�op) and a morphism 푘 : 푌 → 푍 in Psh(u�op), if both ℎ : 푋 → 푌 and 푘 ∘ ℎ : 푋 → 푍 are local isomorphisms, then 푘 : 푌 → 푍 is also a local isomorphism.

191 Specificities

This is straightforward. (Compare the proof of proposition 2.2.13.)

(v). It is not hard to see that morphisms in u�′ that correspond to morphisms in Psh(u�) that are 햩′-locally of u�′-type are the same as morphisms in u�′ that correspond to morphisms in Psh(u�op) that are Zariski-locally of u�′-type. Moreover, by lemma 2.1.16, u� is equivalent to the category ′ ′ op ′ of 햩 -locally 휅-presentable 햩 -sheaves on u� . Thus, (u�, u� ) is indeed ′ ′ the 휅-ary gros pretopos associated with (u�, u� , 햩 ). ⧫

3.5.19(c) Proposition. is a class of étale morphisms in . (i) u� u�햿 ′ is the class of local homeomorphisms in . (ii) u� u�햿 is a unary basis for . (iii) u� (u�햿 , u�) is a unary basis for ′ . (iv) u� (u�햿 , u� )

Proof. (i). See proposition 3.5.14.

(ii). By lemma 3.5.15, étale monomorphisms in u�햿 correspond to open immersions in Psh(u�op), because the presheaf image of a monomorphism of fppf-sheaves is always fppf-closed. Since fppf-locally surjective morphisms are also fpqc-locally surjective, and Zariski-locally surjective morphisms are also fppf-locally surjective, we may apply proposition 3.5.16 and lemma 3.5.17 to deduce that local homeomorphisms in op u�햿 correspond to local isomorphisms in Psh(u� ). (iii). Use proposition 2.3.2.

(iv). In view of lemma 2.1.16, it suﬃces to verify the following:

• For every object 푌 in u� and every fppf-subsheaf 푉 ⊆ h푌 , if 푉 is fppf-

locally 휅-presentable and the inclusion 푈 ↪ h푌 is an open immersion, then there exist an object 푋 in u� and a fppf-locally surjective morph-

ism 푝 : h푋 ↠ 푈 such that the composite h푋 ↠ 푈 ↪ h푌 is a local isomorphism.

This is a straightforward consequence of lemma 3.5.15. ■

192 3.5. Schemes

′ 3.5.20 ¶ With the usual abuses of notation, we may speak of (u�, u� )-extents in ′ prima facie both u� and u�햿 . Although these are different notions , fortu- nately, they coincide. To prove this, we will require a preliminary result.

Lemma. Let 퐹 be a presheaf on u�op. The following are equivalent: (i) There exist an object 푌 in u� and a laminar morphism 푋 → 푌 in ′ such that . u� h푋 ≅ 퐹 (ii) There exist an object 푌 in u� and a laminar morphism 푋 → 푌 in such that . u�햿 h푋 ≅ 퐹

Proof. Let 햪 be the finitary extensive coverage on u�op. We have the ′ op op Yoneda representations u� → Sh(u� , 햪) and u�햿 → Sh(u� , 햪), and by proposition 1.5.18 and theorem 2.1.14, both preserve 휅-ary coproducts. Thus, it suﬃces to prove the claim with ‘laminar morphism’ replaced with ‘open embedding’.

(i) ⇒ (ii). Let 푓 : 푋 → 푌 be an open embedding in u�′ where 푌 is an ′ ′ object in u�. Then there is a (u�, u� )-atlas of 푋 in u� , say (푋,̃ 푝). Let

(푅, 푑0, 푑1) be a kernel pair of 푓 ∘ 푝 : 푋̃ → 푌 in u�. Since 푓 : 푋 → 푌 is ′ a monomorphism in u� , (푅, 푑0, 푑1) is also a kernel pair of 푝 : 푋̃ ↠ 푋 in u�′. Moreover, by proposition 3.5.16, 푓 ∘푝 : 푋̃ → 푌 is a member of u�′, so ′ ′ both projections 푑0, 푑1 : 푅 → 푋̃ are also members of u� . Let 푞 : 푋̃ ↠ 푌 ′ be an exact quotient of (푅, 푑0, 푑1) in u�햿 and let 푚 : 푌 → 푌 be the

unique morphism in u�햿 such that 푚 ∘ 푞 = 푓 ∘ 푝. By proposition 3.5.19(c), ′ ′ 푞 : 푋̃ ↠ 푌 is an étale morphism in u�햿 , so 푚 : 푌 → 푌 is also an étale

morphism in u�햿 . On the other hand, since (푅, 푑0, 푑1) is also a kernel pair ′ of 푚∘푞, 푚 : 푌 → 푌 is a monomorphism in u�햿 , so it is an open embedding ′ in u�햿 . Hence, 푞 : 푋̃ ↠ 푌 is a local homeomorphism in u�햿 . But then

lemma 3.5.17 says that h푞 : h푋̃ → h푌 ′ is Zariski-locally surjective, so we

indeed have h푋 ≅ h푌 ′.

(ii) ⇒ (i). Let 푓 : 푋 → 푌 be an open embedding in u�햿 where 푌 is an ′ object in u�. Then there is a (u�, u� )-atlas of 푋 in u�햿 , say (푋,̃ 푝). Let ′ (푅, 푑0, 푑1) be a kernel pair of 푓 ∘ 푝 : 푋̃ → 푌 in u�, let 푞 : 푋̃ → 푌 be ′ ′ an exact quotient of (푅, 푑0, 푑1) in u� , and let 푚 : 푌 → 푌 be the unique

193 Specificities

morphism in u�′ such that 푚 ∘ 푞 = 푓 ∘ 푝. By the same argument as before, 푚 : 푌 ′ → 푌 is an open embedding in u�′. Since 푝 : 푋̃ ↠ 푋 is a member ′ of u� , h푝 : h푋̃ → h푋 is Zariski-locally surjective, so we have h푋 ≅ h푌 ′, as desired. ■

Proposition. Let 퐹 be a presheaf on u�op. The following are equivalent: There is a ′ -extent in ′ such that . (i) (u�, u� ) 푋 u� h푋 ≅ 퐹 There is a ′ -extent in such that . (ii) (u�, u� ) 푋 u�햿 h푋 ≅ 퐹 Proof. ′ For ease of notation, we will identify u� and u�햿 with their respective essential images in Psh(u�op).

′ ′ (i) ⇒ (ii). Let 푋 be a (u�, u� )-extent in u� . By definition, there is a ′ ′ (u�, u� )-atlas of 푋 in u� , say (푋,̃ 푝), and by proposition 2.3.13, there is a laminar effective epimorphism ̃푝: 푈 → 푋̃ in u�′ such that 푝 ∘ ̃푝: 푈 → 푋 ′ is also a laminar effective epimorphism in u� . Let (푅, 푑0, 푑1) be a kernel ′ pair of 푝 ∘ ̃푝: 푈 → 푋 in u� . Then the projections 푑0, 푑1 : 푅 → 푈 are laminar morphisms in u�′, so by lemma 3.5.20, both 푅 and 푈 are

also objects in u�햿 . Consider exact quotients of (푅, 푑0, 푑1) in u�햿 . By ′ lemma 2.2.14(c), (푅, 푑0, 푑1) is a tractable equivalence relation in u� , and

it follows that (푅, 푑0, 푑1) is also a tractable equivalence relation in u�햿 .

Thus, by lemma 3.5.17, any exact quotient of (푅, 푑0, 푑1) in u�햿 is also an ′ exact quotient of (푅, 푑0, 푑1) in u� , and therefore 푋 is in u�햿 , as desired. (ii) ⇒ (i). The same argument (mutatis mutandis) works. ■

′ ′ Remark. In other words, the essential images of Xt(u�, u� ) ⊆ u� and ′ op Xt(u�, u� ) ⊆ u�햿 in Psh(u� ) coincide. Thus, up to equivalence, there is ′ no ambiguity in the notation Xt(u�, u� ).

′ 3.5.21 ¶ It is more or less clear how to connect our definition of ‘(u�, u� )-extent’ with the functor-of-points definition of ‘scheme’ found in e.g. Demazure and Gabriel [1970], which is known to be equivalent to the definition of ‘scheme’ in terms of locally ringed spaces. We will take this for granted and instead focus on making a more precise statement about the kind of ′ schemes that can be obtained as (u�, u� )-extents.

194 3.5. Schemes

3.5.21(a) Definition. An u�-atlas of a 퐾-scheme 푋 is a set Φ with the following properties:

• Every element of Φ is an open subscheme of 푋. • For every 푈 ∈ Φ, 푈 is isomorphic to the aﬃne 퐾-scheme corresponding to some object in u�.

• 푋 = ⋃푈∈Φ 푈.

3.5.21(b) Definition.A 퐾-scheme 푋 is of (u�, 휅)-type if there is a set Φ with the following properties:

• Φ is a 휅-small u�-atlas.

• For every (푈0, 푈1) ∈ Φ × Φ, 푈0 ∩ 푈1 admits a 휅-small u�-atlas.

Theorem. Let ℳ be the essential image in Psh(u�op) of the metacategory of 퐾-schemes of (u�, 휅)-type. Then ℳ is also the essential image ′ of Xt(u�, u� ).

Proof. Omitted. (Compare the proof of theorem 2.4.13.) ◊

[3] 3.5.22 Example. Let u� be the category of finitely presented 퐾-algebras. It is straightforward to verify the following:

(i) A 퐾-scheme admits an u�-atlas if and only if it is locally of finite presentation.

(ii) A 퐾-scheme admits a finite u�-atlas if and only if it is locally of finite presentation and quasicompact.

(iii) A 퐾-scheme is of (u�, ℵ0)-type if and only if it is of finite presentation, i.e. locally of finite presentation, quasicompact, and quasisepar- ated.

′ In particular, if 휅 = ℵ0, then Xt(u�, u� ) is equivalent to the metacategory of 퐾-schemes of finite presentation, by theorem 3.5.21.

[3] More precisely, let u� be the category of finitely presentable 퐾-algebras whose underlying set is hereditarily 휆-small for some cardinal 휆 such that the underlying set of 퐾 is 휆-small.

195 Specificities

3.5.23 Example. Let 퐾 be a field and let u� be the category defined as follows:

• The objects are pairs (푚, 퐼) where 푚 is a natural number and 퐼 is a

finite subset of the polynomial ring 퐾[푥1, … , 푥푚]. • The morphisms (푚, 퐼) → (푛, 퐽) are the 퐾-algebra homomorphisms

퐾[푥1, … , 푥푚]/(퐼) → 퐾[푥1, … , 푥푛]/(퐽), where (퐼) and (퐽) are the ideals generated by 퐼 and 퐽, respectively. • Composition and identities are inherited.

