Notes on Restricted Inverse Limits of Categories
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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. -
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). -
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1
Limits Commutative Algebra May 11 2020 1. Direct Limits Definition 1: A directed set I is a set with a partial order ≤ such that for every i; j 2 I there is k 2 I such that i ≤ k and j ≤ k. Let R be a ring. A directed system of R-modules indexed by I is a collection of R modules fMi j i 2 Ig with a R module homomorphisms µi;j : Mi ! Mj for each pair i; j 2 I where i ≤ j, such that (i) for any i 2 I, µi;i = IdMi and (ii) for any i ≤ j ≤ k in I, µi;j ◦ µj;k = µi;k. We shall denote a directed system by a tuple (Mi; µi;j). The direct limit of a directed system is defined using a universal property. It exists and is unique up to a unique isomorphism. Theorem 2 (Direct limits). Let fMi j i 2 Ig be a directed system of R modules then there exists an R module M with the following properties: (i) There are R module homomorphisms µi : Mi ! M for each i 2 I, satisfying µi = µj ◦ µi;j whenever i < j. (ii) If there is an R module N such that there are R module homomorphisms νi : Mi ! N for each i and νi = νj ◦µi;j whenever i < j; then there exists a unique R module homomorphism ν : M ! N, such that νi = ν ◦ µi. The module M is unique in the sense that if there is any other R module M 0 satisfying properties (i) and (ii) then there is a unique R module isomorphism µ0 : M ! M 0. -
A Few Points in Topos Theory
A few points in topos theory Sam Zoghaib∗ Abstract This paper deals with two problems in topos theory; the construction of finite pseudo-limits and pseudo-colimits in appropriate sub-2-categories of the 2-category of toposes, and the definition and construction of the fundamental groupoid of a topos, in the context of the Galois theory of coverings; we will take results on the fundamental group of étale coverings in [1] as a starting example for the latter. We work in the more general context of bounded toposes over Set (instead of starting with an effec- tive descent morphism of schemes). Questions regarding the existence of limits and colimits of diagram of toposes arise while studying this prob- lem, but their general relevance makes it worth to study them separately. We expose mainly known constructions, but give some new insight on the assumptions and work out an explicit description of a functor in a coequalizer diagram which was as far as the author is aware unknown, which we believe can be generalised. This is essentially an overview of study and research conducted at dpmms, University of Cambridge, Great Britain, between March and Au- gust 2006, under the supervision of Martin Hyland. Contents 1 Introduction 2 2 General knowledge 3 3 On (co)limits of toposes 6 3.1 The construction of finite limits in BTop/S ............ 7 3.2 The construction of finite colimits in BTop/S ........... 9 4 The fundamental groupoid of a topos 12 4.1 The fundamental group of an atomic topos with a point . 13 4.2 The fundamental groupoid of an unpointed locally connected topos 15 5 Conclusion and future work 17 References 17 ∗e-mail: [email protected] 1 1 Introduction Toposes were first conceived ([2]) as kinds of “generalised spaces” which could serve as frameworks for cohomology theories; that is, mapping topological or geometrical invariants with an algebraic structure to topological spaces. -
Groups and Categories
\chap04" 2009/2/27 i i page 65 i i 4 GROUPS AND CATEGORIES This chapter is devoted to some of the various connections between groups and categories. If you already know the basic group theory covered here, then this will give you some insight into the categorical constructions we have learned so far; and if you do not know it yet, then you will learn it now as an application of category theory. We will focus on three different aspects of the relationship between categories and groups: 1. groups in a category, 2. the category of groups, 3. groups as categories. 4.1 Groups in a category As we have already seen, the notion of a group arises as an abstraction of the automorphisms of an object. In a specific, concrete case, a group G may thus consist of certain arrows g : X ! X for some object X in a category C, G ⊆ HomC(X; X) But the abstract group concept can also be described directly as an object in a category, equipped with a certain structure. This more subtle notion of a \group in a category" also proves to be quite useful. Let C be a category with finite products. The notion of a group in C essentially generalizes the usual notion of a group in Sets. Definition 4.1. A group in C consists of objects and arrows as so: m i G × G - G G 6 u 1 i i i i \chap04" 2009/2/27 i i page 66 66 GROUPSANDCATEGORIES i i satisfying the following conditions: 1. -
Monoidal Functors, Equivalence of Monoidal Categories
