The Theory of Quasi-Categories and Its Applications
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Frobenius and the Derived Centers of Algebraic Theories
Frobenius and the derived centers of algebraic theories William G. Dwyer and Markus Szymik October 2014 We show that the derived center of the category of simplicial algebras over every algebraic theory is homotopically discrete, with the abelian monoid of components isomorphic to the center of the category of discrete algebras. For example, in the case of commutative algebras in characteristic p, this center is freely generated by Frobenius. Our proof involves the calculation of homotopy coherent centers of categories of simplicial presheaves as well as of Bousfield localizations. Numerous other classes of examples are dis- cussed. 1 Introduction Algebra in prime characteristic p comes with a surprise: For each commutative ring A such that p = 0 in A, the p-th power map a 7! ap is not only multiplica- tive, but also additive. This defines Frobenius FA : A ! A on commutative rings of characteristic p, and apart from the well-known fact that Frobenius freely gen- erates the Galois group of the prime field Fp, it has many other applications in 1 algebra, arithmetic and even geometry. Even beyond fields, Frobenius is natural in all rings A: Every map g: A ! B between commutative rings of characteris- tic p (i.e. commutative Fp-algebras) commutes with Frobenius in the sense that the equation g ◦ FA = FB ◦ g holds. In categorical terms, the Frobenius lies in the center of the category of commutative Fp-algebras. Furthermore, Frobenius freely generates the center: If (PA : A ! AjA) is a family of ring maps such that g ◦ PA = PB ◦ g (?) holds for all g as above, then there exists an integer n > 0 such that the equa- n tion PA = (FA) holds for all A. -
Types Are Internal Infinity-Groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau
Types are internal infinity-groupoids Antoine Allioux, Eric Finster, Matthieu Sozeau To cite this version: Antoine Allioux, Eric Finster, Matthieu Sozeau. Types are internal infinity-groupoids. 2021. hal- 03133144 HAL Id: hal-03133144 https://hal.inria.fr/hal-03133144 Preprint submitted on 5 Feb 2021 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Types are Internal 1-groupoids Antoine Allioux∗, Eric Finstery, Matthieu Sozeauz ∗Inria & University of Paris, France [email protected] yUniversity of Birmingham, United Kingdom ericfi[email protected] zInria, France [email protected] Abstract—By extending type theory with a universe of defini- attempts to import these ideas into plain homotopy type theory tionally associative and unital polynomial monads, we show how have, so far, failed. This appears to be a result of a kind of to arrive at a definition of opetopic type which is able to encode circularity: all of the known classical techniques at some point a number of fully coherent algebraic structures. In particular, our approach leads to a definition of 1-groupoid internal to rely on set-level algebraic structures themselves (presheaves, type theory and we prove that the type of such 1-groupoids is operads, or something similar) as a means of presenting or equivalent to the universe of types. -
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. -
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. -
Simplicial Sets, Nerves of Categories, Kan Complexes, Etc
SIMPLICIAL SETS, NERVES OF CATEGORIES, KAN COMPLEXES, ETC FOLING ZOU These notes are taken from Peter May's classes in REU 2018. Some notations may be changed to the note taker's preference and some detailed definitions may be skipped and can be found in other good notes such as [2] or [3]. The note taker is responsible for any mistakes. 1. simplicial approach to defining homology Defnition 1. A simplical set/group/object K is a sequence of sets/groups/objects Kn for each n ≥ 0 with face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n and degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n satisfying certain commutation equalities. Images of degeneracy maps are said to be degenerate. We can define a functor: ordered abstract simplicial complex ! sSet; K 7! Ks; where s Kn = fv0 ≤ · · · ≤ vnjfv0; ··· ; vng (may have repetition) is a simplex in Kg: s s Face maps: di : Kn ! Kn−1; 0 ≤ i ≤ n is by deleting vi; s s Degeneracy maps: si : Kn ! Kn+1; 0 ≤ i ≤ n is by repeating vi: In this way it is very straightforward to remember the equalities that face maps and degeneracy maps have to satisfy. The simplical viewpoint is helpful in establishing invariants and comparing different categories. For example, we are going to define the integral homology of a simplicial set, which will agree with the simplicial homology on a simplical complex, but have the virtue of avoiding the barycentric subdivision in showing functoriality and homotopy invariance of homology. This is an observation made by Samuel Eilenberg. To start, we construct functors: F C sSet sAb ChZ: The functor F is the free abelian group functor applied levelwise to a simplical set. -
