Comprehensive Factorisation Systems
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Partial Mal'tsevness and Partial Protomodularity
PARTIAL MAL’TSEVNESS AND PARTIAL PROTOMODULARITY DOMINIQUE BOURN Abstract. We introduce the notion of Mal’tsev reflection which allows us to set up a partial notion of Mal’tsevness with respect to a class Σ of split epimorphisms stable under pullback and containing the isomorphisms, and we investigate what is remaining of the properties of the global Mal’tsev context. We introduce also the notion of partial protomodularity in the non-pointed context. Introduction A Mal’tsev category is a category in which any reflexive relation is an equivalence relation, see [9] and [10]. The categories Gp of groups and K-Lie of Lie K-algebras are major examples of Mal’tsev categories. The terminology comes from the pi- oneering work of Mal’tsev in the varietal context [16] which was later on widely developped in [18]. In [3], Mal’tsev categories were characterized in terms of split epimorphisms: a finitely complete category D is a Mal’tsev one if and only if any pullback of split epimorphisms in D: t¯ X′ o X O g¯ / O ′ ′ f s f s t Y ′ o Y g / is such that the pair (s′, t¯) is jointly extremally epic. More recently the same kind of property was observed, but only for a certain class Σ of split epimorphisms (f,s) which is stable under pullback and contains arXiv:1507.02886v1 [math.CT] 10 Jul 2015 isomorphisms. By the classes of Schreier or homogeneous split epimorphisms in the categories Mon of monoids and SRng of semi-rings [8], by the classes of puncturing or acupuncturing split epimorphims in the category of quandles [6], by the class of split epimorphic functors with fibrant splittings in the category CatY of categories with a fixed set of objects Y [5]. -
Category Theory
Michael Paluch Category Theory April 29, 2014 Preface These note are based in part on the the book [2] by Saunders Mac Lane and on the book [3] by Saunders Mac Lane and Ieke Moerdijk. v Contents 1 Foundations ....................................................... 1 1.1 Extensionality and comprehension . .1 1.2 Zermelo Frankel set theory . .3 1.3 Universes.....................................................5 1.4 Classes and Gödel-Bernays . .5 1.5 Categories....................................................6 1.6 Functors .....................................................7 1.7 Natural Transformations. .8 1.8 Basic terminology . 10 2 Constructions on Categories ....................................... 11 2.1 Contravariance and Opposites . 11 2.2 Products of Categories . 13 2.3 Functor Categories . 15 2.4 The category of all categories . 16 2.5 Comma categories . 17 3 Universals and Limits .............................................. 19 3.1 Universal Morphisms. 19 3.2 Products, Coproducts, Limits and Colimits . 20 3.3 YonedaLemma ............................................... 24 3.4 Free cocompletion . 28 4 Adjoints ........................................................... 31 4.1 Adjoint functors and universal morphisms . 31 4.2 Freyd’s adjoint functor theorem . 38 5 Topos Theory ...................................................... 43 5.1 Subobject classifier . 43 5.2 Sieves........................................................ 45 5.3 Exponentials . 47 vii viii Contents Index .................................................................. 53 Acronyms List of categories. Ab The category of small abelian groups and group homomorphisms. AlgA The category of commutative A-algebras. Cb The category Func(Cop,Sets). Cat The category of small categories and functors. CRings The category of commutative ring with an identity and ring homomor- phisms which preserve identities. Grp The category of small groups and group homomorphisms. Sets Category of small set and functions. Sets Category of small pointed set and pointed functions. -
Quasi-Categories Vs Simplicial Categories
