Notes on Topos Theory
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Sheaves and Homotopy Theory
SHEAVES AND HOMOTOPY THEORY DANIEL DUGGER The purpose of this note is to describe the homotopy-theoretic version of sheaf theory developed in the work of Thomason [14] and Jardine [7, 8, 9]; a few enhancements are provided here and there, but the bulk of the material should be credited to them. Their work is the foundation from which Morel and Voevodsky build their homotopy theory for schemes [12], and it is our hope that this exposition will be useful to those striving to understand that material. Our motivating examples will center on these applications to algebraic geometry. Some history: The machinery in question was invented by Thomason as the main tool in his proof of the Lichtenbaum-Quillen conjecture for Bott-periodic algebraic K-theory. He termed his constructions `hypercohomology spectra', and a detailed examination of their basic properties can be found in the first section of [14]. Jardine later showed how these ideas can be elegantly rephrased in terms of model categories (cf. [8], [9]). In this setting the hypercohomology construction is just a certain fibrant replacement functor. His papers convincingly demonstrate how many questions concerning algebraic K-theory or ´etale homotopy theory can be most naturally understood using the model category language. In this paper we set ourselves the specific task of developing some kind of homotopy theory for schemes. The hope is to demonstrate how Thomason's and Jardine's machinery can be built, step-by-step, so that it is precisely what is needed to solve the problems we encounter. The papers mentioned above all assume a familiarity with Grothendieck topologies and sheaf theory, and proceed to develop the homotopy-theoretic situation as a generalization of the classical case. -
Topos Theory
Topos Theory Olivia Caramello Sheaves on a site Grothendieck topologies Grothendieck toposes Basic properties of Grothendieck toposes Subobject lattices Topos Theory Balancedness The epi-mono factorization Lectures 7-14: Sheaves on a site The closure operation on subobjects Monomorphisms and epimorphisms Exponentials Olivia Caramello The subobject classifier Local operators For further reading Topos Theory Sieves Olivia Caramello In order to ‘categorify’ the notion of sheaf of a topological space, Sheaves on a site Grothendieck the first step is to introduce an abstract notion of covering (of an topologies Grothendieck object by a family of arrows to it) in a category. toposes Basic properties Definition of Grothendieck toposes Subobject lattices • Given a category C and an object c 2 Ob(C), a presieve P in Balancedness C on c is a collection of arrows in C with codomain c. The epi-mono factorization The closure • Given a category C and an object c 2 Ob(C), a sieve S in C operation on subobjects on c is a collection of arrows in C with codomain c such that Monomorphisms and epimorphisms Exponentials The subobject f 2 S ) f ◦ g 2 S classifier Local operators whenever this composition makes sense. For further reading • We say that a sieve S is generated by a presieve P on an object c if it is the smallest sieve containing it, that is if it is the collection of arrows to c which factor through an arrow in P. If S is a sieve on c and h : d ! c is any arrow to c, then h∗(S) := fg | cod(g) = d; h ◦ g 2 Sg is a sieve on d. -
Noncommutative Stacks
Noncommutative Stacks Introduction One of the purposes of this work is to introduce a noncommutative analogue of Artin’s and Deligne-Mumford algebraic stacks in the most natural and sufficiently general way. We start with quasi-coherent modules on fibered categories, then define stacks and prestacks. We define formally smooth, formally unramified, and formally ´etale cartesian functors. This provides us with enough tools to extend to stacks the glueing formalism we developed in [KR3] for presheaves and sheaves of sets. Quasi-coherent presheaves and sheaves on a fibered category. Quasi-coherent sheaves on geometric (i.e. locally ringed topological) spaces were in- troduced in fifties. The notion of quasi-coherent modules was extended in an obvious way to ringed sites and toposes at the moment the latter appeared (in SGA), but it was not used much in this generality. Recently, the subject was revisited by D. Orlov in his work on quasi-coherent sheaves in commutative an noncommutative geometry [Or] and by G. Laumon an L. Moret-Bailly in their book on algebraic stacks [LM-B]. Slightly generalizing [R4], we associate with any functor F (regarded as a category over a category) the category of ’quasi-coherent presheaves’ on F (otherwise called ’quasi- coherent presheaves of modules’ or simply ’quasi-coherent modules’) and study some basic properties of this correspondence in the case when the functor defines a fibered category. Imitating [Gir], we define the quasi-topology of 1-descent (or simply ’descent’) and the quasi-topology of 2-descent (or ’effective descent’) on the base of a fibered category (i.e. -