The Simplicial Parallel of Sheaf Theory Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 10, No 4 (1968), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES YUH-CHING CHEN Costacks - The simplicial parallel of sheaf theory Cahiers de topologie et géométrie différentielle catégoriques, tome 10, no 4 (1968), p. 449-473 <http://www.numdam.org/item?id=CTGDC_1968__10_4_449_0> © Andrée C. Ehresmann et les auteurs, 1968, tous droits réservés. L’accès aux archives de la revue « Cahiers de topologie et géométrie différentielle catégoriques » implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ CAHIERS DE TOPOLOGIE ET GEOMETRIE DIFFERENTIELLE COSTACKS - THE SIMPLICIAL PARALLEL OF SHEAF THEORY by YUH-CHING CHEN I NTRODUCTION Parallel to sheaf theory, costack theory is concerned with the study of the homology theory of simplicial sets with general coefficient systems. A coefficient system on a simplicial set K with values in an abe - . lian category 8 is a functor from K to Q (K is a category of simplexes) ; it is called a precostack; it is a simplicial parallel of the notion of a presheaf on a topological space X . A costack on K is a « normalized» precostack, as a set over a it is realized simplicial K/" it is simplicial « espace 8ta18 ». The theory developed here is functorial; it implies that all >> homo- logy theories are derived functors. Although the treatment is completely in- dependent of Topology, it is however almost completely parallel to the usual sheaf theory, and many of the same theorems will be found in it though the proofs are usually quite different. -
Giraud's Theorem
Giraud’s Theorem 25 February and 4 March 2019 The Giraud’s axioms allow us to determine when a category is a topos. Lurie’s version indicates the equivalence between the axioms, the left exact localizations, and Grothendieck topologies, i.e. Proposition 1. [2] Let X be a category. The following conditions are equivalent: 1. The category X is equivalent to the category of sheaves of sets of some Grothendieck site. 2. The category X is equivalent to a left exact localization of the category of presheaves of sets on some small category C. 3. Giraud’s axiom are satisfied: (a) The category X is presentable. (b) Colimits in X are universal. (c) Coproducts in X are disjoint. (d) Equivalence relations in X are effective. 1 Sheaf and Grothendieck topologies This section is based in [4]. The continuity of a reald-valued function on topological space can be determined locally. Let (X;t) be a topological space and U an open subset of X. If U is S covered by open subsets Ui, i.e. U = i2I Ui and fi : Ui ! R are continuos functions then there exist a continuos function f : U ! R if and only if the fi match on all the overlaps Ui \Uj and f jUi = fi. The previuos paragraph is so technical, let us see a more enjoyable example. Example 1. Let (X;t) be a topological space such that X = U1 [U2 then X can be seen as the following diagram: X s O e U1 tU1\U2 U2 o U1 O O U2 o U1 \U2 If we form the category OX (the objects are the open sets and the morphisms are given by the op inclusion), the previous paragraph defined a functor between OX and Set i.e it sends an open set U to the set of all continuos functions of domain U and codomain the real numbers, F : op OX ! Set where F(U) = f f j f : U ! Rg, so the initial diagram can be seen as the following: / (?) FX / FU1 ×F(U1\U2) FU2 / F(U1 \U2) then the condition of the paragraph states that the map e : FX ! FU1 ×F(U1\U2) FU2 given by f 7! f jUi is the equalizer of the previous diagram. -
Homotopy Coherent Structures
HOMOTOPY COHERENT STRUCTURES EMILY RIEHL Abstract. Naturally occurring diagrams in algebraic topology are commuta- tive up to homotopy, but not on the nose. It was quickly realized that very little can be done with this information. Homotopy coherent category theory arose out of a desire to catalog the higher homotopical information required to restore constructibility (or more precisely, functoriality) in such “up to homo- topy” settings. The first lecture will survey the classical theory of homotopy coherent diagrams of topological spaces. The second lecture will revisit the free resolutions used to define homotopy coherent diagrams and prove that they can also be understood as homotopy coherent realizations. This explains why diagrams valued in homotopy coherent nerves or more general 1-categories are automatically homotopy coherent. The final lecture will venture into ho- motopy coherent algebra, connecting the newly discovered notion of homotopy coherent adjunction to the classical cobar and bar resolutions for homotopy coherent algebras. Contents Part I. Homotopy coherent diagrams 2 I.1. Historical motivation 2 I.2. The shape of a homotopy coherent diagram 3 I.3. Homotopy coherent diagrams and homotopy coherent natural transformations 7 Part II. Homotopy coherent realization and the homotopy coherent nerve 10 II.1. Free resolutions are simplicial computads 11 II.2. Homotopy coherent realization and the homotopy coherent nerve 13 II.3. Further applications 17 Part III. Homotopy coherent algebra 17 III.1. From coherent homotopy theory to coherent category theory 17 III.2. Monads in category theory 19 III.3. Homotopy coherent monads 19 III.4. Homotopy coherent adjunctions 21 Date: July 12-14, 2017. -
