Enriched Model Categories and an Application to Additive Endomorphism Spectra
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Fibrations and Yoneda's Lemma in An
Journal of Pure and Applied Algebra 221 (2017) 499–564 Contents lists available at ScienceDirect Journal of Pure and Applied Algebra www.elsevier.com/locate/jpaa Fibrations and Yoneda’s lemma in an ∞-cosmos Emily Riehl a,∗, Dominic Verity b a Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USA b Centre of Australian Category Theory, Macquarie University, NSW 2109, Australia a r t i c l e i n f o a b s t r a c t Article history: We use the terms ∞-categories and ∞-functors to mean the objects and morphisms Received 14 October 2015 in an ∞-cosmos: a simplicially enriched category satisfying a few axioms, reminiscent Received in revised form 13 June of an enriched category of fibrant objects. Quasi-categories, Segal categories, 2016 complete Segal spaces, marked simplicial sets, iterated complete Segal spaces, Available online 29 July 2016 θ -spaces, and fibered versions of each of these are all ∞-categories in this sense. Communicated by J. Adámek n Previous work in this series shows that the basic category theory of ∞-categories and ∞-functors can be developed only in reference to the axioms of an ∞-cosmos; indeed, most of the work is internal to the homotopy 2-category, astrict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi- categories, we recapture precisely the same category theory developed by Joyal and Lurie, although our definitions are 2-categorical in natural, making no use of the combinatorial details that differentiate each model. In this paper, we introduce cartesian fibrations, a certain class of ∞-functors, and their groupoidal variants. -
Arxiv:2001.09075V1 [Math.AG] 24 Jan 2020
A topos-theoretic view of difference algebra Ivan Tomašić Ivan Tomašić, School of Mathematical Sciences, Queen Mary Uni- versity of London, London, E1 4NS, United Kingdom E-mail address: [email protected] arXiv:2001.09075v1 [math.AG] 24 Jan 2020 2000 Mathematics Subject Classification. Primary . Secondary . Key words and phrases. difference algebra, topos theory, cohomology, enriched category Contents Introduction iv Part I. E GA 1 1. Category theory essentials 2 2. Topoi 7 3. Enriched category theory 13 4. Internal category theory 25 5. Algebraic structures in enriched categories and topoi 41 6. Topos cohomology 51 7. Enriched homological algebra 56 8. Algebraicgeometryoverabasetopos 64 9. Relative Galois theory 70 10. Cohomologyinrelativealgebraicgeometry 74 11. Group cohomology 76 Part II. σGA 87 12. Difference categories 88 13. The topos of difference sets 96 14. Generalised difference categories 111 15. Enriched difference presheaves 121 16. Difference algebra 126 17. Difference homological algebra 136 18. Difference algebraic geometry 142 19. Difference Galois theory 148 20. Cohomologyofdifferenceschemes 151 21. Cohomologyofdifferencealgebraicgroups 157 22. Comparison to literature 168 Bibliography 171 iii Introduction 0.1. The origins of difference algebra. Difference algebra can be traced back to considerations involving recurrence relations, recursively defined sequences, rudi- mentary dynamical systems, functional equations and the study of associated dif- ference equations. Let k be a commutative ring with identity, and let us write R = kN for the ring (k-algebra) of k-valued sequences, and let σ : R R be the shift endomorphism given by → σ(x0, x1,...) = (x1, x2,...). The first difference operator ∆ : R R is defined as → ∆= σ id, − and, for r N, the r-th difference operator ∆r : R R is the r-th compositional power/iterate∈ of ∆, i.e., → r r ∆r = (σ id)r = ( 1)r−iσi. -
1.Introduction Given a Class Σ of Morphisms of a Category X, We Can Construct a Cat- Egory of Fractions X[Σ−1] Where All Morphisms of Σ Are Invertible
Pre-Publicac¸´ oes˜ do Departamento de Matematica´ Universidade de Coimbra Preprint Number 15–41 A CALCULUS OF LAX FRACTIONS LURDES SOUSA Abstract: We present a notion of category of lax fractions, where lax fraction stands for a formal composition s∗f with s∗s = id and ss∗ ≤ id, and a corresponding calculus of lax fractions which generalizes the Gabriel-Zisman calculus of frac- tions. 1.Introduction Given a class Σ of morphisms of a category X, we can construct a cat- egory of fractions X[Σ−1] where all morphisms of Σ are invertible. More X X Σ−1 precisely, we can define a functor PΣ : → [ ] which takes the mor- Σ phisms of to isomorphisms, and, moreover, PΣ is universal with respect to this property. As shown in [13], if Σ admits a calculus of fractions, then the morphisms of X[Σ−1] can be expressed by equivalence classes of cospans (f,g) of morphisms of X with g ∈ Σ, which correspond to the for- mal compositions g−1f . We recall that categories of fractions are closely related to reflective sub- categories and orthogonality. In particular, if A is a full reflective subcat- egory of X, the class Σ of all morphisms inverted by the corresponding reflector functor – equivalently, the class of all morphisms with respect to which A is orthogonal – admits a left calculus of fractions; and A is, up to equivalence of categories, a category of fractions of X for Σ. In [3] we presented a Finitary Orthogonality Deduction System inspired by the left calculus of fractions, which can be looked as a generalization of the Implicational Logic of [20], see [4]. -