Although u� is not literally a subcategory of the metacategory of 퐾-algebras, there is an evident fully faithful functor which will suﬃce for our purposes. [4] Let u�1 be the category of T1-spaces 푋 such that the set of points of 푋 is hereditarily 휆-small, where 휆 is an uncountable regular cardinal ≥ 휅 such that the underlying set of 퐾 is hereditarily 휆-small. Given an object (푚, 퐼) in u�, let 퐹 (푚, 퐼) be the subspace of 퐾푚 defined as follows,

푚 . 퐹 (푚, 퐼) = {(푥1, … , 푥푚) ∈ 퐾 | ∀휑 ∈ 퐼 휑(푥1, … , 푥푚) = 0}

[5] where 퐾푚 is equipped with the classical Zariski topology. This defines op a functor 퐹 : u� → u�1.

Let u�1 be the class of local homeomorphisms of topological spaces

in u�1, and let 햩1 be the usual coverage on u�1. It can be shown that 퐹 : op u� → u�1 preserves finitary coproducts, so by theorem 1.5.20, we have

an induced functor 퐹 : u� → u�1 that preserves 휅-ary coproducts. It can ′ also be shown that 퐹 : u� → u�1 sends members of u� to local homeomorphisms of topological spaces and sends pullbacks of members of u�′ to pullbacks of local homeomorphisms of topological spaces. Furthermore, ′ 퐹 : u� → u�1 sends 햩 -covering morphisms in u� to surjective continuous maps. Hence, by lemmas 2.4.2(c) and 2.4.2(e), we have a 휅-ary admiss- ′ ′ ible functor 퐹 : (u�, u� , 햩 ) → (u�1, u�1, 햩1). Since (u�1, u�1, 햩1) is effect- ′ ive, by theorem 2.5.7, we have an induced functor 퐹 : Xt(u�, u� ) → u�1

extending 퐹 : u� → u�1. [4] — i.e. topological spaces in which every point is closed. [5] — i.e. the topology in which every 퐹 (푚, 퐼) is a closed subset of 퐾푚.

196 3.5. Schemes

We may think of the above as a functorial way of assigning a T1-space of 퐾-rational points to every 퐾-scheme locally of finite type that respects the geometry of schemes. In the case where 퐾 is an algebraically closed field, this can be identified with the subspace of closed points oftheusual underlying topological space of a scheme.

197

Appendix A Generalities

a.1 Presheaves

Synopsis. We set up notation and terminology for working with presheaves.

a.1.1 ※ Throughout this section, u� is an arbitrary category.

a.1.2 Definition.A presheaf on u� is a contravariant functor from u� to Set. More concretely, a presheaf 퐴 on u� consists of the following data:

• For each object 푋 in u�, a set 퐴(푋).

• For each morphism 푓 : 푋 → 푌 in u�, a map 퐴(푌 ) → 퐴(푋) sending each 푎 ∈ 퐴(푌 ) to 푎 ⋅ 푓 ∈ 퐴(푋).

Moreover, these data are required to satisfy the following condition:

• For every object 푋 in u� and every 푎 ∈ 퐴(푋), we have 푎 ⋅ id푋 = 푎.

• For every composable pair 푓 : 푋 → 푌 and 푔 : 푌 → 푍 in u� and every 푎 ∈ 퐴(푍), we have (푎 ⋅ 푔) ⋅ 푓 = 푎 ⋅ (푔 ∘ 푓). a.1.2(a) Example. For each set 퐾, we have the constant presheaf Δ퐾 defined as follows:

• For each object 푋 in u�, Δ퐾(푋) = 퐾.

• For each morphism 푓 : 푋 → 푌 in u� and each 푘 ∈ Δ퐾(푌 ), 푘 ⋅ 푓 = 푘.

For simplicity, we write ∅ instead of Δ∅ and 1 instead of Δ1.

199 Generalities

a.1.2(b) Example. For each object 푆 in u�, we have a presheaf h푆 defined as follows:

• For each object 푋 in u�, h푆(푋) = u�(푋, 푆).

• For each morphism 푓 : 푋 → 푌 in u� and each 푔 ∈ h푆(푌 ), 푔 ⋅푓 = 푔 ∘푓.

a.1.3 Definition.A morphism of presheaves is a natural transformation of contravariant functors. More concretely, given presheaves 퐴 and 퐵 on u�, a morphism ℎ : 퐴 → 퐵 consists of the following data:

• For each object 푋 in u�, a map ℎ : 퐴(푋) → 퐵(푋).

Moreover, these data are required to satisfy the following condition:

• For all morphisms 푓 : 푋 → 푌 in u� and all 푎 ∈ 퐴(푌 ), we have ℎ(푎 ⋅ 푓) = ℎ(푎) ⋅ 푓.

We write Psh(u�) for the metacategory of presheaves on u�.

Example. Let 푋 be an object in u� and let 푎 ∈ 퐴(푋). There is a unique

morphism 휀푎 : h푋 → 퐴 such that 휀푎(id푋) = 푎, namely the one defined

by 휀푎(푥) = 푎 ⋅ 푥 for each morphism 푥 : 푇 → 푋 in u�. In fact, every

morphism h푋 → 퐴 is of this form for some 푎 ∈ 퐴(푋): this is the Yoneda lemma.

a.1.4 ¶ Let 퐴 be a presheaf on u�.

a.1.4(a) Definition. An element of 퐴 is a pair (푋, 푎) where 푋 is an object in u� and 푎 ∈ 퐴(푋).

a.1.4(b) Definition. The category of elements of 퐴 is the category El(퐴) defined as follows:

• The objects are the elements of 퐴. ′ • The morphisms (푋, 푎 ) → (푌 , 푎) are morphisms 푓 : 푋 → 푌 in u� [1] such that 푎 ⋅ 푓 = 푎′. • Composition and identities are inherited from u�.

[1] Strictly speaking, this is an abuse of notation, as hom-sets are supposed to be disjoint.

200 a.1. Presheaves

The canonical projection 푃 : El(퐴) → u� is the functor sending each ′ object (푋, 푎) in El(퐴) to 푋 in u� and each morphism 푓 : (푋, 푎 ) → (푌 , 푎) in El(퐴) to 푓 : 푋 → 푌 in u�.

The tautological cocone 휆 : h푃 ⇒ Δ퐴 is defined at the object (푋, 푎)

in El(퐴) to be the unique morphism h푋 → 퐴 sending id푋 to 푎.

Example. Let 푆 be an object in u�. The slice category u�∕푆 is El(h푆).

Proposition. Let 퐴 be a presheaf on u�, let 푃 : El(퐴) → u� be the canonical projection, and let be the tautological cocone. Then 휆 : h푃 ⇒ Δ퐴 휆 is a colimiting cocone in Psh(u�).

Proof. See e.g. Theorem 1 in [CWM, Ch. III, §7] or Proposition 1 in [ML–M, Ch. I, §5]. □

a.1.5 Definition. Let 퐴 be a presheaf on u�.A subpresheaf of 퐴 is a presheaf 퐴′ on u� that satisfies the following conditions:

• For each object 푋 in u�, 퐴′(푋) ⊆ 퐴(푋).

• For each morphism 푓 : 푋 → 푌 in u�, the following diagram commutes: 퐴′(푌 ) 퐴(푌 )

(−)⋅푓 (−)⋅푓 퐴′(푋) 퐴(푋) i.e. for each 푎 ∈ 퐴(푌 ), if 푎 ∈ 퐴′(푌 ), then 푎 ⋅ 푓 ∈ 퐴′(푋).

We write 퐴′ ⊆ 퐴 for ‘퐴′ is a subpresheaf of 퐴’.

Remark. The set of subpresheaves of any given presheaf, partially ordered by componentwise inclusion, is a complete lattice with meet (resp. join) given by componentwise intersection (resp. union). a.1.5(a) Example. Let ℎ0 : 퐴0 → 퐵 and ℎ1 : 퐴1 → 퐵 be morphisms in Psh(u�).

The pullback of ℎ0 and ℎ1 is the subpresheaf Pb(ℎ0, ℎ1) ⊆ 퐴0 × 퐴1 defined as follows:

Pb(ℎ0, ℎ1)(푇 ) = {(푎0, 푎1) ∈ 퐴0(푇 ) × 퐴1(푇 ) | ℎ0(푎0) = ℎ1(푎1)}

201 Generalities

a.1.5(b) Example. Let ℎ0, ℎ1 : 퐴 → 퐵 be a parallel pair of morphisms in Psh(u�).

The equaliser of ℎ0 and ℎ1 is the subpresheaf Eq(ℎ0, ℎ1) ⊆ 퐴 defined as follows:

Eq(ℎ0, ℎ1)(푇 ) = {푎 ∈ 퐴(푇 ) | ℎ0(푎) = ℎ1(푎)}

a.1.5(c) Example. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). The kernel relation of ℎ is the presheaf Kr(ℎ) defined as follows:

Kr(ℎ) = Pb(ℎ, ℎ)

a.1.6 Definition. Let 퐴 be a presheaf on u�.A representation of 퐴 is an object (푋, 푎) in El(퐴) that satisfies the following conditions:

′ ′ ′ • For every object (푋 , 푎 ) in El(퐴), there is a morphism 푥 : 푋 → 푋 in u� such that 푎 ⋅ 푥 = 푎′.

′ • Given a parallel pair 푥0, 푥1 : 푋 → 푋 of morphisms in u�, if 푎 ⋅ 푥0 =

푎 ⋅ 푥1, then 푥0 = 푥1. A representable presheaf is a presheaf that admits a representation.

Example. For each object 푆 in u�, the presheaf h푆 is tautologically representable.

Recognition Lemma. Let 퐴 be a presheaf on u�, let 푋 be an object in u�, and let 푎 be principles for an element of 퐴(푋). The following are equivalent: representations is a representation for . of presheaves (i) (푋, 푎) 퐴 (ii) (푋, 푎) is a terminal object in El(퐴). We have an isomorphism in given by . (iii) h푋 → 퐴 Psh(u�) 푥 ↦ 푎 ⋅ 푥

Proof. Straightforward. ⧫

a.1.7 Definition. Let 퐴 be a presheaf on u�.A familial representation of 퐴 is a subset Φ ⊆ ob El(퐴) that satisfies the following conditions:

′ ′ • For every object (푋 , 푎 ) in El(퐴), there is a unique element (푋, 푎) of ′ ′ Φ such that there is a morphism (푋 , 푎 ) → (푋, 푎) in El(퐴).

