Partial Mal'tsevness and Partial Protomodularity
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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. -
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. -
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. -
Arxiv:1808.09738V4 [Math.CT] 12 Nov 2020 Oueycoe N Rsre Ne Lsdiae.Se[ See Images
A CHARACTERISATION OF THE CATEGORY OF COMPACT HAUSDORFF SPACES VINCENZO MARRA AND LUCA REGGIO Abstract. We provide a characterisation of the category KH of compact Hausdorff spaces and continuous maps by means of categorical properties only. To this aim we introduce a notion of filtrality for coherent categories, relating certain lattices of subob- jects to their Boolean centers. Our main result reads as follows: Up to equivalence, KH is the unique non-trivial well-pointed pretopos which is filtral and admits all set-indexed copowers of its terminal object. 1. Introduction Several characterisations of the class of compact Hausdorff spaces are available in the liter- ature. For example, de Groot’s Theorem asserts that compact Hausdorff spaces form the only non-trivial, productive and closed-hereditary class of topological spaces which are ab- solutely closed and preserved under closed images. See [Wattel, 1968, p. 51], and [Franklin & Thomas, 1970] for a categorical translation of this result. The category KH of compact Hausdorff spaces and continuous maps has been widely investigated in categorical topo- logy. There, the characterisation of subcategories of the category of topological spaces is an important concern. In this direction, Herrlich and Strecker showed that KH is the unique non-trivial full epireflective subcategory of Hausdorff spaces that is varietal in the sense of [Linton, 1966]. See [Herrlich & Strecker, 1971], and also [Richter, 1996]. A common feature of these characterisations is that they are all relative to an ambient class of topological spaces. To the best of our knowledge, the only abstract characterisa- tions of the category of compact Hausdorff spaces were provided in [Richter, 1991, Richter, 1992]. -
Arxiv:2104.01816V2 [Math.AT] 12 Apr 2021
The Monadic Tower for ∞-Categories Lior Yanovski∗ Abstract Every right adjoint functor between presentable ∞-categories is shown to decompose canoni- cally as a coreflection, followed by, possibly transfinitely many, monadic functors. Furthermore, the coreflection part is given a presentation in terms of a functorial iterated colimit. Background material, examples, and the relation to homology localization and completion are discussed as well. arXiv:2104.01816v2 [math.AT] 12 Apr 2021 Weltchronik in Versen, Szene: Der Turmbau zu Babel (c. 1370s). Depiction of the construction of the tower of Babel. ∗Max Planck Institute for Mathematics. 1 Contents 1 Introduction 2 2 Conservativity and Generation6 3 Monads and Monadicity 10 4 The Monadic Tower 16 5 Convergence 18 References 23 1 Introduction Background The theory of monads is a general categorical framework that axiomatizes the notion of an algebraic structure. Loosely speaking, a monad T on an ∞-category C encodes a collection of formal oper- ations with “arities in C ” and certain relations among their compositions. For an object X ∈ C , a structure of a T -algebra on X is a realization of this collection of formal operations as actual operations on X, such that the required relations hold (see [BMW12], for an elaboration of this perspective). This framework encompasses many of the familiar algebraic structures in ordinary and ∞-category theory, including, for example, all those which arise from ∞-operads.1 The functor from the ∞-category of T -algebras back to C , which forgets the T -algebra structure, is always a conservative right adjoint. A functor to C that is equivalent to such a forgetful functor, for some monad T on C , is called monadic. -
Topoi of Parametrized Objects