Arxiv:2008.10677V3 [Math.CT] 9 Jul 2021 Space Ha Hoyepiil Ea Ihtewr Fj Ea N194 in Leray J
On sheaf cohomology and natural expansions ∗ Ana Luiza Tenório, IME-USP, [email protected] Hugo Luiz Mariano, IME-USP, [email protected] July 12, 2021 Abstract In this survey paper, we present Čech and sheaf cohomologies – themes that were presented by Koszul in University of São Paulo ([42]) during his visit in the late 1950s – we present expansions for categories of generalized sheaves (i.e, Grothendieck toposes), with examples of applications in other cohomology theories and other areas of mathematics, besides providing motivations and historical notes. We conclude explaining the difficulties in establishing a cohomology theory for elementary toposes, presenting alternative approaches by considering constructions over quantales, that provide structures similar to sheaves, and indicating researches related to logic: constructive (intuitionistic and linear) logic for toposes, sheaves over quantales, and homological algebra. 1 Introduction Sheaf Theory explicitly began with the work of J. Leray in 1945 [46]. The nomenclature “sheaf” over a space X, in terms of closed subsets of a topological space X, first appeared in 1946, also in one of Leray’s works, according to [21]. He was interested in solving partial differential equations and build up a strong tool to pass local properties to global ones. Currently, the definition of a sheaf over X is given by a “coherent family” of structures indexed on the lattice of open subsets of X or as étale maps (= local homeomorphisms) into X. Both arXiv:2008.10677v3 [math.CT] 9 Jul 2021 formulations emerged in the late 1940s and early 1950s in Cartan’s seminars and, in modern terms, they are intimately related by an equivalence of categories. -
Lecture Notes on Sheaves, Stacks, and Higher Stacks
Lecture Notes on Sheaves, Stacks, and Higher Stacks Adrian Clough September 16, 2016 2 Contents Introduction - 26.8.2016 - Adrian Cloughi 1 Grothendieck Topologies, and Sheaves: Definitions and the Closure Property of Sheaves - 2.9.2016 - NeˇzaZager1 1.1 Grothendieck topologies and sheaves.........................2 1.1.1 Presheaves...................................2 1.1.2 Coverages and sheaves.............................2 1.1.3 Reflexive subcategories.............................3 1.1.4 Coverages and Grothendieck pretopologies..................4 1.1.5 Sieves......................................6 1.1.6 Local isomorphisms..............................9 1.1.7 Local epimorphisms.............................. 10 1.1.8 Examples of Grothendieck topologies..................... 13 1.2 Some formal properties of categories of sheaves................... 14 1.3 The closure property of the category of sheaves on a site.............. 16 2 Universal Property of Sheaves, and the Plus Construction - 9.9.2016 - Kenny Schefers 19 2.1 Truncated morphisms and separated presheaves................... 19 2.2 Grothendieck's plus construction........................... 19 2.3 Sheafification in one step................................ 19 2.3.1 Canonical colimit................................ 19 2.3.2 Hypercovers of height 1............................ 19 2.4 The universal property of presheaves......................... 20 2.5 The universal property of sheaves........................... 20 2.6 Sheaves on a topological space............................ 21 3 Categories -
Notes on Categorical Logic
Notes on Categorical Logic Anand Pillay & Friends Spring 2017 These notes are based on a course given by Anand Pillay in the Spring of 2017 at the University of Notre Dame. The notes were transcribed by Greg Cousins, Tim Campion, L´eoJimenez, Jinhe Ye (Vincent), Kyle Gannon, Rachael Alvir, Rose Weisshaar, Paul McEldowney, Mike Haskel, ADD YOUR NAMES HERE. 1 Contents Introduction . .3 I A Brief Survey of Contemporary Model Theory 4 I.1 Some History . .4 I.2 Model Theory Basics . .4 I.3 Morleyization and the T eq Construction . .8 II Introduction to Category Theory and Toposes 9 II.1 Categories, functors, and natural transformations . .9 II.2 Yoneda's Lemma . 14 II.3 Equivalence of categories . 17 II.4 Product, Pullbacks, Equalizers . 20 IIIMore Advanced Category Theoy and Toposes 29 III.1 Subobject classifiers . 29 III.2 Elementary topos and Heyting algebra . 31 III.3 More on limits . 33 III.4 Elementary Topos . 36 III.5 Grothendieck Topologies and Sheaves . 40 IV Categorical Logic 46 IV.1 Categorical Semantics . 46 IV.2 Geometric Theories . 48 2 Introduction The purpose of this course was to explore connections between contemporary model theory and category theory. By model theory we will mostly mean first order, finitary model theory. Categorical model theory (or, more generally, categorical logic) is a general category-theoretic approach to logic that includes infinitary, intuitionistic, and even multi-valued logics. Say More Later. 3 Chapter I A Brief Survey of Contemporary Model Theory I.1 Some History Up until to the seventies and early eighties, model theory was a very broad subject, including topics such as infinitary logics, generalized quantifiers, and probability logics (which are actually back in fashion today in the form of con- tinuous model theory), and had a very set-theoretic flavour. -