202 a.1. Presheaves

′ • For each (푋, 푎) ∈ Φ, given a parallel pair 푥0, 푥1 : 푋 → 푋 of morph-

isms in u�, if 푎 ⋅ 푥0 = 푎 ⋅ 푥1, then 푥0 = 푥1. A familially representable presheaf is a presheaf that admits a familial representation.

Lemma. Let 퐴 be a presheaf on u�. The following are equivalent:

(i) 퐴 is a familially representable presheaf on u�.

(ii) 퐴 is a disjoint union of a family of representable presheaves on u�.

Proof. Straightforward. ⧫

a.1.8 ¶ Let u� be a category. a.1.8(a) Definition.A family 푋 of objects in u� is a map 푋 : idx 푋 → ob u�. a.1.8(b) Definition. Let 푋 and 푌 be families of objects in u�.A matrix of morphisms 푋 → 푌 in u� is a family 푓 (of morphisms in u�) that satisfies the following axioms:

• idx 푓 ⊆ (idx 푌 ) × (idx 푋). • For each 푖 ∈ idx 푋, there is a unique 푗 ∈ idx 푌 such that (푗, 푖) ∈ idx 푓. • For each (푗, 푖) ∈ idx 푓, 푓(푗, 푖) is a morphism 푋(푖) → 푌 (푗) in u�. a.1.8(c) Definition. The metacategory of families of objects in u� is the metacategory Fam(u�) defined as follows: • The objects are the families of objects in u�. [2] • The morphisms 푋 → 푌 are matrices of morphisms 푋 → 푌 in u�. • Composition is defined like matrix multiplication, and identities are given by the evident matrices.

Remark. The evident projection idx : Fam(u�) → Set is a split Grothen- dieck fibration.

[2] Strictly speaking, this is an abuse of notation: hom-sets are supposed to be disjoint.

203 Generalities

Properties of Proposition. the metacat- (i) There is a unique functor 훾 : u� → Fam(u�) with the following egory of families properties:

• For each object 푋 in u�, we have idx 훾(푋) = {∗} and 훾(푋)(∗) = 푋. • For each morphism 푓 : 푋 → 푌 in u�, we have 훾(푓)(∗, ∗) = 푓. (ii) Moreover, 훾 : u� → Fam(u�) is fully faithful. (iii) There is a unique functor ⨿ : Fam(u�) → Psh(u�) with the following properties: • For each object in , we have . 푋 Fam(u�) ⨿푋 = ∐푖∈idx 푋 h푋(푖) • For each morphism 푓 : 푋 → 푌 in Fam(u�), if (푗, 푖) ∈ idx 푓, then the diagram in Psh(u�) shown below commutes,

h푋(푖) ⨿푋

h푓(푗,푖) ⨿푓

h푌 (푗) ⨿푌

where the horizontal arrows are the evident coproduct injections.

(iv) Moreover, ⨿ : Fam(u�) → Psh(u�) is fully faithful and essentially surjective onto the full submetacategory spanned by the familially representable presheaves on u�. (v) Furthermore, ⨿ : Fam(u�) → Psh(u�) is isomorphic to the Yoneda representation induced by 훾 : u� → Fam(u�).

Proof. Straightforward. (For (iv), use lemma a.1.7.) ⧫

a.1.9 Definition.A discrete fibration or discrete cartesian fibration is a functor 푃 : ℰ → u� with the following property:

• For each object 퐸 in ℰ and each morphism 푓 : 푋 → 푃 (퐸) in u�, there exist a unique object 푓 ∗퐸 and a unique morphism 푓̃ : 푓 ∗퐸 → 퐸 such ∗ that 푃 (푓 퐸) = 푋 and 푃 (푓)̃ = 푓. Dually, a discrete opfibration or discrete cocartesian fibration is a functor 푃 : ℰ → u� such that 푃 op : ℰ op → u� op is a discrete fibration.

204 a.1. Presheaves

Recognition Lemma. Let 푃 : ℰ → u� be a functor. The following are equivalent: principle for (i) 푃 : ℰ → u� is a discrete fibration. discrete fibrations (ii) The diagram below is a pullback square in Set: codom mor ℰ ob ℰ mor 푃 ob 푃 mor ob u� codom u� Proof. Immediate. ⧫

a.1.10 ¶ Every morphism ℎ : 퐴 → 퐵 in Psh(u�) induces an evident functor Functoriality El(ℎ) : El(퐴) → El(퐵) making the diagram below commute, of the category El(ℎ) of elements El(퐴) El(퐵)

푃퐴 푃퐵 u� u� where the vertical arrows are the respective canonical projections. Thus,

we have a functor El : Psh(u�) → Cat∕u�. Moreover:

The equivalence Proposition. of presheaves and is fully faithful, and the essential image is (i) El : Psh(u�) → Cat∕u� discrete fibrations the full subcategory spanned by the discrete fibrations with codomain u�. admits a left adjoint, namely the evident (ii) El : Psh(u�) → Cat∕u� functor sending each object in |−| : Cat∕u� → Psh(u�) (ℰ, 푃 ) Cat∕u� to the presheaf defined as follows:

• For each object 푋 in u�, |(ℰ, 푃 )|(푋) is the set of connected com- ponents of the comma category (푋 ↓ 푃 ). • For each morphism 푓 : 푋 → 푌 in u� and each object (퐸, 푔) in (푌 ↓ 푃 ), [(퐸, 푔)] ⋅ 푓 = [(퐸, 푔 ∘ 푓)], where [−] denotes the connected component. admits a right adjoint, namely the evident (iii) El : Psh(u�) → Cat∕u� functor sending each object in to Γ: Cat∕u� → Psh(u�) (ℰ, 푃 ) Cat∕u� the presheaf defined as follows: • For each object in , is the set of functors 푋 u� Γ(ℰ, 푃 )(푋) u�∕푋 → ℰ making the evident triangle commute.

205 Generalities

• For each morphism 푓 : 푋 → 푌 in u�, the action of 푓 is precompos- ition (in ) by the functor given by postcomposition Cat u�∕푋 → u�∕푌 (in u�).

Proof. Straightforward. ⧫

a.1.11(a) Definition.A sieve of u� is a full subcategory u�′ such that the inclusion u�′ ↪ u� is a discrete fibration.

a.1.11(b) Definition.A sieve on an object 푆 in u� is a sieve of the slice category

u�∕푆.

Remark. Explicitly, a sieve of a category u� is a full subcategory u�′ such that, for every morphism 푓 : 푋 → 푌 in u�, if 푌 is in u�′, then 푋 is also in u�′. Thus, a sieve of u� is essentially the same thing as a subpresheaf of the terminal presheaf 1. Similarly, a sieve on an object 푆 is essentially the same thing as a

subpresheaf of the representable presheaf h푆.

a.1.12(a) Definition. The sieve ↓(Φ) ⊆ u� generated by a subset Φ ⊆ ob u� is The sieve gener- defined as follows: ated by a set of objects • For every object 푇 in u�, 푇 is in ↓(Φ) if and only if there is some morphism 푥 : 푇 → 푋 in u� with 푋 ∈ Φ.

a.1.12(b) Definition.A generating set of elements of a presheaf 퐴 on u� is a subset Φ ⊆ ob El(퐴) such that ↓(Φ) = El(퐴).

a.1.12(c) Definition. Let 휅 be a regular cardinal. A presheaf 퐴 on u� is 휅-generable if it admits a 휅-small generating set of elements.

Example. The principal sieve generated by a morphism 푓 : 푋 → 푌 in u� is the sieve ↓⟨푓⟩ on 푌 where (푇 , 푦) is in ↓⟨푓⟩ if and only if 푦 : 푇 → 푌 factors through 푓 : 푋 → 푌 . By construction, {(푋, 푓)} is a generating set of elements of the presheaf on u� corresponding to ↓⟨푓⟩.

206 a.1. Presheaves

a.1.13 Definition. Let 퐹 : u� → u� be a functor and let u� be a sieve of u�. The pullback sieve 퐹 ∗u� is the sieve of u� defined as follows:

• For every object 푋 in u�, 푋 is in 퐹 ∗u� if and only if 퐹 푋 is in u�.

Example. Let 푓 : 푋 → 푌 be a morphism in u�. There is an evident

functor Σ푓 : u�∕푋 → u�∕푌 defined on objects by (푇 , 푥) ↦ (푇 , 푓 ∘ 푥). For ∗ ∗ simplicity, we write 푓 u� instead of (Σ푓 ) u� in this case.

a.1.14 ¶ Let Φ be a set of objects in u�. Define a category 횫Φ as follows: • The objects are finite lists of elements of Φ of length ≥ 1.

• The morphisms (푋0, … , 푋푚) → (푌0, … , 푌푛) are the monotone maps [3] 훼 : {0, … , 푚} → {0, … , 푛} such that 푌훼(푖) = 푋푖 for 푖 ∈ {0, … , 푚}. • Composition and identities are inherited from Set.

Let u� be the sieve of u� generated by Φ. Assuming that the relevant op products exist in u� , let 푃 : (횫Φ) → u� be the functor that sends each

object (푋0, … , 푋푚) in 횫Φ to the product 푋0 × ⋯ × 푋푚 in u� and each

morphism 훼 : (푋0, … , 푋푚) → (푌0, … , 푌푛) in 횫Φ to the corresponding ∗ projection 훼 : 푌0 × ⋯ × 푌푛 → 푋0 × ⋯ × 푋푚 in u� . The functor op is homotopy cofinal. Lemma. 푃 : (횫Φ) → u�

Proof. Let 푈 be an object in u� . We must verify that the comma category (푈 ↓ 푃 ) is weakly contractible. It is clear that (푈 ↓ 푃 ) is inhabited. More- over, (푈 ↓ 푃 ) is isomorphic to the opposite of the category of simplices of a 0-coskeletal simplicial set; but any inhabited 0-coskeletal simplicial set is a contractible Kan complex, so we are done. ■

a.1.15 ¶ Let 퐴 be a presheaf on u�. a.1.15(a) Definition. An equivalence relation on 퐴 is a subpresheaf 푅 ⊆ 퐴 × 퐴 such that, for every object 푋 in u�, 푅(푋) is an equivalence relation on 퐴(푋).

[3] Strictly speaking, this is an abuse of notation, as hom-sets are supposed to be disjoint.