Quasi-categories vs Simplicial categories Andr´eJoyal January 07 2007 Abstract We show that the coherent nerve functor from simplicial categories to simplicial sets is the right adjoint in a Quillen equivalence between the model category for simplicial categories and the model category for quasi-categories. Introduction A quasi-category is a simplicial set which satisfies a set of conditions introduced by Boardman and Vogt in their work on homotopy invariant algebraic structures [BV]. A quasi-category is often called a weak Kan complex in the literature. The category of simplicial sets S admits a Quillen model structure in which the cofibrations are the monomorphisms and the fibrant objects are the quasi- categories [J2]. We call it the model structure for quasi-categories. The resulting model category is Quillen equivalent to the model category for complete Segal spaces and also to the model category for Segal categories [JT2]. The goal of this paper is to show that it is also Quillen equivalent to the model category for simplicial categories via the coherent nerve functor of Cordier. We recall that a simplicial category is a category enriched over the category of simplicial sets S. To every simplicial category X we can associate a category X0 enriched over the homotopy category of simplicial sets Ho(S). A simplicial functor f : X → Y is called a Dwyer-Kan equivalence if the functor f 0 : X0 → Y 0 is an equivalence of Ho(S)-categories. It was proved by Bergner, that the category of (small) simplicial categories SCat admits a Quillen model structure in which the weak equivalences are the Dwyer-Kan equivalences [B1]. -
Universality of Multiplicative Infinite Loop Space Machines
UNIVERSALITY OF MULTIPLICATIVE INFINITE LOOP SPACE MACHINES DAVID GEPNER, MORITZ GROTH AND THOMAS NIKOLAUS Abstract. We establish a canonical and unique tensor product for commutative monoids and groups in an ∞-category C which generalizes the ordinary tensor product of abelian groups. Using this tensor product we show that En-(semi)ring objects in C give rise to En-ring spectrum objects in C. In the case that C is the ∞-category of spaces this produces a multiplicative infinite loop space machine which can be applied to the algebraic K-theory of rings and ring spectra. The main tool we use to establish these results is the theory of smashing localizations of presentable ∞-categories. In particular, we identify preadditive and additive ∞-categories as the local objects for certain smashing localizations. A central theme is the stability of algebraic structures under basechange; for example, we show Ring(D ⊗ C) ≃ Ring(D) ⊗ C. Lastly, we also consider these algebraic structures from the perspective of Lawvere algebraic theories in ∞-categories. Contents 0. Introduction 1 1. ∞-categories of commutative monoids and groups 4 2. Preadditive and additive ∞-categories 6 3. Smashing localizations 8 4. Commutative monoids and groups as smashing localizations 11 5. Canonical symmetric monoidal structures 13 6. More functoriality 15 7. ∞-categories of semirings and rings 17 8. Multiplicative infinite loop space theory 19 Appendix A. Comonoids 23 Appendix B. Algebraic theories and monadic functors 23 References 26 0. Introduction The Grothendieck group K0(M) of a commutative monoid M, also known as the group completion, is the universal abelian group which receives a monoid map from M. -
Monads of Effective Descent Type and Comonadicity
Theory and Applications of Categories, Vol. 16, No. 1, 2006, pp. 1–45. MONADS OF EFFECTIVE DESCENT TYPE AND COMONADICITY BACHUKI MESABLISHVILI Abstract. We show, for an arbitrary adjunction F U : B→Awith B Cauchy complete, that the functor F is comonadic if and only if the monad T on A induced by the adjunction is of effective descent type, meaning that the free T-algebra functor F T : A→AT is comonadic. This result is applied to several situations: In Section 4 to give a sufficient condition for an exponential functor on a cartesian closed category to be monadic, in Sections 5 and 6 to settle the question of the comonadicity of those functors whose domain is Set,orSet, or the category of modules over a semisimple ring, in Section 7 to study the effectiveness of (co)monads on module categories. Our final application is a descent theorem for noncommutative rings from which we deduce an important result of A. Joyal and M. Tierney and of J.-P. Olivier, asserting that the effective descent morphisms in the opposite of the category of commutative unital rings are precisely the pure monomorphisms. 1. Introduction Let A and B be two (not necessarily commutative) rings related by a ring homomorphism i : B → A. The problem of Grothendieck’s descent theory for modules with respect to i : B → A is concerned with the characterization of those (right) A-modules Y for which there is X ∈ ModB andanisomorphismY X⊗BA of right A-modules. Because of a fun- damental connection between descent and monads discovered by Beck (unpublished) and B´enabou and Roubaud [6], this problem is equivalent to the problem of the comonadicity of the extension-of-scalars functor −⊗BA :ModB → ModA. -
