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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. -
Congruences Between Derivatives of Geometric L-Functions
1 Congruences between Derivatives of Geometric L-Functions David Burns with an appendix by David Burns, King Fai Lai and Ki-Seng Tan Abstract. We prove a natural equivariant re¯nement of a theorem of Licht- enbaum describing the leading terms of Zeta functions of curves over ¯nite ¯elds in terms of Weil-¶etalecohomology. We then use this result to prove the validity of Chinburg's (3)-Conjecture for all abelian extensions of global function ¯elds, to prove natural re¯nements and generalisations of the re- ¯ned Stark conjectures formulated by, amongst others, Gross, Tate, Rubin and Popescu, to prove a variety of explicit restrictions on the Galois module structure of unit groups and divisor class groups and to describe explicitly the Fitting ideals of certain Weil-¶etalecohomology groups. In an appendix coau- thored with K. F. Lai and K-S. Tan we also show that the main conjectures of geometric Iwasawa theory can be proved without using either crystalline cohomology or Drinfeld modules. 1991 Mathematics Subject Classi¯cation: Primary 11G40; Secondary 11R65; 19A31; 19B28. Keywords and Phrases: Geometric L-functions, leading terms, congruences, Iwasawa theory 2 David Burns 1. Introduction The main result of the present article is the following Theorem 1.1. The central conjecture of [6] is valid for all global function ¯elds. (For a more explicit statement of this result see Theorem 3.1.) Theorem 1.1 is a natural equivariant re¯nement of the leading term formula proved by Lichtenbaum in [27] and also implies an extensive new family of integral congruence relations between the leading terms of L-functions associated to abelian characters of global functions ¯elds (see Remark 3.2). -
Arxiv:1705.02246V2 [Math.RT] 20 Nov 2019 Esyta Ulsubcategory Full a That Say [15]
WIDE SUBCATEGORIES OF d-CLUSTER TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels, cokernels, and extensions. Wide subcategories provide a significant interface between representation theory and combinatorics. If Φ is a finite dimensional algebra, then each functorially finite wide subcategory of mod(Φ) is of the φ form φ∗ mod(Γ) in an essentially unique way, where Γ is a finite dimensional algebra and Φ −→ Γ is Φ an algebra epimorphism satisfying Tor (Γ, Γ) = 0. 1 Let F ⊆ mod(Φ) be a d-cluster tilting subcategory as defined by Iyama. Then F is a d-abelian category as defined by Jasso, and we call a subcategory of F wide if it is closed under sums, summands, d- kernels, d-cokernels, and d-extensions. We generalise the above description of wide subcategories to this setting: Each functorially finite wide subcategory of F is of the form φ∗(G ) in an essentially φ Φ unique way, where Φ −→ Γ is an algebra epimorphism satisfying Tord (Γ, Γ) = 0, and G ⊆ mod(Γ) is a d-cluster tilting subcategory. We illustrate the theory by computing the wide subcategories of some d-cluster tilting subcategories ℓ F ⊆ mod(Φ) over algebras of the form Φ = kAm/(rad kAm) . Dedicated to Idun Reiten on the occasion of her 75th birthday 1. Introduction Let d > 1 be an integer. This paper introduces and studies wide subcategories of d-abelian categories as defined by Jasso. The main examples of d-abelian categories are d-cluster tilting subcategories as defined by Iyama. -
Classification of Finite Abelian Groups
Math 317 C1 John Sullivan Spring 2003 Classification of Finite Abelian Groups (Notes based on an article by Navarro in the Amer. Math. Monthly, February 2003.) The fundamental theorem of finite abelian groups expresses any such group as a product of cyclic groups: Theorem. Suppose G is a finite abelian group. Then G is (in a unique way) a direct product of cyclic groups of order pk with p prime. Our first step will be a special case of Cauchy’s Theorem, which we will prove later for arbitrary groups: whenever p |G| then G has an element of order p. Theorem (Cauchy). If G is a finite group, and p |G| is a prime, then G has an element of order p (or, equivalently, a subgroup of order p). ∼ Proof when G is abelian. First note that if |G| is prime, then G = Zp and we are done. In general, we work by induction. If G has no nontrivial proper subgroups, it must be a prime cyclic group, the case we’ve already handled. So we can suppose there is a nontrivial subgroup H smaller than G. Either p |H| or p |G/H|. In the first case, by induction, H has an element of order p which is also order p in G so we’re done. In the second case, if ∼ g + H has order p in G/H then |g + H| |g|, so hgi = Zkp for some k, and then kg ∈ G has order p. Note that we write our abelian groups additively. Definition. Given a prime p, a p-group is a group in which every element has order pk for some k. -
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. -
Op → Sset in Simplicial Sets, Or Equivalently a Functor X : ∆Op × ∆Op → Set