207 Generalities

a.1.15(b) Definition. The quotient of 퐴 by an equivalence relation 푅 is the presheaf 퐴/푅 defined as follows,

(퐴/푅)(푋) = 퐴(푋)/푅(푋) ′ ′ 퐴 ⋅ 푓 = {푎 ⋅ 푓 | 푎 ∈ 퐴 }

where 푋 is an arbitrary object in u�, 푓 : 푋0 → 푋1 is an arbitrary morphism ′ in u�, and 퐴 is an 푅(푋1)-equivalence class in 퐴(푋1).

Remark. 퐴/푅 is indeed a well-defined presheaf on u�, because 푅 is a subpresheaf of 퐴 × 퐴.

208 a.2. Coverages

a.2 Coverages

Synopsis. We define some terminology related to coverages, a variation on the notion of Grothendieck topology, and we record some basic results.

Prerequisites. §1.1, a.1.

a.2.1 ※ Throughout this section, u� is an arbitrary category.

a.2.2 ¶ Let 퐴 be a presheaf on u�, let 푋 be an object in u�, and let u� be a sieve on 푋.

Definition. The separation condition (resp. sheaf condition) on 퐴 with respect to u� is the following: • For every commutative square in Cat of the form below,

u� El(퐴)

u�∕푋 u�

where El(퐴) → u� and u�∕푋 → u� are the projections, there is at most

one (resp. exactly one) functor u�∕푋 → El(퐴) making both evident triangles commute.

Example. 퐴 always satisfies the sheaf condition with respect to the maximal sieve on 푋.

Let be the subpresheaf of corresponding to . The The sheaf condi- Lemma. 푆 h푋 u� ⊆ u�∕푋 tion as right following are equivalent: orthogonality (i) 퐴 satisfies the separation condition (resp. sheaf condition) with respect to u� . (ii) The map

HomPsh(u�)(h푋, 퐴) → HomPsh(u�)(푆, 퐴)

induced by the inclusion is injective (resp. bijective). 푆 ↪ h푋

Proof. Apply proposition a.1.10. ■

209 Generalities

a.2.3 ¶ Let 퐴 be a presheaf on u�, let 푋 be an object in u�, let u� and u� ′ be ′ sieves on 푋, and let 푆 and 푆 be the corresponding subpresheaves of h푋.

a.2.3(a) Lemma. Assume the following hypotheses: Local character • 퐴 satisfies the separation condition with respect to u� . of separation • For every object in , satisfies the separation condition with conditions (푈, 푥) u� 퐴 respect to 푥∗u� ′. Then 퐴 satisfies the separation condition with respect to u� ′ as well.

Proof. Let 0̄푠 ,1 ̄푠 : h푋 → 퐴 be morphisms in Psh(u�) such that the diagram below commutes:

′ 푆 h푋

1̄푠

h푋 퐴 0̄푠

We must show that 0̄푠 =1 ̄푠 . Let (푈, 푥) be an object in u� . By definition, we have the following commutative diagram:

∗ ′ 푥 푆 h푈 h푥

′ h푈 푆 h푋

1̄푠 h푥 h푋 퐴 0̄푠

Since 퐴 satisfies the separation condition with respect to 푥∗u� ′, we deduce

that 0̄푠 ∘ h푥 =1 ̄푠 ∘ h푥. Hence, the diagram below commutes:

푆 h푋

1̄푠

h푋 퐴 0̄푠

But 퐴 also satisfies the separation condition with respect to u� , so 0̄푠 =1 ̄푠 , as required. ■

210 a.2. Coverages

a.2.3(b) Corollary. Assume the following hypotheses: Upward- • u� ⊆ u� ′. closedness • satisfies the separation condition with respect to . of separation 퐴 u� conditions Then 퐴 also satisfies the separation condition with respect to u� ′.

Proof. If (푈, 푥) is an object in u� , then 푥∗u� ′ is the maximal sieve on 푈. Thus, the claim is a special case of lemma a.2.3(a). ■

a.2.3(c) Lemma. Assume the following hypotheses: Local char- • 퐴 satisfies the sheaf condition with respect to u� . acter of sheaf • Forevery object in , satisfies the sheaf condition with respect conditions (푈, 푥) u� 퐴 to 푥∗u� ′. • For every object (푈, 푥) in u� ′, 퐴 satisfies the separation condition with respect to 푥∗u� . Then 퐴 satisfies the sheaf condition with respect to u� ′ as well.

Proof. Let 푠′ : 푆′ → 퐴 be a morphism in Psh(u�). By lemma a.2.3(a), 퐴 satisfies the separation condition with respect to u� ′, so any extension of ′ ′ 푠 along the inclusion 푆 ↪ h푋 is unique if it exists; but it remains to be shown that such an extension exists. First, we will construct a morphism 푠 : 푆 → 퐴 such that the restriction to 푆 ∩ 푆′ agrees with the restriction of 푠′ : 푆′ → 퐴. Let (푈, 푥) be an object in u� . Since 퐴 satisfies the sheaf condition with ∗ ′ respect to 푥 u� , there is a unique morphism 푠(푈,푥) : h푈 → 퐴 making the following diagram commute:

푠′ 푥∗푆′ 푆′ 퐴

푠(푈,푥)

h푈 h푋 h푥 Let 푢 : 푉 → 푈 be a morphism in u�. Then the diagram below commutes,

푠′ 푢∗푥∗푆′ 푥∗푆′ 푆′ 퐴

푠(푈,푥)

h푉 h푈 h푢

211 Generalities

and since 퐴 satisfies the separation condition with respect to 푢∗푥∗u� ′, it

follows that 푠(푈,푥) ∘h푢 = 푠(푉 ,푥∘푢). Thus, we have a well-defined morphism

푠 : 푆 → 퐴 given by 푠(푥) = 푠(푈,푥)(id푈 ) for each object (푈, 푥) of u� . Now, since 퐴 satisfies the sheaf condition with respect to u� , there is

a (unique) morphism ̄푠: h푋 → 퐴 that extends 푠 : 푆 → 퐴. It remains to ′ ′ be shown that ̄푠: h푋 → 퐴 extends 푠 : 푆 → 퐴. Let (푈, 푥) be an object in u� ′. Clearly, the restriction of ̄푠 to 푆 ∩ 푆′ agrees with the restriction of 푠′, so the following diagram commutes,

푠′ 푥∗푆 푆′ 퐴

̄푠

h푈 h푋 h푥

and since 퐴 satisfies the separation condition with respect to 푥∗u� , it follows that (̄푠 푥) = 푠′(푥). Thus, ̄푠 is indeed an extension of 푠′. ■

a.2.3(d) Corollary. Assume the following hypotheses: Upward- • u� ⊆ u� ′. closedness of sheaf conditions • 퐴 satisfies the sheaf condition with respect to u� . • For every object (푈, 푥) in u� ′, 퐴 satisfies the separation condition with respect to 푥∗u� . Then 퐴 also satisfies the sheaf condition with respect to u� ′.

Proof. If (푈, 푥) is an object in u� , then 푥∗u� ′ is the maximal sieve on 푈. Thus, the claim is a special case of lemma a.2.3(c). ■

a.2.4 ¶ Let ℬ be a set of morphisms in u� and let 푋 be an object in u�.

Definition.A ℬ-sink on 푋 is a subset Φ of ob u�∕푋 with the following property:

• For every (푈, 푥) ∈ Φ, 푥 : 푈 → 푋 is a member of ℬ. A sink on 푋 is a (mor u�)-sink on 푋.

212 a.2. Coverages

Refinement Lemma. Let Φ and Φ′ be sinks on an object 푋 in u�. The following are of sinks equivalent:

′ (i) ↓(Φ ) ⊆ ↓(Φ). (ii) For every (푈, 푥) ∈ Φ, there is a commutative square in u� of the form below, ′ 푉 푢 푈 ′ 푢 푥′

푈 푥 푋 ′ ′ ′ where (푈 , 푥 ) ∈ Φ .

Proof. Straightforward. ⧫

a.2.5 ¶ Let 푋 be an object in u�.

a.2.5(a) Definition. A sieve u� on 푋 is strict-epimorphic if it has the following property:

• For every object 푌 in u�, the representable presheaf h푌 satisfies the sheaf condition with respect to u� .

a.2.5(b) Definition. A sieve u� on 푋 is universally strict-epimorphic if it has the following property: ∗ • For every object (푇 , 푥) in u�∕푋, the pullback sieve 푥 u� is a strict- epimorphic sieve on 푇 .

a.2.5(c) Definition. A sink Φ on 푋 is strict-epimorphic (resp. universally strict-epimorphic) if the sieve ↓(Φ) is strict-epimorphic (resp. universally strict-epimorphic).

a.2.6 ¶ Let 퐴 be a presheaf on u�, let 푋 be an object in u�, let Φ be a sink on 푋,

and let Γ(Φ, 퐴) be the subset of ∏(푇 ,푥)∈Φ 퐴(푇 ) consisting of the elements

(푎(푇 ,푥) | (푇 , 푥) ∈ Φ) with the following property: • For every commutative square in u� of the form below,

푡1 푈 푇1

푡0 푥1 푇 푋 0 푥0

213 Generalities

if both and are in , then . (푇0, 푥0) (푇1, 푥1) Φ 푎(푇0,푥0) ⋅ 푡0 = 푎(푇1,푥1) ⋅ 푡1 Clearly, we have the following map:

Φ∗ : 퐴(푋) → Γ(Φ, 퐴) 푎 ↦ (푎 ⋅ 푥 | (푇 , 푥) ∈ Φ)

The sheaf Lemma. The following are equivalent: condition with (i) 퐴 satisfies the separation condition (resp. sheaf condition) with respect to a sink respect to the sieve ↓(Φ) on 푋 generated by Φ. (ii) The map Φ∗ : 퐴(푋) → Γ(Φ, 퐴) is injective (resp. bijective).

Proof. Straightforward. (Compare lemma a.2.2.) ⧫

a.2.7(a) Definition.A tree on an object 푋 in u� is a set Φ with the following properties:

• The elements of Φ are finite sequences (푓1, … , 푓푚) of morphisms in

u� such that 푚 ≥ 0 and dom 푓푖 = codom 푓푖+1 for 0 < 푖 < 푚. • The empty sequence is in Φ.

• If (푓1, … , 푓푚, 푓푚+1) ∈ Φ, then (푓1, … , 푓푚) ∈ Φ.