Lecture Notes on Simplicial Homotopy Theory
Lectures on Homotopy Theory The links below are to pdf files, which comprise my lecture notes for a first course on Homotopy Theory. I last gave this course at the University of Western Ontario during the Winter term of 2018. The course material is widely applicable, in fields including Topology, Geometry, Number Theory, Mathematical Pysics, and some forms of data analysis. This collection of files is the basic source material for the course, and this page is an outline of the course contents. In practice, some of this is elective - I usually don't get much beyond proving the Hurewicz Theorem in classroom lectures. Also, despite the titles, each of the files covers much more material than one can usually present in a single lecture. More detail on topics covered here can be found in the Goerss-Jardine book Simplicial Homotopy Theory, which appears in the References. It would be quite helpful for a student to have a background in basic Algebraic Topology and/or Homological Algebra prior to working through this course. J.F. Jardine Office: Middlesex College 118 Phone: 519-661-2111 x86512 E-mail: [email protected] Homotopy theories Lecture 01: Homological algebra Section 1: Chain complexes Section 2: Ordinary chain complexes Section 3: Closed model categories Lecture 02: Spaces Section 4: Spaces and homotopy groups Section 5: Serre fibrations and a model structure for spaces Lecture 03: Homotopical algebra Section 6: Example: Chain homotopy Section 7: Homotopical algebra Section 8: The homotopy category Lecture 04: Simplicial sets Section 9: -
Protomodular Aspect of the Dual of a Topos
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector ARTICLE IN PRESS Advances in Mathematics 187 (2004) 240–255 http://www.elsevier.com/locate/aim Protomodular aspect of the dual of a topos Dominique Bourn Centre Universitaire de la Mi-voix, Lab. d’Analyse, Ge´ome´trie et Alge`bre, Universite´ du Littoral, 50 rue Ferdinand Buisson, BP699, 62228 Calais Cedex, France Received 3 March 2003; accepted 15 September 2003 Communicated by Ross Street Abstract The structure of the dual of a topos is investigated under its aspect of a Barr exact and protomodular category. In particular the normal monomorphisms in the fibres of the fibration of pointed objects are characterized, and the change of base functors with respect to this same fibration are shown to reflect those normal monomorphisms. r 2003 Elsevier Inc. All rights reserved. MSC: 18B25; 18D30; 08B10 Keywords: Topos; Fibrations; Mal’cev and protomodular categories 0. Introduction It is well known that, given a topos E; the subobject classifier O is endowed with an internal structure of Heyting Algebra, and that consequently the contravariant ‘‘power-set’’ functor OðÞ : Eop-E takes its values in the category HeytE of internal Heyting algebras in E: But the category Heyt of Heyting Algebras shares with the categories Gp of groups or Rg of rings the property of being protomodular. This property [3] says that the fibration of pointed objects p : PtC-C; whose fibre at X is the category PtX C of split epimorphisms with codomain X; has its change of base functors conservative (i.e. -
A Simplicial Set Is a Functor X : a Op → Set, Ie. a Contravariant Set-Valued