Homotopical Categories: from Model Categories to ( ,)-Categories ∞
HOMOTOPICAL CATEGORIES: FROM MODEL CATEGORIES TO ( ;1)-CATEGORIES 1 EMILY RIEHL Abstract. This chapter, written for Stable categories and structured ring spectra, edited by Andrew J. Blumberg, Teena Gerhardt, and Michael A. Hill, surveys the history of homotopical categories, from Gabriel and Zisman’s categories of frac- tions to Quillen’s model categories, through Dwyer and Kan’s simplicial localiza- tions and culminating in ( ;1)-categories, first introduced through concrete mod- 1 els and later re-conceptualized in a model-independent framework. This reader is not presumed to have prior acquaintance with any of these concepts. Suggested exercises are included to fertilize intuitions and copious references point to exter- nal sources with more details. A running theme of homotopy limits and colimits is included to explain the kinds of problems homotopical categories are designed to solve as well as technical approaches to these problems. Contents 1. The history of homotopical categories 2 2. Categories of fractions and localization 5 2.1. The Gabriel–Zisman category of fractions 5 3. Model category presentations of homotopical categories 7 3.1. Model category structures via weak factorization systems 8 3.2. On functoriality of factorizations 12 3.3. The homotopy relation on arrows 13 3.4. The homotopy category of a model category 17 3.5. Quillen’s model structure on simplicial sets 19 4. Derived functors between model categories 20 4.1. Derived functors and equivalence of homotopy theories 21 4.2. Quillen functors 24 4.3. Derived composites and derived adjunctions 25 4.4. Monoidal and enriched model categories 27 4.5. -
SHORT INTRODUCTION to ENRICHED CATEGORIES FRANCIS BORCEUX and ISAR STUBBE Département De Mathématique, Université Catholique
SHORT INTRODUCTION TO ENRICHED CATEGORIES FRANCIS BORCEUX and ISAR STUBBE Departement de Mathematique, Universite Catholique de Louvain, 2 Ch. du Cyclotron, B-1348 Louvain-la-Neuve, Belgium. e-mail: [email protected] — [email protected] This text aims to be a short introduction to some of the basic notions in ordinary and enriched category theory. With reasonable detail but always in a compact fashion, we have brought together in the rst part of this paper the denitions and basic properties of such notions as limit and colimit constructions in a category, adjoint functors between categories, equivalences and monads. In the second part we pass on to enriched category theory: it is explained how one can “replace” the category of sets and mappings, which plays a crucial role in ordinary category theory, by a more general symmetric monoidal closed category, and how most results of ordinary category theory can be translated to this more general setting. For a lack of space we had to omit detailed proofs, but instead we have included lots of examples which we hope will be helpful. In any case, the interested reader will nd his way to the references, given at the end of the paper. 1. Ordinary categories When working with vector spaces over a eld K, one proves such theorems as: for all vector spaces there exists a base; every vector space V is canonically included in its bidual V ; every linear map between nite dimensional based vector spaces can be represented as a matrix; and so on. But where do the universal quantiers take their value? What precisely does “canonical” mean? How can we formally “compare” vector spaces with matrices? What is so special about vector spaces that they can be based? An answer to these questions, and many more, can be formulated in a very precise way using the language of category theory. -
Abelian Categories
Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists. -
A Model Structure for Quasi-Categories