• If (푓1, … , 푓푚) ∈ Φ (and 푚 > 0), then codom 푓1 = 푋.

a.2.7(b) Definition.A leaf of a tree Φ on an object 푋 in u� is (푓1, … , 푓푚) ∈ Φ with the following property:

• For every (푔1, … , 푔푛) ∈ Φ, if 푚 ≤ 푛 and (푓1, … , 푓푚) = (푔1, … , 푔푚), then 푚 = 푛.

a.2.7(c) Definition. A tree Φ on an object 푋 in u� is proper if it satisfies the following condition:

• Every element of Φ is a prefix of some leaf of Φ.

a.2.7(d) Definition. The composite of a tree Φ on an object 푋 in u� is the sink Φ◦ on 푋 defined as follows:

◦ Φ = {(dom 푓푚, 푓1 ∘ ⋯ ∘ 푓푚) | (푓1, … , 푓푚) is a leaf of Φ}

214 a.2. Coverages

Remark. In particular, if Φ contains only the empty sequence, then Φ◦ =

{(푋, id푋)}.

a.2.8 ¶ Following [Johnstone, 2002, §C2.1], it is convenient to introduce the following variation on the notion of Grothendieck topology. a.2.8(a) Definition.A coverage on u� consists of the following data: • For each object 푋 in u�, a set 햩(푋) of sinks on 푋. These data are required to satisfy the following conditions:

• For every object 푋 in u�, {(푋, id푋)} is a member of 햩(푋). • For every morphism 푓 : 푋 → 푌 in u�, if Ψ ∈ 햩(푌 ), then there is Φ ∈ 햩(푋) such that ↓(Φ) ⊆ 푓 ∗ ↓(Ψ). a.2.8(b) Definition. A coverage 햩 on u� is upward-closed if it has the following property:

′ • For every object 푋 in u�, given subsets Φ and Φ of ob u�∕푋 such that ′ ′ ↓(Φ ) ⊆ ↓(Φ), if Φ ∈ 햩(푋), then Φ ∈ 햩(푋) as well. a.2.8(c) Definition. A coverage 햩 on u� is composition-closed if it has the following property:

• For every object 푋 in u�, given Φ ∈ 햩(푋) and Ψ(푈,푥) ∈ 햩(푈) for each (푈, 푥) ∈ Φ, we have:

{(푉 , 푥 ∘ 푢) | (푈, 푥) ∈ Φ, (푉 , 푢) ∈ Ψ(푈,푥)} ∈ 햩(푋) a.2.8(d) Definition. A coverage is saturated if it is both upward-closed and composition-closed. a.2.8(e) Example. The trivial coverage on u� is the coverage 햩 where 햩(푋) =

{{(푋, id푋)}}. This coverage is composition-closed, but it is not upward- closed in general. a.2.8(f) Example. The chaotic coverage on u� is the coverage 햩 where a sink is in 햩(푋) if and only if it contains some (푈, 푥) where 푥 : 푈 → 푋 is a split epimorphism in u�. This coverage is saturated.

215 Generalities

a.2.9 ※ For the remainder of this section, 햩 is an arbitrary coverage on u�.

a.2.10 Definition.A 햩-tree on an object 푋 in u� is a tree Φ on 푋 with the following properties:

• Φ is a proper tree on 푋. • The set

{(푈, 푥) ∈ ob u�∕푋 | (푥) ∈ Φ}

is either empty or a member of 햩(푋).

• For every (푓1, … , 푓푚) ∈ Φ (where 푚 > 0), the set

(푉 , 푢) ∈ ob u� 푓 , … , 푓 , 푢 ∈ Φ { ∕dom 푓푚 | ( 1 푚 ) }

is either empty or a member of 햩(dom 푓푚).

Recognition Lemma. The following are equivalent: principle for (i) 햩 is a composition-closed coverage on u�. composition- closed coverages (ii) For every object 푋 in u�, the composite of every 햩-tree on 푋 is a member of 햩(푋).

Proof. Straightforward. ⧫

The composition- Proposition. For each object 푋 in u�, let 퐽(̄ 푋) be the set of all sinks closure of on 푋 of the form Φ◦ for some 햩-tree Φ on 푋. Then 퐽 ̄ is the smallest a coverage composition-closed coverage on u� that contains 햩.

Proof. Straightforward. ⧫

a.2.11(a) Definition. The canonical coverage on u� is the coverage 햩 on u� where 햩(푋) is the set of universally strict-epimorphic sinks on 푋.

a.2.11(b) Definition.A subcanonical coverage on u� is a coverage 햩 on u� such that every element of 햩(푋) is a universally strict-epimorphic sink on 푋.

216 a.2. Coverages

Remark. It is clear that the canonical coverage is indeed a coverage, and (by definition) it is the largest subcanonical coverage. Moreover, by lemma a.2.3(c) (and corollary a.2.3(d)), the canonical coverage is a saturated coverage.

Recognition prin- Lemma. Let 햩 be an upward-closed coverage on u�. The following are ciple for subca- equivalent: nonical upward- closed coverages (i) 햩 is a subcanonical coverage on u�. (ii) Forevery object 푋 in u�, every element of 햩(푋) is a strict-epimorphic sink on 푋.

Proof. Straightforward. ⧫

a.2.12 ¶ Let 퐵 be a presheaf on u�.

Definition. A subpresheaf 퐴 ⊆ 퐵 on u� is 햩-closed if it has the following property:

• For every element (푋, 푏) of 퐵 and every Φ ∈ 햩(푋), if 푏 ⋅ 푥 ∈ 퐴(푈) for every (푈, 푥) ∈ Φ, then 푏 ∈ 퐴(푋).

Example. Of course, 퐵 itself is a 햩-closed subpresheaf of 퐵.

Remark. The class of 햩-closed subpresheaves of 퐵 is closed under arbitrary intersections.

a.2.13 ¶ Let ℎ : 퐴 → 퐵 be a morphism of presheaves on u� and let 퐵′ be the subpresheaf of 퐵 defined as follows:

• For every element (푋, 푏) of 퐵, 푏 ∈ 퐵′(푋) if and only if there is a 햩-tree Φ on 푋 such that, for every (푇 , 푥) ∈ Φ◦, there is 푎 ∈ 퐴(푇 ) such that ℎ(푎) = 푏 ⋅ 푥.

Definition. The 햩-closed support of ℎ : 퐴 → 퐵 is the subpresheaf 퐵′ ⊆ 퐵 defined above.

217 Generalities

Properties of the Proposition. closed support (i) 퐵′ as defined above is indeed a subpresheaf of 퐵. of a morphism (ii) 퐵′ is the smallest 햩-closed subpresheaf of 퐵 containing the image of presheaves of ℎ : 퐴 → 퐵.

Proof. Straightforward. (For (i), use proposition a.2.10.) ⧫

a.2.14 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is 햩-locally surjective if the 햩-closed support of ℎ : 퐴 → 퐵 is 퐵 itself.

Properties of Proposition. local surjective (i) Every epimorphism in Psh(u�) is 햩-locally surjective. morphisms (ii) The class of 햩-locally surjective morphisms of presheaves on u� is a class of fibrations in Psh(u�).

(iii) The class of 햩-locally surjective morphisms of presheaves on u� is closed under (possibly infinitary) coproduct in Psh(u�).

(iv) Given morphisms ℎ : 퐴 → 퐵 and 푘 : 퐵 → 퐶 in Psh(u�), if the composite 푘 ∘ ℎ : 퐴 → 퐶 is 햩-locally surjective, then 푘 : 퐵 → 퐶 is also 햩-locally surjective.

Proof. Straightforward. ⧫

a.2.15(a) Definition. A subpresheaf of a presheaf on u� is 햩-dense if the inclusion is 햩-locally surjective.

a.2.15(b) Definition. A sieve on an object 푋 in u� is 햩-covering if the correspond-

ing subpresheaf of h푋 is 햩-dense.

We write CSv햩(푋) for the set of 햩-covering sieves on 푋, partially ordered by inclusion.

a.2.15(c) Definition. A sink Φ on an object 푋 in u� is 햩-covering if the sieve ↓(Φ) is 햩-covering.

a.2.15(d) Definition. A morphism 푥 : 푈 → 푋 in u� is 햩-covering if the principal sieve ↓⟨푥⟩ is 햩-covering.

218 a.2. Coverages

Example. Assuming 햩 is the trivial coverage on u�, a sink on 푋 is 햩- covering if and only if it contains some (푈, 푥) where 푥 : 푈 → 푋 is a split epimorphism in u�.

Properties of Proposition. For each object 푋 in u�, let 햩(̂ 푋) be the set of 햩-covering covering sinks sinks on 푋. Then 햩̂ is the smallest saturated coverage on u� containing 햩.

Proof. Apply proposition a.2.14. ■

a.2.16 ¶ Let 퐴 be a presheaf on u�.

Definition.A 햩-local generating set of elements of 퐴 is a set Φ of elements of 퐴 with the following property: • For every 햩-closed subpresheaf 퐴′ ⊆ 퐴, if Φ is contained in the set of elements of 퐴′, then 퐴′ = 퐴.

Example. Clearly, the set of elements of 퐴 itself is a 햩-local generating set of elements.

Local gener- Lemma. Let 휅 be a regular cardinal and assume the following hypotheses: ating sets of • For every object 푋 in u�, every 햩-covering sink on 푋 contains a 휅-small locally gener- 햩-covering sink. able presheaves • 퐴 admits a 휅-small 햩-local generating set of elements. Then, for every 햩-local generating set Φ of elements of 퐴, there is a 휅- small subset Φ′ ⊆ Φ such that Φ′ is also a 햩-local generating set of elements of 퐴.

Proof. Let Θ be a 휅-small 햩-local generating set of elements. For each , there is a -small subset ′ such that is con- (푋, 푎) ∈ Θ 휅 Φ(푋,푎) ⊆ Φ (푋, 푎) tained in every -closed subpresheaf of containing ′ . Thus, taking 햩 퐴 Φ(푋,푎) ′ ′ , we obtain the desired -small -local generating Φ = ⋃(푋,푎)∈Θ Φ(푋,푎) 휅 햩 set of elements of 퐴. ■

a.2.17 Definition. A morphism ℎ : 퐴 → 퐵 in Psh(u�) is 햩-locally injective if

the relative diagonal Δℎ : 퐴 → Kr(ℎ) is 햩-locally surjective.

219 Generalities

Properties of Proposition. locally injective (i) Every monomorphism in Psh(u�) is 햩-locally injective. morphisms (ii) The class of 햩-locally injective morphisms of presheaves on u� is a class of separated fibrations in Psh(u�). (iii) Given morphisms ℎ : 퐴 → 퐵 and 푘 : 퐵 → 퐶 in Psh(u�), if the composite 푘 ∘ ℎ : 퐴 → 퐶 is 햩-locally injective, then ℎ : 퐴 → 퐵 is also 햩-locally injective.

Proof. (i) and (ii). Apply proposition 1.1.7 to proposition a.2.14.