Contents 9 Simplicial sets 1 10 The simplex category and realization 10 11 Model structure for simplicial sets 15 9 Simplicial sets A simplicial set is a functor X : Dop ! Set; ie. a contravariant set-valued functor defined on the ordinal number category D. One usually writes n 7! Xn. Xn is the set of n-simplices of X. A simplicial map f : X ! Y is a natural transfor- mation of such functors. The simplicial sets and simplicial maps form the category of simplicial sets, denoted by sSet — one also sees the notation S for this category. If A is some category, then a simplicial object in A is a functor A : Dop ! A : Maps between simplicial objects are natural trans- formations. 1 The simplicial objects in A and their morphisms form a category sA . Examples: 1) sGr = simplicial groups. 2) sAb = simplicial abelian groups. 3) s(R − Mod) = simplicial R-modules. 4) s(sSet) = s2Set is the category of bisimplicial sets. Simplicial objects are everywhere. Examples of simplicial sets: 1) We’ve already met the singular set S(X) for a topological space X, in Section 4. S(X) is defined by the cosimplicial space (covari- ant functor) n 7! jDnj, by n S(X)n = hom(jD j;X): q : m ! n defines a function ∗ n q m S(X)n = hom(jD j;X) −! hom(jD j;X) = S(X)m by precomposition with the map q : jDmj ! jDmj. The assignment X 7! S(X) defines a covariant func- tor S : CGWH ! sSet; called the singular functor. -
Invertible Objects in Motivic Homotopy Theory
Invertible Objects in Motivic Homotopy Theory Tom Bachmann M¨unchen2016 Invertible Objects in Motivic Homotopy Theory Tom Bachmann Dissertation an der Fakult¨atf¨urMathematik, Informatik und Statistik der Ludwig{Maximilians{Universit¨at M¨unchen vorgelegt von Tom Bachmann aus Chemnitz M¨unchen, den 19. Juli 2016 Erstgutachter: Prof. Dr. Fabien Morel Zweitgutachter: Prof. Dr. Marc Levine Drittgutachter: Prof. Dr. math. Oliver R¨ondigs Tag der m¨undlichen Pr¨ufung:18.11.2016 Eidesstattliche Versicherung Bachmann, Tom Name, Vorname Ich erkläre hiermit an Eides statt, dass ich die vorliegende Dissertation mit dem Thema Invertible Objects in Motivic Homotopy Theory selbständig verfasst, mich außer der angegebenen keiner weiteren Hilfsmittel bedient und alle Erkenntnisse, die aus dem Schrifttum ganz oder annähernd übernommen sind, als solche kenntlich gemacht und nach ihrer Herkunft unter Bezeichnung der Fundstelle einzeln nachgewiesen habe. Ich erkläre des Weiteren, dass die hier vorgelegte Dissertation nicht in gleicher oder in ähnlicher Form bei einer anderen Stelle zur Erlangung eines akademischen Grades eingereicht wurde. München, 1.12.2016 Ort, Datum Unterschrift Doktorandin/Doktorand Eidesstattliche Versicherung Stand: 31.01.2013 vi Abstract If X is a (reasonable) base scheme then there are the categories of interest in stable motivic homotopy theory SH(X) and DM(X), constructed by Morel-Voevodsky and others. These should be thought of as generalisations respectively of the stable homotopy category SH and the derived category of abelian groups D(Ab), which are studied in classical topology, to the \world of smooth schemes over X". Just like in topology, the categories SH(X); DM(X) are symmetric monoidal: there is a bifunctor (E; F ) 7! E ⊗ F satisfying certain properties; in particular there is a unit 1 satisfying E ⊗ 1 ' 1 ⊗ E ' E for all E. -
On the Geometric Realization of Dendroidal Sets
Fabio Trova On the Geometric Realization of Dendroidal Sets Thesis advisor: Prof. Ieke Moerdijk Leiden University MASTER ALGANT University of Padova Et tu ouvriras parfois ta fenˆetre, comme ¸ca,pour le plaisir. Et tes amis seront bien ´etonn´esde te voir rire en regardant le ciel. Alors tu leur diras: “Oui, les ´etoiles,¸came fait toujours rire!” Et ils te croiront fou. Je t’aurai jou´eun bien vilain tour. A Irene, Lorenzo e Paolo a chi ha fatto della propria vita poesia a chi della poesia ha fatto la propria vita Contents Introduction vii Motivations and main contributions........................... vii 1 Category Theory1 1.1 Categories, functors, natural transformations...................1 1.2 Adjoint functors, limits, colimits..........................4 1.3 Monads........................................7 1.4 More on categories of functors............................ 10 1.5 Monoidal Categories................................. 13 2 Simplicial Sets 19 2.1 The Simplicial Category ∆ .............................. 19 2.2 The category SSet of Simplicial Sets........................ 21 2.3 Geometric Realization................................ 23 2.4 Classifying Spaces.................................. 