14 1.4. Monoidal functors, equivalence of monoidal categories. As we have explained, monoidal categories are a categorification of monoids. Now we pass to categorification of morphisms between monoids, namely monoidal functors. 0 0 0 0 0 Definition 1.4.1. Let (C; ⊗; 1; a; ι) and (C ; ⊗ ; 1 ; a ; ι ) be two monoidal 0 categories. A monoidal functor from C to C is a pair (F; J) where 0 0 ∼ F : C ! C is a functor, and J = fJX;Y : F (X) ⊗ F (Y ) −! F (X ⊗ Y )jX; Y 2 Cg is a natural isomorphism, such that F (1) is isomorphic 0 to 1 . and the diagram (1.4.1) a0 (F (X) ⊗0 F (Y )) ⊗0 F (Z) −−F− (X−)−;F− (Y− )−;F− (Z!) F (X) ⊗0 (F (Y ) ⊗0 F (Z)) ? ? J ⊗0Id ? Id ⊗0J ? X;Y F (Z) y F (X) Y;Z y F (X ⊗ Y ) ⊗0 F (Z) F (X) ⊗0 F (Y ⊗ Z) ? ? J ? J ? X⊗Y;Z y X;Y ⊗Z y F (aX;Y;Z ) F ((X ⊗ Y ) ⊗ Z) −−−−−−! F (X ⊗ (Y ⊗ Z)) is commutative for all X; Y; Z 2 C (“the monoidal structure axiom”). A monoidal functor F is said to be an equivalence of monoidal cate gories if it is an equivalence of ordinary categories. Remark 1.4.2. It is important to stress that, as seen from this defini tion, a monoidal functor is not just a functor between monoidal cate gories, but a functor with an additional structure (the isomorphism J) satisfying a certain equation (the monoidal structure axiom). -
Math 395: Category Theory Northwestern University, Lecture Notes
Math 395: Category Theory Northwestern University, Lecture Notes Written by Santiago Can˜ez These are lecture notes for an undergraduate seminar covering Category Theory, taught by the author at Northwestern University. The book we roughly follow is “Category Theory in Context” by Emily Riehl. These notes outline the specific approach we’re taking in terms the order in which topics are presented and what from the book we actually emphasize. We also include things we look at in class which aren’t in the book, but otherwise various standard definitions and examples are left to the book. Watch out for typos! Comments and suggestions are welcome. Contents Introduction to Categories 1 Special Morphisms, Products 3 Coproducts, Opposite Categories 7 Functors, Fullness and Faithfulness 9 Coproduct Examples, Concreteness 12 Natural Isomorphisms, Representability 14 More Representable Examples 17 Equivalences between Categories 19 Yoneda Lemma, Functors as Objects 21 Equalizers and Coequalizers 25 Some Functor Properties, An Equivalence Example 28 Segal’s Category, Coequalizer Examples 29 Limits and Colimits 29 More on Limits/Colimits 29 More Limit/Colimit Examples 30 Continuous Functors, Adjoints 30 Limits as Equalizers, Sheaves 30 Fun with Squares, Pullback Examples 30 More Adjoint Examples 30 Stone-Cech 30 Group and Monoid Objects 30 Monads 30 Algebras 30 Ultrafilters 30 Introduction to Categories Category theory provides a framework through which we can relate a construction/fact in one area of mathematics to a construction/fact in another. The goal is an ultimate form of abstraction, where we can truly single out what about a given problem is specific to that problem, and what is a reflection of a more general phenomenom which appears elsewhere. -
Arxiv:1911.00818V2 [Math.CT] 12 Jul 2021 Oc Eerhlbrtr,Teus Oenet Rcrei M Carnegie Or Shou Government, Express and U.S
A PRACTICAL TYPE THEORY FOR SYMMETRIC MONOIDAL CATEGORIES MICHAEL SHULMAN Abstract. We give a natural-deduction-style type theory for symmetric monoidal cat- egories whose judgmental structure directly represents morphisms with tensor products in their codomain as well as their domain. The syntax is inspired by Sweedler notation for coalgebras, with variables associated to types in the domain and terms associated to types in the codomain, allowing types to be treated informally like “sets with elements” subject to global linearity-like restrictions. We illustrate the usefulness of this type the- ory with various applications to duality, traces, Frobenius monoids, and (weak) Hopf monoids. Contents 1 Introduction 1 2 Props 9 3 On the admissibility of structural rules 11 4 The type theory for free props 14 5 Constructing free props from type theory 24 6 Presentations of props 29 7 Examples 31 1. Introduction 1.1. Type theories for monoidal categories. Type theories are a powerful tool for reasoning about categorical structures. This is best-known in the case of the internal language of a topos, which is a higher-order intuitionistic logic. But there are also weaker type theories that correspond to less highly-structured categories, such as regular logic for arXiv:1911.00818v2 [math.CT] 12 Jul 2021 regular categories, simply typed λ-calculus for cartesian closed categories, typed algebraic theories for categories with finite products, and so on (a good overview can be found in [Joh02, Part D]). This material is based on research sponsored by The United States Air Force Research Laboratory under agreement number FA9550-15-1-0053. -
A Short Introduction to Category Theory
A SHORT INTRODUCTION TO CATEGORY THEORY September 4, 2019 Contents 1. Category theory 1 1.1. Definition of a category 1 1.2. Natural transformations 3 1.3. Epimorphisms and monomorphisms 5 1.4. Yoneda Lemma 6 1.5. Limits and colimits 7 1.6. Free objects 9 References 9 1. Category theory 1.1. Definition of a category. Definition 1.1.1. A category C is the following data (1) a class ob(C), whose elements are called objects; (2) for any pair of objects A; B of C a set HomC(A; B), called set of morphisms; (3) for any three objects A; B; C of C a function HomC(A; B) × HomC(B; C) −! HomC(A; C) (f; g) −! g ◦ f; called composition of morphisms; which satisfy the following axioms (i) the sets of morphisms are all disjoints, so any morphism f determines his domain and his target; (ii) the composition is associative; (iii) for any object A of C there exists a morphism idA 2 Hom(A; A), called identity morphism, such that for any object B of C and any morphisms f 2 Hom(A; B) and g 2 Hom(C; A) we have f ◦ idA = f and idA ◦ g = g. Remark 1.1.2. The above definition is the definition of what is called a locally small category. For a general category we should admit that the set of morphisms is not a set. If the class of object is in fact a set then the category is called small. For a general category we should admit that morphisms form a class. -