Left Kan Extensions Preserving Finite Products
Left Kan Extensions Preserving Finite Products Panagis Karazeris, Department of Mathematics, University of Patras, Patras, Greece Grigoris Protsonis, Department of Mathematics, University of Patras, Patras, Greece 1 Introduction In many situations in mathematics we want to know whether the left Kan extension LanjF of a functor F : C!E, along a functor j : C!D, where C is a small category and E is locally small and cocomplete, preserves the limits of various types Φ of diagrams. The answer to this general question is reducible to whether the left Kan extension LanyF of F , along the Yoneda embedding y : C! [Cop; Set] preserves those limits (see [12] x3). Such questions arise in the classical context of comparing the homotopy of simplicial sets to that of spaces but are also vital in comparing, more generally, homotopical notions on various combinatorial models (e.g simplicial, bisimplicial, cubical, globular, etc, sets) on the one hand, and various \realizations" of them as spaces, categories, higher categories, simplicial categories, relative categories etc, on the other (see [8], [4], [15], [2]). In the case where E = Set the answer is classical and well-known for the question of preservation of all finite limits (see [5], Chapter 6), the question of finite products (see [1]) as well as for the case of various types of finite connected limits (see [11]). The answer to such questions is also well-known in the case E is a Grothendieck topos, for the class of all finite limits or all finite connected limits (see [14], chapter 7 x 8 and [9]). -
Fields Lectures: Simplicial Presheaves
Fields Lectures: Simplicial presheaves J.F. Jardine∗ January 31, 2007 This is a cleaned up and expanded version of the lecture notes for a short course that I gave at the Fields Institute in late January, 2007. I expect the cleanup and expansion processes to continue for a while yet, so the interested reader should check the web site http://www.math.uwo.ca/∼jardine periodi- cally for updates. Contents 1 Simplicial presheaves and sheaves 2 2 Local weak equivalences 4 3 First local model structure 6 4 Other model structures 12 5 Cocycle categories 13 6 Sheaf cohomology 16 7 Descent 22 8 Non-abelian cohomology 25 9 Presheaves of groupoids 28 10 Torsors and stacks 30 11 Simplicial groupoids 35 12 Cubical sets 43 13 Localization 48 ∗This research was supported by NSERC. 1 1 Simplicial presheaves and sheaves In all that follows, C will be a small Grothendieck site. Examples include the site op|X of open subsets and open covers of a topo- logical space X, the site Zar|S of Zariski open subschemes and open covers of a scheme S, or the ´etale site et|S, again of a scheme S. All of these sites have “big” analogues, like the big sites Topop,(Sch|S)Zar and (Sch|S)et of “all” topological spaces with open covers, and “all” S-schemes T → S with Zariski and ´etalecovers, respectivley. Of course, there are many more examples. Warning: All of the “big” sites in question have (infinite) cardinality bounds on the objects which define them so that the sites are small, and we don’t talk about these bounds. -
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. -
Free Models of T-Algebraic Theories Computed As Kan Extensions Paul-André Melliès, Nicolas Tabareau
Free models of T-algebraic theories computed as Kan extensions Paul-André Melliès, Nicolas Tabareau To cite this version: Paul-André Melliès, Nicolas Tabareau. Free models of T-algebraic theories computed as Kan exten- sions. 2008. hal-00339331 HAL Id: hal-00339331 https://hal.archives-ouvertes.fr/hal-00339331 Preprint submitted on 17 Nov 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Free models of T -algebraic theories computed as Kan extensions Paul-André Melliès Nicolas Tabareau ∗ Abstract One fundamental aspect of Lawvere’s categorical semantics is that every algebraic theory (eg. of monoid, of Lie algebra) induces a free construction (eg. of free monoid, of free Lie algebra) computed as a Kan extension. Unfortunately, the principle fails when one shifts to linear variants of algebraic theories, like Adams and Mac Lane’s PROPs, and similar PROs and PROBs. Here, we introduce the notion of T -algebraic theory for a pseudomonad T — a mild generalization of equational doctrine — in order to describe these various kinds of “algebraic theories”. Then, we formulate two conditions (the first one combinatorial, the second one algebraic) which ensure that the free model of a T -algebraic theory exists and is computed as an Kan extension.