Arxiv:2012.08669V1 [Math.CT] 15 Dec 2020 2 Preface
Sheaf Theory Through Examples (Abridged Version) Daniel Rosiak December 12, 2020 arXiv:2012.08669v1 [math.CT] 15 Dec 2020 2 Preface After circulating an earlier version of this work among colleagues back in 2018, with the initial aim of providing a gentle and example-heavy introduction to sheaves aimed at a less specialized audience than is typical, I was encouraged by the feedback of readers, many of whom found the manuscript (or portions thereof) helpful; this encouragement led me to continue to make various additions and modifications over the years. The project is now under contract with the MIT Press, which would publish it as an open access book in 2021 or early 2022. In the meantime, a number of readers have encouraged me to make available at least a portion of the book through arXiv. The present version represents a little more than two-thirds of what the professionally edited and published book would contain: the fifth chapter and a concluding chapter are missing from this version. The fifth chapter is dedicated to toposes, a number of more involved applications of sheaves (including to the \n- queens problem" in chess, Schreier graphs for self-similar groups, cellular automata, and more), and discussion of constructions and examples from cohesive toposes. Feedback or comments on the present work can be directed to the author's personal email, and would of course be appreciated. 3 4 Contents Introduction 7 0.1 An Invitation . .7 0.2 A First Pass at the Idea of a Sheaf . 11 0.3 Outline of Contents . 20 1 Categorical Fundamentals for Sheaves 23 1.1 Categorical Preliminaries . -
A Grothendieck Site Is a Small Category C Equipped with a Grothendieck Topology T
Contents 5 Grothendieck topologies 1 6 Exactness properties 10 7 Geometric morphisms 17 8 Points and Boolean localization 22 5 Grothendieck topologies A Grothendieck site is a small category C equipped with a Grothendieck topology T . A Grothendieck topology T consists of a collec- tion of subfunctors R ⊂ hom( ;U); U 2 C ; called covering sieves, such that the following hold: 1) (base change) If R ⊂ hom( ;U) is covering and f : V ! U is a morphism of C , then f −1(R) = fg : W ! V j f · g 2 Rg is covering for V. 2) (local character) Suppose R;R0 ⊂ hom( ;U) and R is covering. If f −1(R0) is covering for all f : V ! U in R, then R0 is covering. 3) hom( ;U) is covering for all U 2 C . 1 Typically, Grothendieck topologies arise from cov- ering families in sites C having pullbacks. Cover- ing families are sets of maps which generate cov- ering sieves. Suppose that C has pullbacks. A topology T on C consists of families of sets of morphisms ffa : Ua ! Ug; U 2 C ; called covering families, such that 1) Suppose fa : Ua ! U is a covering family and y : V ! U is a morphism of C . Then the set of all V ×U Ua ! V is a covering family for V. 2) Suppose ffa : Ua ! Vg is covering, and fga;b : Wa;b ! Uag is covering for all a. Then the set of composites ga;b fa Wa;b −−! Ua −! U is covering. 3) The singleton set f1 : U ! Ug is covering for each U 2 C . -
Separating Discs G-Topologies and Grothendieck Topologies
18.727, Topics in Algebraic Geometry (rigid analytic geometry) Kiran S. Kedlaya, fall 2004 More on G-topologies, part 1 (of 2) Leftover from last time: separating discs Here’s a more precise answer to Andre’s question about separating discs (which you need in order to do the reduction of the theorem identifying AF to the connected case). I’ll leave it to you to work out the (easy) reduction of the general case to this specific case. Proposition 1. Given r1 < r2 and c > 0, there exists a rational function f ∈ K(x) such that sup{|f(x) − 1| : |x| ≤ r1} ≤ c, sup{|f(x)| : |x| ≥ r2} ≤ c. Proof. For simplicity, I’m going to consider the special case where there exists r ∈ |K∗| with ∗ r1 < r < r2, and leave the general case as an exercise. Choose a ∈ K with |a| = r, and put g(x) = a/(a − x). Then for |x| ≤ r1, |g(x) − 1| = |x/(a − x)| = |x|/r1 < 1 while for |x| ≥ r2, |g(x)| = |a/(a − x)| = r/|x| < 1. N N Now take N large enough that (r/r2) ≤ c. Then g has the desired bound on the outer disc; in fact, so does any polynomial in gN with integer coefficients. On the other hand, we N also have that |g − 1| ≤ r1/r < 1 on the inner disc; so we can take f = (1 − gN )M − 1 M for M so large that (r1/r) ≤ c. G-topologies and Grothendieck topologies The notion of a G-topology is a special case of the concept of a Grothendieck topology. -