A MODEL STRUCTURE FOR QUASI-CATEGORIES EMILY RIEHL DISCUSSED WITH J. P. MAY 1. Introduction Quasi-categories live at the intersection of homotopy theory with category theory. In particular, they serve as a model for (1; 1)-categories, that is, weak higher categories with n-cells for each natural number n that are invertible when n > 1. Alternatively, an (1; 1)-category is a category enriched in 1-groupoids, e.g., a topological space with points as 0-cells, paths as 1-cells, homotopies of paths as 2-cells, and homotopies of homotopies as 3-cells, and so forth. The basic data for a quasi-category is a simplicial set. A precise definition is given below. For now, a simplicial set X is given by a diagram in Set o o / X o X / X o ··· 0 o / 1 o / 2 o / o o / with certain relations on the arrows. Elements of Xn are called n-simplices, and the arrows di : Xn ! Xn−1 and si : Xn ! Xn+1 are called face and degeneracy maps, respectively. Intuition is provided by simplical complexes from topology. There is a functor τ1 from the category of simplicial sets to Cat that takes a simplicial set X to its fundamental category τ1X. The objects of τ1X are the elements of X0. Morphisms are generated by elements of X1 with the face maps defining the source and target and s0 : X0 ! X1 picking out the identities. Composition is freely generated by elements of X1 subject to relations given by elements of X2. More specifically, if x 2 X2, then we impose the relation that d1x = d0x ◦ d2x. -
All (,1)-Toposes Have Strict Univalent Universes
All (1; 1)-toposes have strict univalent universes Mike Shulman University of San Diego HoTT 2019 Carnegie Mellon University August 13, 2019 One model is not enough A (Grothendieck{Rezk{Lurie)( 1; 1)-topos is: • The category of objects obtained by \homotopically gluing together" copies of some collection of \model objects" in specified ways. • The free cocompletion of a small (1; 1)-category preserving certain well-behaved colimits. • An accessible left exact localization of an (1; 1)-category of presheaves. They are a powerful tool for studying all kinds of \geometry" (topological, algebraic, differential, cohesive, etc.). It has long been expected that (1; 1)-toposes are models of HoTT, but coherence problems have proven difficult to overcome. Main Theorem Theorem (S.) Every (1; 1)-topos can be given the structure of a model of \Book" HoTT with strict univalent universes, closed under Σs, Πs, coproducts, and identity types. Caveats for experts: 1 Classical metatheory: ZFC with inaccessible cardinals. 2 We assume the initiality principle. 3 Only an interpretation, not an equivalence. 4 HITs also exist, but remains to show universes are closed under them. Towards killer apps Example 1 Hou{Finster{Licata{Lumsdaine formalized a proof of the Blakers{Massey theorem in HoTT. 2 Later, Rezk and Anel{Biedermann{Finster{Joyal unwound this manually into a new( 1; 1)-topos-theoretic proof, with a generalization applicable to Goodwillie calculus. 3 We can now say that the HFLL proof already implies the (1; 1)-topos-theoretic result, without manual translation. (Modulo closure under HITs.) Outline 1 Type-theoretic model toposes 2 Left exact localizations 3 Injective model structures 4 Remarks Review of model-categorical semantics We can interpret type theory in a well-behaved model category E : Type theory Model category Type Γ ` A Fibration ΓA Γ Term Γ ` a : A Section Γ ! ΓA over Γ Id-type Path object . -
Enriched Categories
Enriched Categories Ross Tate May 16, 2018 Definition (Enriched Category over a Multicategory). For a multicategory M, an M-enriched category is made of: Objects A set Ob of objects Morphisms For each A and B in Ob, an object Hom(A; B) of M Identities For each A in Ob, a nullary multimorphism idA :[] !M Hom(A; A) Compositions For each A, B, and C in Ob, a multimorphism compA;B;C : [Hom(A; B); Hom(B; C)] !M Hom(A; C) Identity With the property that comp and comp equal id aidA;idHom(A;B) A;B;C aid Hom(A;B);idB A;B;C Hom(A;B) Associativity And the property that comp equals comp acompA;B;C ;idHom(C;D) A;C;D aidHom(A;B);compB;C;D A;B;D Example. A Set-enriched category is simply a category. Ob is the set of objects of the category, and Hom(A; B) is the set of morphisms from A to B. idA is the identity morphism id A, and compA;B;C is the composition function mapping f : A ! B and g : B ! C to f ; g : A ! C. The identity and associativity requirements of Set-enriched categories are exactly the identity and associativity requirements of categories. Example. Many examples of enriched categories are enriched over a multicategory M with an obvious faithful functor to Set. In such cases, M-enriched categories can be viewed as categories whose sets of morphisms are enriched with additional M-structure, and whose identities and compositions respect that structure. -
Locally Cartesian Closed Categories, Coalgebras, and Containers