(iii). By hypothesis, the relative diagonal Δ푘∘ℎ : 퐴 → Kr(푘 ∘ ℎ) is 햩- locally surjective; but Kr(ℎ) ⊆ Kr(푘 ∘ ℎ), so by lemma 1.1.3, the relative

diagonal Δℎ : 퐴 → Kr(ℎ) is also 햩-locally surjective. ■

a.2.18 Definition. A morphism in Psh(u�) is 햩-locally bijective if it is both 햩-locally injective and 햩-locally surjective.

Locally injective, Lemma. Let ℎ : 퐴 → 퐵 and 푘 : 퐵 → 퐶 be morphisms in Psh(u�). locally surjective, (i) If ℎ : 퐴 → 퐵 is 햩-locally surjective and 푘 ∘ ℎ : 퐴 → 퐶 is 햩-locally and locally injective, then is also -locally injective. bijective 푘 : 퐵 → 퐶 햩 morphisms (ii) If 푘 : 퐵 → 퐶 is 햩-locally injective and 푘 ∘ ℎ : 퐴 → 퐶 is 햩-locally surjective, then ℎ : 퐴 → 퐵 is also 햩-locally surjective.

Proof. (i). We have the following commutative square in Psh(u�), Δ 퐴 푘∘ℎ Kr(푘 ∘ ℎ)

ℎ 퐵 Kr(푘) Δ푘 and by proposition a.2.14, Kr(푘 ∘ ℎ) → Kr(푘) is 햩-locally surjective. But

Δ푘∘ℎ : 퐴 → Kr(푘 ∘ ℎ) is also 햩-locally surjective, so it follows that Δ푘 : 퐵 → Kr(푘) is 햩-locally surjective, as required.

(ii). We have the following pullback square in Psh(u�), 푞 Pb(푘 ∘ ℎ, 푘) 퐵

푝 푘 퐴 퐶 푘∘ℎ

220 a.2. Coverages

and since 푘 ∘ ℎ : 퐴 → 퐶 is 햩-locally surjective, 푞 : Pb(푘 ∘ ℎ, 푘) → 퐵 is also 햩-locally surjective. On the other hand, by the pullback pasting lemma, we also have a pullback square in Psh(u�) of the form below,

ℎ 퐴 퐵

⟨id퐴,ℎ⟩ Δ푘 Pb(푘 ∘ ℎ, 푘) Kr(푘)

and since 푘 : 퐵 → 퐶 is 햩-locally injective, ⟨id퐴, ℎ⟩ : 퐴 → Pb(푘 ∘ ℎ, 푘)

is 햩-locally surjective. But ℎ ∘ 푝 ∘ ⟨푖푑퐴, ℎ⟩ = ℎ = 푞 ∘ ⟨푖푑퐴, ℎ⟩, and the latter is the composite of two 햩-locally surjective morphisms in Psh(u�), so ℎ : 퐴 → 퐵 is also 햩-locally surjective. ■

Properties of Proposition. locally bijective (i) Every isomorphism in Psh(u�) is 햩-locally bijective. morphisms of The class of -locally bijective morphisms in has the 2- presheaves (ii) 햩 Psh(u�) out-of-6 property.

Proof. Apply propositions a.2.14 and a.2.17 and lemma a.2.18. ■

a.2.19 Definition.A 햩-weak pullback diagram in Psh(u�) is a commutative diagram in Psh(u�) of the form below,

푃 퐴1

ℎ1

퐴0 퐵 ℎ0

where the induced morphism 푃 → Pb(ℎ0, ℎ1) is 햩-locally surjective.

Weak pullback Lemma. Consider a commutative diagram in Psh(u�) of the form below: pasting lemma 퐴″ 퐴′ 퐴

퐵″ 퐵′ 퐵

(i) If both squares are 햩-weak pullback diagrams in Psh(u�), then the outer rectangle is also a 햩-weak pullback diagram in Psh(u�).

221 Generalities

(ii) If the right square is a pullback diagram in Psh(u�) and the outer rectangle is a 햩-weak pullback diagram in Psh(u�), then the left square is also a 햩-weak pullback diagram in Psh(u�).

Proof. Straightforward. (Use proposition a.2.14 and the ordinary pullback pasting lemma.) ⧫

222 a.3. Sheaves

a.3 Sheaves

Synopsis. We recall some of the basic theory of sheaves on a site.

Prerequisites. §1.1, a.1, a.2.

a.3.1 ※ Throughout this section, u� is a category and 햩 is a coverage on u�.

a.3.2 Definition. A presheaf 퐴 on u� is 햩-separated if it has the following property:

• For every object 푋 in u� and every Φ ∈ 햩(푋), 퐴 satisfies the separation condition with respect to the sieve ↓(Φ) on 푋.

Recognition prin- Lemma. Let 퐴 be a presheaf on u�. The following are equivalent: ciple for separ- (i) 퐴 is a 햩-separated presheaf on u�. ated presheaves The image of the diagonal is a -closed subpre- (ii) Δ퐴 : 퐴 → 퐴 × 퐴 햩 sheaf of 퐴 × 퐴.

Proof. In view of lemma a.2.2, this is a special case of lemma 1.1.16. ■

a.3.3 ¶ Let 퐴 be a presheaf on u�, let 푅 be an equivalence relation on 퐴, and let 퐴/푅 be the quotient presheaf.

Quotients by Lemma. The following are equivalent: closed equival- (i) 푅 is a 햩-closed subpresheaf of 퐴 × 퐴. ence relations (ii) 퐴/푅 is a 햩-separated presheaf on u�.

Proof. (i) ⇒ (ii). Let 푋 be an object in u�, let 0̄푎 ,1 ̄푎 : h푋 → 퐴/푅 be morphisms in Psh(u�), and let 푞 : 퐴 → 퐴/푅 be the quotient morphism.

Since 푞 : 퐴 → 퐴/푅 is an epimorphism, we may choose 푎0, 푎1 : h푋 → 퐴

such that 0̄푎 = 푞 ∘ 푎0 and 1̄푎 = 푞 ∘ 푎1. (Here, we are using the Yoneda lemma.) Suppose u� is a member of 햩(푋) such that we have a commutative square in Psh(u�) of the form below,

푆u� h푋

1̄푎

h푋 퐴 푅 0̄푎 /

223 Generalities

where 푆u� is the subpresheaf of h푋 corresponding to u� . We then have the following commutative diagram in Psh(u�),

푆u� Kr(푞) 퐴/푅

Δ 퐴/푅

h푋 퐴 × 퐴 (퐴/푅) × (퐴/푅) ⟨푎0,푎1⟩ 푞×푞

where the dashed arrow exists because the right half is a pullback square. By construction, Kr(푞) is 푅, and 푅 is 햩-closed in 퐴 × 퐴, so we must have

푎0 = 푎1. Hence, 0̄푎 =1 ̄푎 , as required.

(ii) ⇒ (i). This is a consequence of lemma a.3.2 and the fact that the preimage of a 햩-closed subpresheaf is also a 햩-closed subpresheaf. ■

Remark. In particular, every 햩-closed equivalence relation on 퐴 is of the form Kr(ℎ) for some epimorphism ℎ : 퐴 → 퐵 in Psh(u�) where 퐵 is a 햩-separated presheaf on u�.

a.3.4 Lemma. Let 퐴 be a presheaf on u�, let 푅 be a subpresheaf of 퐴 × 퐴, and The closure of a let 푅̄ be the smallest 햩-closed subpresheaf of 퐴 × 퐴 such that 푅 ⊆ 푅̄. binary relation on a presheaf (i) If 푅 is a reflexive relation on 퐴, then 푅̄ is also a reflexive relation on 퐴.

(ii) If 푅 is a symmetric relation on 퐴, then 푅̄ is also a symmetric relation on 퐴.

(iii) If 푅 is a transitive relation on 퐴, then 푅̄ is also a transitive relation on 퐴.

Proof. (i) and (ii). Straightforward.

(iii). Let 푅 ×퐴 푅 and 푅̄ ×퐴 푅̄ be the subpresheaves of 퐴 × 퐴 × 퐴 defined as follows:

푅 ×퐴 푅 = (푅 × 퐴) ∩ (퐴 × 푅)

푅̄ ×퐴 푅̄ = (푅̄ × 퐴) ∩ (퐴 × 푅̄)

224 a.3. Sheaves

Clearly, 푅 ×퐴 푅 ⊆ 푅̄ ×퐴 푅̄; moreover, by proposition a.2.14, 푅 ×퐴 푅 is

a 햩-dense subpresheaf of 푅̄ ×퐴 푅̄. Let 푑1 : 푅̄ ×퐴 푅̄ → 퐴 × 퐴 be defined as follows:

푑1(푎0, 푎1, 푎2) = (푎0, 푎2) We then have a commutative diagram in Psh(u�) of the form below:

푅 ×퐴 푅 푅̄

푅̄ ×퐴 푅̄ 퐴 × 퐴 푑1

Since 푅 ×퐴 푅 is 햩-dense in 푅̄ ×퐴 푅̄ and 푅̄ is 햩-closed in 퐴 × 퐴, 푑1 :

푅̄ ×퐴 푅̄ → 퐴 × 퐴 factors through the inclusion 푅̄ ↪ 퐴 × 퐴. Thus, 푅̄ is indeed a transitive relation on 퐴. ■

a.3.5 Definition.A 햩-sheaf on u� is a presheaf 퐴 on u� with the following property:

• For every object 푋 in u� and every Φ ∈ 햩(푋), 퐴 satisfies the sheaf condition with respect to the sieve ↓(Φ) on 푋.

We write Sh(u�, 햩) for the full subcategory of Psh(u�) spanned by the 햩-sheaves.

Lemma. Let 퐴 be a presheaf on u�. The following are equivalent:

(i) 퐴 is a 햩-sheaf on u�.

(ii) 퐴 satisfies the sheaf condition with respect to every 햩-covering sieve on 푋.

Proof. (i) ⇒ (ii). Apply lemma a.2.3(c) and corollary a.2.3(b).

(ii) ⇒ (i). Immediate. ■

a.3.6 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Psh(u�). Assuming 퐵 is 햩- Locally injective separated, the following are equivalent: morphisms of (i) is a monomorphism in Psh . presheaves ℎ : 퐴 → 퐵 (u�) with separ- (ii) ℎ : 퐴 → 퐵 is a 햩-locally injective morphism in Psh(u�). ated codomain 225 Generalities

Proof. (i) ⇒ (ii). See proposition a.2.17.