Notes/Exercises on the Functor of Points. Due Monday, Febrary 8 in Class. for a More Careful Treatment of This Material, Consult
Notes/exercises on the functor of points. Due Monday, Febrary 8 in class. For a more careful treatment of this material, consult the last chapter of Eisenbud and Harris' book, The Geometry of Schemes, or Section 9.1.6 in Vakil's notes, although I would encourage you to try to work all this out for yourself. Let C be a category. For each object X 2 Ob(C), define a contravariant functor FX : C!Sets by setting FX (S) = Hom(S; X) and for each morphism in C, f : S ! T define FX (f) : Hom(S; X) ! Hom(T;X) by composition with f. Given a morphism g : X ! Y , we get a natural transformation of functors, FX ! FY defined by mapping Hom(S; X) ! Hom(S; Y ) using composition with g. Lemma 1 (Yoneda's Lemma). Every natural transformation φ : FX ! FY is induced by a unique morphism f : X ! Y . Proof: Given a natural transformation φ, set fφ = φX (id). Exercise 1. Check that φ is the natural transformation induced by fφ. This implies that the functor F that takes an object X of C to the functor FX and takes a morphism in C to the corresponding natural transformation of functors defines a fully faithful functor from C to the category whose objects are functors from C to Sets and whose morphisms are natural transformations of functors. In particular, X is uniquely determined by FX . However, not all functors are of the form FX . Definition 1. A contravariant functor F : C!Sets is set to be representable if ∼ there exists an object X in C such that F = FX . -
Topics in Algebraic Geometry - Talk 3 the Etale´ Site
Topics in Algebraic Geometry - Talk 3 The Etale´ Site Pol van Hoften University of Utrecht, 4001613 [email protected] February 16, 2016 1 Introduction and motivating examples Today, we are going to talk about Grothendieck topologies. A Grothendieck topology is something that (generalizes/axiomatizes) the notion of an open cover of a topological space. 1.1 Set Theory All the categories we work with will be small. If you really want to know how to get a small (and nice) category of schemes, here is a reference [Sta16, Tag 000H]. 1.2 Sheaves on a topological space The first motivating example, and the example to keep in mind during the rest of the lecture, is that of sheaves on a topological space. Let X be a topological space, then there is a partially ordered set of open sets Op(X), which can be viewed as a category. Recall that a presheaf on X is precisely a presheaf on this category Op(X), i.e., a contravariant functor Op(X) ! Set. Let F be a presheaf, then we call F a sheaf if for every open set U and every open cover fUαgα2A of U the following holds: • For all s 2 F (U) such that s = t for all α then s = t; Uα Uα • If sα 2 F (Uα) for all α and sα = sβ then there is an s 2 F (U) such that s = sα (this s Uα,β Uα,β Uα is unique by the first property). Equivalently, the following sequence of sets is an equalizer Y Y F (U) F (Uα) F (Uβ,γ); α β,γ 1 where the parallel arrows are defined as follows: For every pair (β; γ) we have maps Q pβ α2A Uα Uβ pγ Uγ Uβ,γ Q which induces two maps into β,γ F (Uβ,γ).The proof of this will be an exercise. -
HOMOTOPY SPECTRA and DIOPHANTINE EQUATIONS Yuri I. Manin
HOMOTOPY SPECTRA AND DIOPHANTINE EQUATIONS Yuri I. Manin -I- CONTENTS 0. Introduction and summary .................................................................... II 1. Homotopy spectra: a brief presentation ................................................ XIII 2. Diophantine equations: distribution of rational points on algebraic varieties ................................... XIX 3. Rational points, sieves, and assemblers ................................................ XXIV 4. Anticanonical heights and points count ............................................. XXXVI 5. Sieves \beyond heights" ? ..................................................................... XLIX References ..................................................................................................... LIII -II- 0. INTRODUCTION AND SUMMARY • For a long stretch of time in the history of mathematics, Number Theory and Topology formed vast, but disjoint domains of mathematical knowledge. Emmanuel Peyre reminds us in [Pe19] that the Babylonian clay tablet Plimpton 322 (about 1800 BC) contained a list of integer solutions of the \Diophantine" equation a2+b2 = c2: archetypal theme of number theory, named after Diophantus of Alexandria (about 250 BC). Babilonian Clay Tablet Plimpton 322 -III- • \Topology" was born much later, but arguably, its cousin { modern measure theory, { goes back to Archimedes, author of Psammites (\Sand Reckoner"), who was approximately a contemporary of Diophantus. In modern language, Archimedes measures the volume of observable universe