TOPOI OF PARAMETRIZED OBJECTS MARC HOYOIS Abstract. We give necessary and sufficient conditions on a presentable 1-category C so that families of objects of C form an 1-topos. In particular, we prove a conjecture of Joyal that this is the case whenever C is stable. Let X be an 1-topos and let C be a sheaf of 1-categories on X. We denote by Z C ! X X R the cartesian fibration classified by C. An object in X C is thus a pair (U; c) with U 2 X and c 2 C(U). The question we are interested in is the following: R Question 1. When is X C an 1-topos? R If C is a presentable 1-category, we will also denote by X C ! X the cartesian fibration classified by the sheaf U 7! ShvC(X=U ) ' C ⊗ X=U : R 1 Joyal calls C an 1-locus if S C is an 1-topos, and he conjectures that any presentable stable 1-category is an 1-locus [Joy15]. The motivating example, due to Biedermann and Rezk, is the 1-category Sp of spectra: there is an equivalence Z fin Sp ' Exc(S∗ ; X); X and the right-hand side is an 1-topos [Lur17a, Remark 6.1.1.11]. More generally, for any small 1-category A with finite colimits and a final object, Z n n Exc∗ (A; S) ' Exc (A; X) X is an 1-topos. This paper gives a partial answer to Question 1 in Theorem 2. As a corollary, we obtain a characteriza- tion of 1-loci (see Corollary 5), similar to Rezk's characterization of 1-topoi, which easily implies Joyal's conjecture (see Example 7). -
On Orthogonality in Locally Presentable Categories Cahiers De Topologie Et Géométrie Différentielle Catégoriques, Tome 42, No 1 (2001), P
CAHIERS DE TOPOLOGIE ET GÉOMÉTRIE DIFFÉRENTIELLE CATÉGORIQUES M. HEBERT J. ADAMEK J. ROSICKÝ More on orthogonality in locally presentable categories Cahiers de topologie et géométrie différentielle catégoriques, tome 42, no 1 (2001), p. 51-80 <http://www.numdam.org/item?id=CTGDC_2001__42_1_51_0> © Andrée C. Ehresmann et les auteurs, 2001, 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 VolumeXLlI-1 (2001) GEDMETRIE DIFFERENTIELLE CATEGORIQUES MORE ON ORTHOGONALITY IN LOCALLY PRESENTABLE CATEGORIES by M. HEBERT, J. ADAMEK* and J. ROSICKy Rdsum6. Nous présentons une solution nouvelle au problbme des sous-categories orthogonales dans les cat6gories localement pr6senta- bles, essentiellement différente de la solution classique de Gabriel et Ulmer. Diverses applications sont donn6es. En particulier nous l’utilisons pour caractériser les classes oméga-orthogonales dans les categories localement finiment pr6sentables, c’est-h-dire leurs sous- categories pleines de la forme EL ou les domaines et codomaines des morphismes de E sont finiment pr6sentables. Nous l’utilisons aussi pour trouver une condition suffisante pour la r6flexivit6 des sous- categories de categories accessibles. Finalement, nous donnons une description des cat6gories de fractions dans les petites categories fini- ment compl6tes. I. Introduction Many "everyday" categories have the following type of presentation: a general locally finitely presentable (LFP) category ,C, representing the signature in some sense, is given, together with a set E of mor- phisms having finitely presentable domains and codomains. -
Topics in Koszul Duality, Michaelmas 2019, Oxford University Lecture 2
Topics in Koszul Duality, Michaelmas 2019, Oxford University Lecture 2: Categorical Background In this lecture, we will take a second look at Morita theory, and illustrate several cate- gorical concepts along the way. This will set the stage for the Barr{Beck theorem, which will be discussed in the next lecture, and for its higher categorical generalisation. Abstract categorical arguments of this kind can often be circumvented when dealing with \discrete" objects such as rings or modules, but become an indispensable tool in the study of \homotopical" objects like differential graded algebras or chain complexes. As we have seen in the overview class last week, Koszul duality forces us into a differential graded setting because certain hom-spaces only become nontrivial after having been derived 2 ∼ (e.g. remember that for R = k[]/ , we had HomR(k; k) = k but R HomR(k; k) = k[x]). 2.1. A reminder on Morita theory. The first result from last class will serve as a toy model; we recall it using Notation 1.7 from last class. ~ Proposition 2.1. Let Q 2 ModR be a left module over an associative ring R such that (1) Q is finite projective, i.e. a direct summand of R⊕n for some n; (2) Q is a generator, which means that the functor HomR(Q; −) is faithful. op Then R and S = EndR(Q) are Morita equivalent, which is witnessed by inverse equivalences ~ ~ Ge : ModR ! ModS ;M 7! HomR(Q; M) ~ ~ Fe : ModS ! ModR;N 7! Q ⊗S N: Proving Proposition 2.1 by hand is a reasonably straightforward, yet tedious, exercise. -
Characterizing Serre Quotients with No Section Functor And