Contents 21 Bisimplicial sets 1 22 Homotopy colimits and limits (revisited) 10 23 Applications, Quillen’s Theorem B 23 21 Bisimplicial sets A bisimplicial set X is a simplicial object X : Dop ! sSet in simplicial sets, or equivalently a functor X : Dop × Dop ! Set: I write Xm;n = X(m;n) for the set of bisimplices in bidgree (m;n) and Xm = Xm;∗ for the vertical simplicial set in horiz. degree m. Morphisms X ! Y of bisimplicial sets are natural transformations. s2Set is the category of bisimplicial sets. Examples: 1) Dp;q is the contravariant representable functor Dp;q = hom( ;(p;q)) 1 on D × D. p;q G q Dm = D : m!p p;q The maps D ! X classify bisimplices in Xp;q. The bisimplex category (D × D)=X has the bisim- plices of X as objects, with morphisms the inci- dence relations Dp;q ' 7 X Dr;s 2) Suppose K and L are simplicial sets. The bisimplicial set Kט L has bisimplices (Kט L)p;q = Kp × Lq: The object Kט L is the external product of K and L. There is a natural isomorphism Dp;q =∼ Dpט Dq: 3) Suppose I is a small category and that X : I ! sSet is an I-diagram in simplicial sets. Recall (Lecture 04) that there is a bisimplicial set −−−!holim IX (“the” homotopy colimit) with vertical sim- 2 plicial sets G X(i0) i0→···→in in horizontal degrees n. The transformation X ! ∗ induces a bisimplicial set map G G p : X(i0) ! ∗ = BIn; i0→···→in i0→···→in where the set BIn has been identified with the dis- crete simplicial set K(BIn;0) in each horizontal de- gree. -
On Universal Properties of Preadditive and Additive Categories
On universal properties of preadditive and additive categories Karoubi envelope, additive envelope and tensor product Bachelor's Thesis Mathias Ritter February 2016 II Contents 0 Introduction1 0.1 Envelope operations..............................1 0.1.1 The Karoubi envelope.........................1 0.1.2 The additive envelope of preadditive categories............2 0.2 The tensor product of categories........................2 0.2.1 The tensor product of preadditive categories.............2 0.2.2 The tensor product of additive categories...............3 0.3 Counterexamples for compatibility relations.................4 0.3.1 Karoubi envelope and additive envelope...............4 0.3.2 Additive envelope and tensor product.................4 0.3.3 Karoubi envelope and tensor product.................4 0.4 Conventions...................................5 1 Preliminaries 11 1.1 Idempotents................................... 11 1.2 A lemma on equivalences............................ 12 1.3 The tensor product of modules and linear maps............... 12 1.3.1 The tensor product of modules.................... 12 1.3.2 The tensor product of linear maps................... 19 1.4 Preadditive categories over a commutative ring................ 21 2 Envelope operations 27 2.1 The Karoubi envelope............................. 27 2.1.1 Definition and duality......................... 27 2.1.2 The Karoubi envelope respects additivity............... 30 2.1.3 The inclusion functor.......................... 33 III 2.1.4 Idempotent complete categories.................... 34 2.1.5 The Karoubi envelope is idempotent complete............ 38 2.1.6 Functoriality.............................. 40 2.1.7 The image functor........................... 46 2.1.8 Universal property........................... 48 2.1.9 Karoubi envelope for preadditive categories over a commutative ring 55 2.2 The additive envelope of preadditive categories................ 59 2.2.1 Definition and additivity....................... -
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. -
Derived Functors and Homological Dimension (Pdf)
DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION George Torres Math 221 Abstract. This paper overviews the basic notions of abelian categories, exact functors, and chain complexes. It will use these concepts to define derived functors, prove their existence, and demon- strate their relationship to homological dimension. I affirm my awareness of the standards of the Harvard College Honor Code. Date: December 15, 2015. 1 2 DERIVED FUNCTORS AND HOMOLOGICAL DIMENSION 1. Abelian Categories and Homology The concept of an abelian category will be necessary for discussing ideas on homological algebra. Loosely speaking, an abelian cagetory is a type of category that behaves like modules (R-mod) or abelian groups (Ab). We must first define a few types of morphisms that such a category must have. Definition 1.1. A morphism f : X ! Y in a category C is a zero morphism if: • for any A 2 C and any g; h : A ! X, fg = fh • for any B 2 C and any g; h : Y ! B, gf = hf We denote a zero morphism as 0XY (or sometimes just 0 if the context is sufficient). Definition 1.2. A morphism f : X ! Y is a monomorphism if it is left cancellative. That is, for all g; h : Z ! X, we have fg = fh ) g = h. An epimorphism is a morphism if it is right cancellative. The zero morphism is a generalization of the zero map on rings, or the identity homomorphism on groups. Monomorphisms and epimorphisms are generalizations of injective and surjective homomorphisms (though these definitions don't always coincide). It can be shown that a morphism is an isomorphism iff it is epic and monic. -
Full Text (PDF Format)