U.U.D.M. Project Report 2013:5 Locally cartesian closed categories, coalgebras, and containers Tilo Wiklund Examensarbete i matematik, 15 hp Handledare: Erik Palmgren, Stockholms universitet Examinator: Vera Koponen Mars 2013 Department of Mathematics Uppsala University Contents 1 Algebras and Coalgebras 1 1.1 Morphisms .................................... 2 1.2 Initial and terminal structures ........................... 4 1.3 Functoriality .................................... 6 1.4 (Co)recursion ................................... 7 1.5 Building final coalgebras ............................. 9 2 Bundles 13 2.1 Sums and products ................................ 14 2.2 Exponentials, fibre-wise ............................. 18 2.3 Bundles, fibre-wise ................................ 19 2.4 Lifting functors .................................. 21 2.5 A choice theorem ................................. 22 3 Enriching bundles 25 3.1 Enriched categories ................................ 26 3.2 Underlying categories ............................... 29 3.3 Enriched functors ................................. 31 3.4 Convenient strengths ............................... 33 3.5 Natural transformations .............................. 41 4 Containers 45 4.1 Container functors ................................ 45 4.2 Natural transformations .............................. 47 4.3 Strengths, revisited ................................ 50 4.4 Using shapes ................................... 53 4.5 Final remarks ................................... 56 i Introduction -
An Integral Model Structure and Truncation Theory for Coherent Group
An integral model structure and truncation theory for coherent group actions Yonatan Harpaz Matan Prasma Abstract In this work we study the homotopy theory of coherent group ac- tions from a global point of view, where we allow both the group and the space acted upon to vary. Using the model of Segal group actions and the model categorical Grothendieck construction we construct a model category encompassing all Segal group actions simultaneously. We then prove a global rectification result in this setting. We proceed to develop a general truncation theory for the model-categorical Grothendieck con- struction and apply it to the case of Segal group actions. We give a simple characterization of n-truncated Segal group actions and show that every Segal group action admits a convergent Postnikov tower. Contents 1 Introduction 2 2 Preliminaries 3 2.1 Modelcategories ............................. 3 2.2 Theintegralmodelstructure...................... 4 2.3 Invarianceofthe integralmodel structure . 5 2.4 Integralmodel categoriesofgroupactions . 7 3 An integral model structure for Segal group actions 10 3.1 AmodelstructureforSegalgroups . 10 arXiv:1506.04117v1 [math.AT] 12 Jun 2015 3.1.1 Relation with Bousfield’s approach . 15 3.2 Construction of the integralmodel structure . 16 3.3 RectificationofSegalgroupactions . 19 4 Truncation theory for integral model categories 21 5 Truncation in the integral model category of Segal group ac- tions 25 5.1 ConvergenceofthePostnikovtower. 31 6 References 34 1 1 Introduction Let S be the category of simplicial sets (which we shall refer to as spaces) and let G be a simplicial group. Homotopy theories of spaces equipped with an action of G are of fundamental interest in algebraic topology. -
Simplicially Enriched Categories and Relative Categories
Simplicially enriched categories and relative categories Talk by Alex Rolle, notes by Luis Scoccola Reading seminar on Higher Category Theory University of Western Ontario, Fall 2016 Classical homotopy theory usually deals with topological spaces and simplicial sets, so it is natural to ask if there is a way to construct quasicategories that model these homotopy theories. The goal of this presentation is to describe methods that let us construct quasicategories out of other objects that also present homotopy theories. In particular we want to be able to turn simplicially enriched categories and model categories into quasicategories. References are: • [Lur09] for an introduction and a many of applications of simplicially enriched categories. • [Ber07] for details about the model structure on the category of simplicially enriched cat- egories. • [GZ12] for the general theory of localization of relative categories. • [DK80] for the simplicial localization of relative categories. 1 Simplicially enriched categories Notation 1. We denote the category of (small) simplicially enriched categories by SCat. We want to define a functor that turns simplicially enriched categories into simplicial sets. As usual it is easier to start defining its right adjoint. Definition 2. Start by defining the functor C : ∆ ,! SCat on objects. For every natural number n construct the category C(n) that has as objects the set f0 : : : ng and as morphisms: ( ; if i > j Hom(i; j) = N(Pi;j) if i ≤ j where Pi;j is the poset of subintervals of [i : : : j] that contain both