CHARACTERIZING SERRE QUOTIENTS WITH NO SECTION FUNCTOR AND APPLICATIONS TO COHERENT SHEAVES MOHAMED BARAKAT AND MARKUS LANGE-HEGERMANN ABSTRACT. We prove an analogon of the the fundamental homomorphism theorem for certain classes of exact and essentially surjective functors of Abelian categories Q : A→B. It states that Q is up to equivalence the Serre quotient A → A/ ker Q, even in cases when the latter does not admit a section functor. For several classes of schemes X, including projective and toric varieties, this characterization applies to the sheafification functor from a certain category A of finitely presented graded modules to the category B = Coh X of coherent sheaves on X. This gives a direct proof that Coh X is a Serre quotient of A. 1. INTRODUCTION An essentially surjective exact functor Q : A → B of Abelian categories induces an essen- tially surjective exact embedding Q : A/C → B, where C is the kernel of Q and A/C the Serre quotient. It is natural to ask if or when Q is an equivalence of categories, i.e., if or when the fundamental homomorphism Theorem is valid for exact functors of Abelian categories. Grothendieck mentioned this question in [Gro57] but did not provide an answer: Thus A/C appears as an abelian category; moreover the identity functor Q : A→A/C is exact (and, in particular, commutes with kernels, cokernels, images, and coimages), Q(A)=0 if and only if A ∈ C, and any object of A/C has the form Q(A) for some A ∈A. These are the facts (which essentially characterize the quotientcategory) which allow us to safely apply the “mod C” language, since this language signifies simply that we are in the quotient abelian category. -
TWO-DIMENSIONAL MONAD THEORY Serve It Only To
Journal of Pure and Applied Algebra 59 (1989) l-41 North-Holland TWO-DIMENSIONAL MONAD THEORY R. BLACKWELL, G.M. KELLY* and A.J. POWER* Pure Mathematics Department, University of Sydney, N.S. W. 2006, Australia Communicated by M. Barr Received 2 June 1987 We consider a 2-monad T with rank on a complete and cocomplete 2-category, and write T-Alg for the 2-category given the T-algebras, the morphisms preserving the structure to within coherent isomorphisms, and the appropriate 2-cells; T-Alg, is the sub-2-category obtained by taking the strict morphisms. We show that T-Alg admits pseudo-limits and certain other limits, and that the inclusion 2-functor T-Alg,- T-Alg has a left adjoint. We introduce the notion of flexible algebra, and use it to prove that T-Alg admits all bicolimits and that the 2-functor T-Alg + S-Alg induced by a monad-map S+ T admits a left biadjoint. 1. Introduction 1.1. This is the first of a series of articles reporting the work of Kelly and various of his colleagues on what we may call two-dimensional universal algebra: the study of structures borne not by a set but by a category, or by a family of categories, and so on; in the context, however, where the morphisms of primary interest are not the strict ones, which preserve the relevant structure on the nose, but those which pre- serve it only to within coherent isomorphisms. This first article is concerned with the two-dimensional aspect of monad theory. -
FIBRATIONS of 8-CATEGORIES Contents Introduction 2 0.1. Main
FIBRATIONS OF 8-CATEGORIES DAVID AYALA & JOHN FRANCIS Abstract. We construct a flagged 8-category Corr of 8-categories and bimodules among them. We prove that Corr classifies exponentiable fibrations. This representability of exponentiable fibra- tions extends that established by Lurie of both coCartesian fibrations and Cartesian fibrations, as they are classified by the 8-category of 8-categories and its opposite, respectively. We introduce the flagged 8-subcategories LCorr and RCorr of Corr, whose morphisms are those bimodules which are left final and right initial, respectively. We identify the notions of fibrations these flagged 8-subcategories classify, and show that these 8-categories carry universal left/right fibrations. Contents Introduction 2 0.1. Main results 3 0.2. Motivation 5 0.3. Questions/problems 7 0.4. Conventions 8 1. Correspondences 9 1.1. Correspondence between two 8-categories 9 1.2. Exponentiable fibrations 10 1.3. Classifying correspondences 14 1.4. Symmetric monoidal structure 24 1.5. Conservative exponentiable correspondences 25 2. Cartesian and coCartesian fibrations 26 2.1. Basics about (co)Cartesian fibrations 27 2.2. Characterizing (co)Cartesian fibrations 32 2.3. (co)Cartesian-replacement 41 2.4. Left fibrations and right fibrations 45 2.5. Sub-left/right fibrations of coCartesian/Cartesian fibrations 50 3. Left final/right initial fibrations 50 3.1. Left final and right initial correspondences 51 3.2. Universal left/right fibrations over LCorr/RCorr 53 3.3. Left/right fibration-replacement 58 4. Corr as bimodules and as bifibrations 59 4.1. Correspondences as bimodules and bifibrations 59 4.2.