(ii) ⇒ (i). By lemma a.3.2, the image of Δ퐵 : 퐵 → 퐵 × 퐵 is a 햩-closed subpresheaf of 퐵 × 퐵. It is clear that the preimage of a 햩-closed subpresheaf is also a 햩-closed subpresheaf, hence Kr(ℎ) is a 햩-closed subpresheaf

of 퐴 × 퐴. But the relative diagonal Δℎ : 퐴 → Kr(ℎ) is 햩-locally surjective, so it is an isomorphism. Thus, ℎ : 퐴 → 퐵 is a monomorphism in Psh(u�). ■

a.3.7 Definition.A 햩-sheaf completion of a presheaf 퐴 on u� is a pair (퐴,̂ 푖) where 퐴̂ is a 햩-sheaf on u� and 푖 : 퐴 → 퐴̂ is a 햩-locally bijective morphism in Psh(u�).

Sheaves are right Proposition. If 퐹 is a 햩-sheaf on u� and ℎ : 퐴 → 퐵 is a 햩-locally bijective orthogonal to morphism in Psh(u�), then locally bijective

morphisms HomPsh(u�)(ℎ, 퐹 ) : HomPsh(u�)(퐵, 퐹 ) → HomPsh(u�)(퐴, 퐹 )

is a bijection.

Proof. Let 푠 : 퐴 → 퐹 be a morphism in Psh(u�). We must show that there is a unique morphism ̄푠: 퐵 → 퐹 in Psh(u�) such that ̄푠∘ ℎ = 푠. Since 퐹 is a 햩-separated presheaf on u�, by lemma a.3.2, Kr(푠) is a 햩- closed subpresheaf of 퐴 × 퐴. On the other hand, the relative diagonal

Δℎ : 퐴 → Kr(ℎ) is 햩-locally surjective, so we have Kr(ℎ) ⊆ Kr(푠). Thus, recalling lemma a.2.18, we may assume without loss of generality that ℎ : 퐴 → 퐵 is a 햩-locally surjective monomorphism in Psh(u�). Let (푋, 푏) be an element of 퐵 and let u� be the sieve on 푋 where (푈, 푥) is in u� if and only if 푏 ⋅ 푥 is in the image of ℎ : 퐴 → 퐵. Then, we have the following pullback square in Psh(u�),

푆 퐴 ℎ h 퐵 푋 푏⋅−

where 푆 ⊆ h푋 is the subpresheaf corresponding to u� , so by proposition a.2.14, u� is a 햩-covering sieve on 푋. Hence, by lemma a.3.5, there

226 a.3. Sheaves

is a unique 푐 ∈ 퐹 (푋) such that, for every (푈, 푥) in u� , we have 푐⋅푥 = 푠(푎) where 푎 is the unique 푎 ∈ 퐴(푈) such that ℎ(푎) = 푏 ⋅ 푥. This defines a morphism ̄푠: 퐵 → 퐹 extending 푠 : 퐴 → 퐹 along ℎ : 퐴 → 퐵, and it is straightforward to check that it is the unique such extension. ■

Sheaf comple- Corollary. Let 퐴 be a presheaf on u�. If (퐴,̂ 푖) is a 햩-sheaf completion tions are of 퐴, then (퐴,̂ 푖) is an initial object in the comma category (퐴 ↓ Sh(u�, 햩)). reflections Proof. This is an immediate consequence of proposition a.3.7. ■

Remark. In particular, 햩-sheaf completions are unique up to unique isomorphism.

a.3.8 ¶ Let 퐴 be a presheaf on u�. For general reasons, there is a 햩-sheaf completion of 퐴, but it is convenient to have a more explicit construction.

Definition. The presheaf of 햩-local sections of 퐴, or Grothendieck plus construction for 퐴 with respect to to 햩, is the presheaf 퐴+ on u� defined as follows,

+ ̌ 0 lim Hom 퐴 (푋) = H햩 (푋, 퐴) = op Psh(u�)(푆u� , 퐴) −−→u� :CSv햩(푋)

where 푆u� is the subpresheaf of h푋 corresponding to the sieve u� . The + unit 휄퐴 : 퐴 → 퐴 is given at each object 푋 in u� by the component of the colimiting cocone corresponding to the maximal sieve on 푋.

Remark. Note that the above indeed defines a presheaf on u�. Moreover, op proposition a.2.14 implies that CSv햩(푋) is a directed poset, so the colimit appearing in the definition is very well behaved. In addition, 퐴+ is clearly functorial in 퐴.

a.3.8(a) Lemma. The evident endofunctor on Psh(u�) defined by 퐴 ↦ 퐴+ preserves limits of finite diagrams.

Proof. Straightforward. (Use the fact that lim : [ℐ, Set] → Set pre- −−→ℐ serves limits of finite diagrams when ℐ is a directed poset.) ⧫

227 Generalities

The unit + is a -locally bijective morphism in a.3.8(b) Lemma. 휄퐴 : 퐴 → 퐴 햩 Psh(u�).

Proof. Let 푋 be an object in u� and let 푎0, 푎1 : h푋 → 퐴 be morphisms in Psh(u�). Using the Yoneda lemma and the explicit construction of filtered

colimits in Set, we see that 휄퐴 ∘ 푎0 = 휄퐴 ∘ 푎1 if and only if there is some 햩-covering sieve u� on 푋 such that the diagram below commutes,

푆u� h푋

푎1 h 퐴 푋 푎0

where 푆u� is the subpresheaf of h푋 corresponding to u� . Thus, the relative diagonal Kr is -locally surjective, i.e. + is a Δ휄퐴 : 퐴 → (휄퐴) 햩 휄퐴 : 퐴 → 퐴 햩-locally injective. + On the other hand, for any morphism 푏 : h푋 → 퐴 in Psh(u�), there

is a 햩-covering sieve u� on 푋 and a morphism 푎 : 푆u� → 퐴 such that the diagram below commutes:

푎 푆u� 퐴

휄퐴

h 퐴+ 푋 푏 + Thus, 휄퐴 : 퐴 → 퐴 is also 햩-locally surjective. ■

a.3.8(c) Lemma. (i) 퐴+ is a 햩-separated presheaf on u�. (ii) If 퐴 is a 햩-separated presheaf on u�, then 퐴+ is a 햩-sheaf on u�.

Proof. See e.g. [ML–M, Ch. III, §5] or Proposition 2.2.6 in [Johnstone, 2002, Part C]. □

a.3.8(d) Proposition. Let 퐴++ be the twice-iterated Grothendieck plus construction and let ++ be . Then ++ is a -sheaf 휂퐴 : 퐴 → 퐴 휄퐴+ ∘ 휄퐴 (퐴 , 휂퐴) 햩 completion of 퐴.

Proof. Apply proposition a.2.18 to lemmas a.3.8(b) and a.3.8(c). ■

228 a.3. Sheaves

a.3.9 ¶ To summarise:

Theorem. (i) Every presheaf on u� has a 햩-sheaf completion. (ii) In particular, the inclusion Sh(u�, 햩) ↪ Psh(u�) has a left adjoint. (iii) Moreover, any such left adjoint preserves finite limits.

Proof. (i). See proposition a.3.8(d).

(ii). Apply corollary a.3.7.

(iii). Apply lemma a.3.8(a) (twice). ■

a.3.10 Lemma. Let ℎ : 퐴 → 퐵 be a morphism in Sh(u�, 햩). The following are Recognition prin- equivalent: ciple for locally (i) is -locally surjective. surjective morph- ℎ : 퐴 → 퐵 햩 isms of sheaves (ii) ℎ : 퐴 → 퐵 is an effective epimorphism in Sh(u�, 햩).

(iii) ℎ : 퐴 → 퐵 is an epimorphism in Sh(u�, 햩).

Proof. (i) ⇒ (ii). Let 퐵′ be the presheaf image of ℎ : 퐴 → 퐵 and let ℎ′ : 퐴 → 퐵′ be the induced morphism. Then 퐵′ is a 햩-dense subpresheaf ′ ++ ++ of 퐵, so the induced morphism (퐵 ) → 퐵 is an isomorphism, by propositions a.3.7 and a.3.8(d). On the other hand, ℎ′ : 퐴 → 퐵′ is ′ ++ ++ ′ ++ an effective epimorphism in Psh(u�), so (ℎ ) : 퐴 → (퐵 ) is an effective epimorphism in Sh(u�, 햩), by theorem a.3.9. Thus, ℎ : 퐴 → 퐵 is also an effective epimorphism in Sh(u�, 햩).

(ii) ⇒ (iii). Immediate.

(iii) ⇒ (i). See Corollary 5 in [ML–M, Ch. III, §7]. ■

Let be a family of -sheaves on , let a.3.11 Lemma. (퐴푖 | 푖 ∈ 퐼) 햩 u� 퐴 = ∐푖∈퐼 퐴푖 Elements of in , and let be the respective coproduct injection. Sh(u�, 햩) ℎ푖 : 퐴푖 → 퐴 coproducts For every element (푋, 푎) of 퐴, there is a 햩-covering sink Φ on 푋 with the of sheaves following property:

• For every , there exist and such that (푈, 푥) ∈ Φ 푖 ∈ 퐼 푎푖 ∈ 퐴푖(푈) . ℎ푖(푎푖) = 푎 ⋅ 푥

229 Generalities

Moreover, if 햩 is composition-closed, then we may also assume that Φ ∈ 햩(푋).

Proof. By theorem a.3.9, the unit of the reflector of Sh(u�, 햩) ⊆ Psh(u�) is componentwise locally bijective. Unwinding definitions, the claim follows. (Recall paragraph a.2.13.) ■

a.3.12 ¶ Let 푋 be an object in u� and let @푋 : Sh(u�, 햩) → Set be the evaluation functor at 푋, i.e. the evident functor given on objects by 퐴 ↦ 퐴(푋).

Assume 햩 is a subcanonical coverage on u�. By the Yoneda lemma, @푋 : Sh(u�, 햩) → Set is a representable functor, so it preserves limits. When does it preserve colimits?

Definition. An object 푋 in u� is 햩-local if every 햩-covering sink on 푋 contains a split epimorphism.

Lemma. The following are equivalent: (i) 푋 is a 햩-local object in u�. has a right adjoint, namely the evident (ii) @푋 : Sh(u�, 햩) → Set functor defined on objects as follows, ∇푋 : Set → Sh(u�, 햩)

u�(푋,−) ∇푋푇 = 푇

with counit given by evaluation at . @푋∇푋푇 → 푇 id푋 preserves colimits. (iii) @푋 : Sh(u�, 햩) → Set

Proof. (i) ⇒ (ii). It is straightforward to verify that we have the following adjunction, @푋 Set ⊥ Psh(u�) ∇푋 where the functors and the counit are defined as above. To complete the

proof, it is enough to verify that ∇푋푇 is a 햩-sheaf on u� for every set 푇 .