28 3 Multicategory Theory 29 3.1 Trees.......................................... 29 3.2 Planar Multicategories................................ 31 3.3 Symmetric multicategories.............................. 34 3.4 (co)completeness of Multicat ............................. 37 3.5 Closed monoidal structure in Multicat ....................... 40 4 Dendroidal Sets 43 4.1 The dendroidal category Ω .............................. 43 4.1.1 Algebraic definition of Ω ........................... 44 4.1.2 Operadic definition of Ω ........................... 45 4.1.3 Equivalence of the definitions........................ 46 4.1.4 Faces and degeneracies............................ 48 4.2 The category dSet of Dendroidal Sets........................ 52 4.3 Nerve of a Multicategory............................... 55 4.4 Closed Monoidal structure on dSet ........................ -
The Multitopic Ω–Category of All Multitopic Ω–Categories by M. Makkai (Mcgill University) September 2, 1999; Corrected: June 5, 2004
The multitopic ω–category of all multitopic ω–categories by M. Makkai (McGill University) September 2, 1999; corrected: June 5, 2004 Introduction. Despite its considerable length, this paper is only an announcement. It gives two definitions: that of "multitopic ω-category", and that of "the (large) multitopic set of all (small) multitopic ω-categories". It also announces the theorem that the latter is a multitopic ω-category. The work has two direct sources. One is the paper [H/M/P] in which, among others, the concept of "multitopic set" was introduced. The other is the author's work on FOLDS, first order logic with dependent sorts. The latter was reported on in [M2]. A detailed account of the work on FOLDS is in [M3]. For the understanding of the present paper, what is contained in [M2] suffices. In fact, section 1 of this paper gives the definitions of all that's needed in this paper; so, probably, there won't be even a need to consult [M2]. The concept of multitopic set, the main contribution of [H/M/P], was, in turn, inspired by the work of J. Baez and J. Dolan [B/D]. Multitopic sets are a variant of opetopic sets of loc. cit. The name "multitopic set" refers to multicategories, a concept originally due to J. Lambek [L], and given an only moderately generalized formulation in [H/M/P]. The earlier "opetopic set" is based on a concept of operad; see [B/D]. I should say that the exact relationship of the two concepts ("multitopic set" and "opetopic set") is still not clarified. -
Ends and Coends
THIS IS THE (CO)END, MY ONLY (CO)FRIEND FOSCO LOREGIAN† Abstract. The present note is a recollection of the most striking and use- ful applications of co/end calculus. We put a considerable effort in making arguments and constructions rather explicit: after having given a series of preliminary definitions, we characterize co/ends as particular co/limits; then we derive a number of results directly from this characterization. The last sections discuss the most interesting examples where co/end calculus serves as a powerful abstract way to do explicit computations in diverse fields like Algebra, Algebraic Topology and Category Theory. The appendices serve to sketch a number of results in theories heavily relying on co/end calculus; the reader who dares to arrive at this point, being completely introduced to the mysteries of co/end fu, can regard basically every statement as a guided exercise. Contents Introduction. 1 1. Dinaturality, extranaturality, co/wedges. 3 2. Yoneda reduction, Kan extensions. 13 3. The nerve and realization paradigm. 16 4. Weighted limits 21 5. Profunctors. 27 6. Operads. 33 Appendix A. Promonoidal categories 39 Appendix B. Fourier transforms via coends. 40 References 41 Introduction. The purpose of this survey is to familiarize the reader with the so-called co/end calculus, gathering a series of examples of its application; the author would like to stress clearly, from the very beginning, that the material presented here makes arXiv:1501.02503v2 [math.CT] 9 Feb 2015 no claim of originality: indeed, we put a special care in acknowledging carefully, where possible, each of the many authors whose work was an indispensable source in compiling this note.