ASIAN J. MATH. © 1997 International Press Vol. 1, No. 2, pp. 330-417, June 1997 009 K-THEORY FOR TRIANGULATED CATEGORIES 1(A): HOMOLOGICAL FUNCTORS * AMNON NEEMANt 0. Introduction. We should perhaps begin by reminding the reader briefly of Quillen's Q-construction on exact categories. DEFINITION 0.1. Let £ be an exact category. The category Q(£) is defined as follows. 0.1.1. The objects of Q(£) are the objects of £. 0.1.2. The morphisms X •^ X' in Q(£) between X,X' G Ob{Q{£)) = Ob{£) are isomorphism classes of diagrams of morphisms in £ X X' \ S Y where the morphism X —> Y is an admissible mono, while X1 —y Y is an admissible epi. Perhaps a more classical way to say this is that X is a subquotient of X''. 0.1.3. Composition is defined by composing subquotients; X X' X' X" \ ^/ and \ y/ Y Y' compose to give X X' X" \ s \ s Y PO Y' \ S Z where the square marked PO is a pushout square. The category Q{£) can be realised to give a space, which we freely confuse with the category. The Quillen if-theory of the exact category £ was defined, in [9], to be the homotopy of the loop space of Q{£). That is, Ki{£)=ni+l[Q{£)]. Quillen proved many nice functoriality properties for his iT-theory, and the one most relevant to this article is the resolution theorem. The resolution theorem asserts the following. 7 THEOREM 0.2. Let F : £ —> J be a fully faithful, exact inclusion of exact categories. -
Coreflective Subcategories
transactions of the american mathematical society Volume 157, June 1971 COREFLECTIVE SUBCATEGORIES BY HORST HERRLICH AND GEORGE E. STRECKER Abstract. General morphism factorization criteria are used to investigate categorical reflections and coreflections, and in particular epi-reflections and mono- coreflections. It is shown that for most categories with "reasonable" smallness and completeness conditions, each coreflection can be "split" into the composition of two mono-coreflections and that under these conditions mono-coreflective subcategories can be characterized as those which are closed under the formation of coproducts and extremal quotient objects. The relationship of reflectivity to closure under limits is investigated as well as coreflections in categories which have "enough" constant morphisms. 1. Introduction. The concept of reflections in categories (and likewise the dual notion—coreflections) serves the purpose of unifying various fundamental con- structions in mathematics, via "universal" properties that each possesses. His- torically, the concept seems to have its roots in the fundamental construction of E. Cech [4] whereby (using the fact that the class of compact spaces is productive and closed-hereditary) each completely regular F2 space is densely embedded in a compact F2 space with a universal extension property. In [3, Appendice III; Sur les applications universelles] Bourbaki has shown the essential underlying similarity that the Cech-Stone compactification has with other mathematical extensions, such as the completion of uniform spaces and the embedding of integral domains in their fields of fractions. In doing so, he essentially defined the notion of reflections in categories. It was not until 1964, when Freyd [5] published the first book dealing exclusively with the theory of categories, that sufficient categorical machinery and insight were developed to allow for a very simple formulation of the concept of reflections and for a basic investigation of reflections as entities themselvesi1). -
Van Kampen Colimits As Bicolimits in Span*
Van Kampen colimits as bicolimits in Span? Tobias Heindel1 and Pawe lSoboci´nski2 1 Abt. f¨urInformatik und angewandte kw, Universit¨atDuisburg-Essen, Germany 2 ECS, University of Southampton, United Kingdom Abstract. The exactness properties of coproducts in extensive categories and pushouts along monos in adhesive categories have found various applications in theoretical computer science, e.g. in program semantics, data type theory and rewriting. We show that these properties can be understood as a single universal property in the associated bicategory of spans. To this end, we first provide a general notion of Van Kampen cocone that specialises to the above colimits. The main result states that Van Kampen cocones can be characterised as exactly those diagrams in that induce bicolimit diagrams in the bicategory of spans Span , C C provided that C has pullbacks and enough colimits. Introduction The interplay between limits and colimits is a research topic with several applica- tions in theoretical computer science, including the solution of recursive domain equations, using the coincidence of limits and colimits. Research on this general topic has identified several classes of categories in which limits and colimits relate to each other in useful ways; extensive categories [5] and adhesive categories [21] are two examples of such classes. Extensive categories [5] have coproducts that are “well-behaved” with respect to pullbacks; more concretely, they are disjoint and universal. Extensivity has been used by mathematicians [4] and computer scientists [25] alike. In the presence of products, extensive categories are distributive [5] and thus can be used, for instance, to model circuits [28] or to give models of specifications [11].