Reflective Subcategories and Dense Subcategories
RENDICONTI del SEMINARIO MATEMATICO della UNIVERSITÀ DI PADOVA LUCIANO STRAMACCIA Reflective subcategories and dense subcategories Rendiconti del Seminario Matematico della Università di Padova, tome 67 (1982), p. 191-198 <http://www.numdam.org/item?id=RSMUP_1982__67__191_0> © Rendiconti del Seminario Matematico della Università di Padova, 1982, tous droits réservés. L’accès aux archives de la revue « Rendiconti del Seminario Matematico della Università di Padova » (http://rendiconti.math.unipd.it/) 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 conte- nir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Reflective Subcategories and Dense Subcategories. LUCIANO STRAMACCIA (*) Introduction. In [M], S. Mardesic defined the notion of a dense subcategory ~ c C, generalizing the situation one has in the Shape Theory of topological spaces, where K = HCW (= the homotopy category of CW-compleges) and C = HTOP ( = the homotopy category of topolo- gical spaces). In [G], E. Giuli observed that « dense subcategories » are a generalization of « reflective subcategories » and characterized (epi-) dense subcategories of TOP. In this paper we prove that the concepts of density and reflectivity are symmetric with respect to the passage to pro-categories; this means that, if K c C, then K is dense in C if and only if pro-K is reflective in pro-C. In order to do this we establish two necessary and sufficient condi- tions for K being dense in C. -
Graded Monoidal Categories Compositio Mathematica, Tome 28, No 3 (1974), P
COMPOSITIO MATHEMATICA A. FRÖHLICH C. T. C. WALL Graded monoidal categories Compositio Mathematica, tome 28, no 3 (1974), p. 229-285 <http://www.numdam.org/item?id=CM_1974__28_3_229_0> © Foundation Compositio Mathematica, 1974, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions géné- rales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infrac- tion 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/ COMPOSITIO MATHEMATICA, Vol. 28, Fasc. 3, 1974, pag. 229-285 Noordhoff International Publishing Printed in the Netherlands GRADED MONOIDAL CATEGORIES A. Fröhlich and C. T. C. Wall Introduction This paper grew out of our joint work on the Brauer group. Our idea was to define the Brauer group in an equivariant situation, also a ’twisted’ version incorporating anti-automorphisms, and give exact sequences for computing it. The theory related to representations of groups by auto- morphisms of various algebraic structures [4], [5], on one hand, and to the theory of quadratic and Hermitian forms (see particularly [17]) on the other. In developing these ideas, we observed that many of the arguments could be developed in a purely abstract setting, and that this clarified the nature of the proofs. The purpose of this paper is to present this setting, together with such theorems as need no further structure. To help motivate the reader, and fix ideas somewhat in tracing paths through the abstractions which follow, we list some of the examples to which the theory will be applied. -
Note on Monoidal Localisation
BULL. AUSTRAL. MATH. SOC. '8D05« I8DI0' I8DI5 VOL. 8 (1973), 1-16. Note on monoidal localisation Brian Day If a class Z of morphisms in a monoidal category A is closed under tensoring with the objects of A then the category- obtained by inverting the morphisms in Z is monoidal. We note the immediate properties of this induced structure. The main application describes monoidal completions in terms of the ordinary category completions introduced by Applegate and Tierney. This application in turn suggests a "change-of- universe" procedure for category theory based on a given monoidal closed category. Several features of this procedure are discussed. 0. Introduction The first step in this article is to apply a reflection theorem ([5], Theorem 2.1) for closed categories to the convolution structure of closed functor categories described in [3]. This combination is used to discuss monoidal localisation in the following sense. If a class Z of morphisms in a symmetric monoidal category 8 has the property that e € Z implies 1D ® e € Z for all objects B € B then the category 8, of fractions of 8 with respect to Z (as constructed in [fl]. Chapter l) is a monoidal category. Moreover, the projection functor P : 8 -* 8, then solves the corresponding universal problem in terms of monoidal functors; hence such a class Z is called monoidal. To each class Z of morphisms in a monoidal category 8 there corresponds a monoidal interior Z , namely the largest monoidal class Received 18 July 1972. The research here reported was supported in part by a grant from the National Science Foundation' of the USA.