It is not hard to see that @푋 : Psh(u�) → Set sends 햩-locally surjective

morphisms in Psh(u�) to surjections. But @푋 : Psh(u�) → Set preserves

monomorphisms, so by lemma a.2.2 and adjointness, ∇푋푇 is indeed a 햩-sheaf on u�.

230 a.3. Sheaves

(ii) ⇒ (iii). Immediate.

(iii) ⇒ (i). Let Φ be a covering sink on 푋. By lemma a.3.10, the induced

morphism ∐(푈,푥)∈Φ h푈 → h푋 in Sh(u�, 햩) is an (effective) epimorphism;

but @푋 : Sh(u�, 햩) → Set preserves coproducts and epimorphisms, so the induced map u�(푈, 푋) → u�(푋, 푋) ∐ (푈,푥)∈Φ is surjective. Hence, there is (푈, 푥) ∈ Φ such that 푥 : 푈 → 푋 is a split epimorphism in u�. ■

a.3.13 ¶ Let u� be a category and let 햪 be a coverage on u�.

Definition.A pre-admissible functor 퐹 : (u�, 햩) → (u�, 햪) is a functor 퐹 : u� → u� with the following property: • For every 햪-sheaf 퐵 on u�, 퐹 ∗퐵 is an 햩-sheaf on u�.

Pushforward of Proposition. If 퐹 : (u�, 햩) → (u�, 햪) is a pre-admissible functor, then the sheaves along restriction functor 퐹 ∗ : Sh(u�, 햪) → Sh(u�, 햩) has a left adjoint. pre-admissible functors Proof. Let 퐴 be a 햩-sheaf on u�. By theorem a.3.9, Sh(u�, 햪) has colimits of all (small) diagrams, so by proposition a.1.4 (and the Yoneda lemma), ∗ there exist a 햪-sheaf 퐹!퐴 on u� and a morphism 휂퐴 : 퐴 → 퐹 퐹!퐴 in Sh(u�, 햩) such that, for every 햪-sheaf 퐵 on u�, the following is a bijection:

∗ HomSh(u�,햪)(퐹!퐴, 퐵) → HomSh(u�,햩)(퐴, 퐹 퐵) ∗ ℎ ↦ 퐹 ℎ ∘ 휂퐴

Indeed, we may take 퐹 퐴 = lim 푘∗h , where 푘∗h is the 햪- ! −−→(푋,푎):El(퐴) 퐹 푋 퐹 푋 sheaf completion of the representable presheaf h퐹 푋 on u�. ■

Pre-admissible Corollary. If 퐹 : (u�, 햩) → (u�, 햪) is a pre-admissible functor, then 퐹 functors preserve sends 햩-covering morphisms in u� to 햪-covering morphisms in u�. covering morphisms Proof. Combine lemmas a.2.18 and a.3.10 with proposition a.3.13. ■

231

Bibliography

De Jong, Aise Johan et al. [Stacks] Stacks Project. url: http://math.columbia.edu/algebraic_geometry/stacks-git.

Demazure, Michel and Pierre Gabriel [1970] Groupes algébriques. Tome I: Géométrie algébrique, généralités, groupes commutatifs. With an appendix, Corps de classes local, by Michiel Hazewinkel. Paris: Masson & Cie, Éditeur, 1970. xxvi+700.

Gran, Marino and Enrico M. Vitale [1998] ‘On the exact completion of the homotopy category’. In: Cahiers Topologie Géom. Différentielle Catég. 39.4 (1998), pp. 287–297. issn: 0008-0004.

Johnstone, Peter T. [2002] Sketches of an elephant: a topos theory compendium. Vol. 1. Oxford Logic Guides 43. New York: The Clarendon Press Oxford University Press, 2002. xxii+468+71. isbn: 0-19-853425-6; Johnstone, Peter T. Sketches of an elephant: a topos theory compendium. Vol. 2. Oxford Logic Guides 44. Oxford: The Clarendon Press Oxford University Press, 2002. i–xxii, 469–1089 and I1–I71. isbn: 0-19-851598-7.

Joyal, André and Ieke Moerdijk [JM] ‘A completeness theorem for open maps’. In: Ann. Pure Appl. Logic 70.1 (1994), pp. 51–86. issn: 0168-0072. doi: 10.1016/0168-0072(94)90069-8.

233 Bibliography

Lurie, Jacob [DAG 5] Derived algebraic geometry V: Structured spaces. 4th May 2009. arXiv: 0905.0459.

Mac Lane, Saunders [CWM] Categories for the working mathematician. Second. Graduate Texts in Mathematics 5. New York: Springer-Verlag, 1998. xii+314. isbn: 0-387-98403-8.

Mac Lane, Saunders and Ieke Moerdijk [ML–M] Sheaves in geometry and logic. A first introduction to topos theory. Corrected. Universitext. New York: Springer-Verlag, 1992. xii+629. isbn: 0-387-97710-4.

Shulman, Michael A. [2012] ‘Exact completions and small sheaves’. In: Theory Appl. Categ. 27 (2012), pp. 97–173. issn: 1201-561X.

234 Index

atlas, 109 bisifted —, 14 — of a scheme, 195 cosifted —, 14 smooth —, 162 quadrable —, 2 sifted —, 14 basis for a gros pretopos, 108 class of separated fibrations, 8 unary —, 108 cocartesian fibration discrete —, 204 calypsis, 31 collection axiom quadrable —, 32 local —, 24 canonical projection — of the category of elements, 200 complex cartesian fibration local —, 56 discrete —, 204 composite category — of a tree, 214 — of elements, see category of coproduct elements strict —, 62 — of families, 203 coverage, 215 coherent —, 36 adapted —, 89 exact —, 56 canonical —, 61, 216 extensive —, 63 chaotic —, 215 regular —, 34 composition-closed —, 215 regulated —, 36 extensive —, 70 category of elements, 200 fppf —, 184 charted object, 139 fpqc —, 184 class of étale morphisms, 102 saturated —, 215 class of fibrations, 2 subcanonical —, 216 class of morphisms superextensive —, 71 — closed under pullback, 2 trivial —, 215

235 Index

unary —, 41 fibrant object, 3 saturated —, 42 fibration, see also class of fibrations summable —, 67 — of simplicial sets upward-closed —, 41 discrete —, 153 upward-closed —, 215 discrete —, see cartesian fibration, Zariski —, 184 discrete — étale —, 184 fork, 50 exact —, 50 descent axiom left-exact —, 50 — for an ecumene, 90 mid-exact —, 50 — for open embeddings in an right-exact —, 50 ecumene, 97 functor ecumene, 89 admissible —, 59, 140 admissible —, 138 pre-admissible —, 231 effective —, 142 regular —, 35 étale —, 95 regulated —, 39 extensive —, 92 regulated —, 91 generating set element — of elements of a presheaf, 206 — of a presheaf, 200 local —, 219 embedding, 31 generator open —, 97 — of a presheaf epimorphism local —, 43 strict —, 40 homomorphism of rings universally —, 40 — of finite presentation, 176 equaliser étale —, 177 — of morphisms of presheaves, faithfully flat —, 175 202 flat —, 174 equivalence relation, 40 fppf —, 178 — on a presheaf, 207 étale —, 94 image tractable —, 99 exact —, 32 exact completion, 58 initial object extent, 110 strict —, 62

family, 203 kernel relation

236 Index

— of morphisms of presheaves, eucalyptic —, 32 202 quadrably —, 32 eunoic —, 93 leaf, 214 genial —, 93 limit laminar —, 113 weak —, 44 perfect —, 9 local homeomorphism, 97 quadrable —, 1 local isomorphism regulated —, 36 — of presheaves, 189 separated —, 5 manifold, 163 morphism of presheaves, 200 map — familially of a given type, 17 perfect —, 150 — locally of a given type, 22 proper —, 147 — of a given type locally on the semiproper —, 146 base, 19 matrix of morphisms, 203 — of a given type semilocally on — of a given type, 17 the base, 18 monomorphism — semilocally of a given type, 21 complemented —, 66 locally bijective —, 220 morphism locally injective —, 219 — locally of a given type, 27 locally surjective —, 218 — of a given type locally on the étale —, 185 base, 21 nerve — of a given type locally on the — of a category, 158 domain, 15 — of a given type semilocally on object the base, 21 — right orthogonal to a morphism, — of a given type semilocally on 10 the domain, 14 charted —, see charted object — right orthogonal to a morphism, local —, 230 10 localic —, 112 — semilocally of a given type, 27 separated —, 4 agathic —, 33 open immersion covering —, 218 — of presheaves, 186 étale —, 93, see also class of étale open quasilocalisation, 182 morphisms opfibration

237 Index

discrete —, see cocartesian — morphism, see morphism, fibration, discrete — quadrable — class of morphisms, see class of plus construction morphisms, quadrable — Grothendieck —, see presheaf, — quotient of local sections — of a presheaf, 208 presentation exact —, 41 — of a presheaf topological — local —, 43 universal —, 145 presheaf, 199 — of local sections, 227 relative point of view, 1 — represented by a manifold, 163 right orthogonal familial representation of —, 202 morphism — to a morphism, see familially representable —, 202 morphism, — right generable —, 206 orthogonal to a morphism locally —, 43 object — to a morphism, see morphism of —, see morphism of object, — right orthogonal to presheaves a morphism presentable — locally —, 43 scheme representable —, 202 — of a given type, 195 representation of —, 202 Segal condition separated —, 223 strict —, 157 pretopos, 63 separated fibration, see also class of gros —, 106 separated fibrations associated —, 138 separation condition, 209 localic —, 78 sheaf, 225 petit —, 107 — presented by a local complex, pullback 57 — of morphisms of presheaves, generable —, 87 201 presentable —, 88 pullback diagram sheaf completion, 226 weak —, 221 sheaf condition, 209 Shulman condition, 44 quadrable sieve

238 Index

— generated by a set of objects, 206 — of a category, 206 — on an object, 206 covering —, 218 principal —, 206 pullback —, 207 strict-epimorphic —, 213 universally —, 213 sink, 212 covering —, 218 strict-epimorphic —, 213 universally —, 213 site — for a pretopos, 77 unary —, 77 slice category, 201 subobject, 31 subpresheaf, 201 closed — — with respect to a coverage, 217 dense —, 218 support closed — of a morphism of presheaves, 217 tautological cocone, 200 topological space — of given type, 170 tree, 214, 216 proper —, 214 union disjoint —, 61 exact —, 34

239