DOCSLIB.ORG

A NOTE on DIRECT PRODUCTS and EXT1 CONTRAVARIANT FUNCTORS 1. Introduction Commuting Properties of Hom and Ext Functors with Resp

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A NOTE on DIRECT PRODUCTS and EXT1 CONTRAVARIANT FUNCTORS 1. Introduction Commuting Properties of Hom and Ext Functors with Resp A NOTE ON DIRECT PRODUCTS AND EXT1 CONTRAVARIANT FUNCTORS FLAVIU POP AND CLAUDIU VALCULESCU Abstract. In this paper we prove, using inequalities between infinite car- dinals, that, if R is an hereditary ring, the contravariant derived functor 1 ExtR(−;G) commutes with direct products if and only if G is an injective R-module. 1. Introduction Commuting properties of Hom and Ext functors with respect to direct sums and direct products are very important in Module Theory. It is well known that, if G is an R-module (in this paper all modules are right R-modules), the covariant Hom-functor HomR(G; −) : Mod-R ! Mod-Z preserves direct products, i.e. the natural homomorphism Y Y φF : HomR(G; Ni) ! HomR(G; Ni) i2I i2I Q induced by the family HomR(G; pi), where pi : j2I Nj ! Ni are the canonical projections associated to direct product, is an isomorphism, for any family of R- modules F = (Ni)i2I . On the other hand, the contravariant Hom-functor HomR(−;G) : Mod-R ! Mod-Z does not preserve, in general, the direct products. More precisely, if F = (Mi)i2I is a family of R-modules, then the natural homomorphism Y Y F : HomR( Mi;G) ! HomR(Mi;G) i2I i2I Q induced by the family HomR(ui;G), where ui : Mi ! j2I Mj are the canonical injections associated to the direct product, is not, in general, an isomorphism. We Q note that if we replace direct products i2I Mi by direct sums ⊕i2I Mi, a similar discussion about commuting properties leads to important notions in Module The- ory: small and self-small modules, respectively slender and self-slender modules, and the structure of these modules can be very complicated (see [3], [9], [11]). The same arguments are valid for the covariant(respectively, the contravari- n n ant) derived functors ExtR(G; −) : Mod-R ! Mod-Z (respectively, ExtR(−;G): Mod-R ! Mod-Z). In the case of hereditary rings, the modules G such that 1 ExtR(G; −) commutes naturally with direct sums are studied in [6] and [17]. For Date: September 11, 2013. 2010 Mathematics Subject Classification. 16D90, 16E30. Key words and phrases. direct product, derived functor, Ext functor, extensions, hereditary ring, injective module, cotorsion group. 1 2 FLAVIU POP AND CLAUDIU VALCULESCU commuting properties with respect to some limits and colimits we refer to [7] and [13, Section 6.1 and Section 6.3]. A survey of recent results about the behavior of these functors is presented in [1] and, although the authors treat the general case, they are focused on the context of abelian groups. In [5], Breaz showed that F is an isomorphism for all families F if and only if G is the trivial module. Following a similar idea, in this paper, we will prove that, 1 Q Q 1 if R is an hereditary ring, the Z-modules ExtR( i2I Mi;G) and i2I ExtR(Mi;G) are naturally isomorphic if and only if G is injective. 2. Main Theorem Throughout this paper, by a ring R we will understand a unital associative ring, an R-module is a right R-module and we will denote by Mod-R the cat- n egory of all right R-modules. Also, we will use ExtR(−;G) for the n-th con- travariant derived functor, where n is a positive integer and G is an R-module. If f : M ! N is an R-homomorphism, then we denote the induced homomorphism n n 0 ExtR(f; G)(respectively, HomR(f; G)) by f (respectively, by f ). If M is an R-module and λ is a cardinal, then M λ(respectively, M (λ)) represents the direct product(respectively, the direct sum) of λ copies of M. Moreover, if F = (Mi)i2I is a family of R-modules then, for all positive integers n, we consider the natural isomorphisms induced by the universal property of direct product n n M Y n χF : ExtR( Mi;G) ! ExtR(Mi;G) i2I i2I respectively, the natural homomorphisms induced by the universal property of di- rect product n n Y Y n F : ExtR( Mi;G) ! ExtR(Mi;G): i2I i2I n n n If F = (M)κ≤λ, then we denote χF (respectively, F ) by χM,λ (respectively, by n M,λ). A ring R is called hereditary if all submodules of projective R-modules are pro- k jective. We note that, R is hereditary if and only if ExtR(M; N) = 0, for all R-modules M and N and for all integers k ≥ 2. In order to prove our main result, we need some cardinal arguments. The first lemma is a well known result(and it was proven, for example, in [4]), and the second lemma is also well known(see, for instance, [15, Lemma 3.1]). Moreover, these results were generalized by Stoviˇcekinˇ [16]. Let X be a set and let A be a family of subsets of X. We say that A is almost disjoint if the intersection of any two distinct elements of A is finite. Lemma 2.1. Let λ be an infinite cardinal. Then there is a family of λ@0 countable almost disjoint subsets of λ. Lemma 2.2. For any cardinal σ, there is an infinite cardinal λ ≥ σ such that λ@0 = 2λ. Consequently, for every cardinal σ, there is an infinite cardinal λ such that σ ≤ λ < λ@0 . We have the following lemma, which is very important for the proof of the main result. A NOTE ON DIRECT PRODUCTS AND EXT1 CONTRAVARIANT FUNCTORS 3 Lemma 2.3. Let R be a ring and let G be an R-module. Let λ be a cardinal and let F = (Mκ)κ≤λ be a family of R-modules. Suppose that 1 1 Y Y 1 F : ExtR( Mκ;G) ! ExtR(Mκ;G) κ≤λ κ≤λ is an isomorphism. If M Y i : Mκ ! Mκ κ≤λ κ≤λ is the natural homomorphism then the induced homomorphism 1 1 Y 1 M i : ExtR( Mκ;G) ! ExtR( Mκ;G) κ≤λ κ≤λ is an isomorphism. Proof. Consider the following commutative diagram 1 1 Q i 1 L ExtR( κ≤λ Mκ;G) −−−−! ExtR( κ≤λ Mκ;G) ? ? 1 ? ?χ1 : F y y F Q 1 id Q 1 κ≤λ ExtR(Mκ;G) −−−−! κ≤λ ExtR(Mκ;G): 1 1 By hypothesis, F is an isomorphism and, since χF is also an isomorphism, it 1 follows that i is an isomorphism too. We also need the following useful lemma. 1 Lemma 2.4. Let R be an hereditary ring and let G be an R-module. If ExtR(M; G) 6= 1 λ (λ) 0 then ExtR(M =M ;G) 6= 0, for any R-module M and for any infinite cardinal λ. 1 Proof. Let M be an R-module with ExtR(M; G) 6= 0 and let λ be an infinite cardinal. The R-homomorphism θ : M ! M λ=M (λ) defined by (λ) (λ) θ(x) = (x)κ≤λ + M = (x; x; x; ··· ) + M is a monomorphism. Then, we have the following short exact sequence 0 ! M −!θ M λ=M (λ) −! Coker(θ) ! 0: Since R is hereditary, the induced sequence 0 0 λ (λ) θ ∆ 0 ! HomR(Coker(θ);G) −! HomR(M =M ;G) −! HomR(M; G) −! 0 1 1 ∆ 1 1 λ (λ) θ 1 ∆ −! ExtR(Coker(θ);G) −! ExtR(M =M ;G) −! ExtR(M; G) −! 0 1 1 λ (λ) 1 is exact, so that θ : ExtR(M =M ;G) ! ExtR(M; G) is an epimorphism. Since 1 1 λ (λ) ExtR(M; G) 6= 0, it follows that ExtR(M =M ;G) 6= 0. Now we give the main result of the paper. Theorem 2.5. Let R be an hereditary ring and let G be an R-module. If M is an R-module, then the following statements are equivalent: 1 1 λ 1 λ (a) M,λ : ExtR(M ;G) ! ExtR(M; G) is an isomorphism, for any cardinal λ; 4 FLAVIU POP AND CLAUDIU VALCULESCU 1 µ (b) ExtR(M ;G) = 0, for any cardinal µ. Proof. Let M be an R-module. 1 µ (a) ) (b) Suppose that there exists a cardinal µ such that ExtR(M ;G) 6= 0. 1 1 Since M,µ is an isomorphism, it follows that ExtR(M; G) 6= 0. We denote by X the R-module M @0 =M (@0). By Lemma 2.2, there is an infinite @0 λ cardinal λ ≥ jHomR(M; G)j such that λ = 2 . We have the following short exact sequence p 0 ! M (λ) −!i M λ −! M λ=M (λ) ! 0; hence the induced sequence 0 0 0 λ (λ) p λ i (λ) ∆ 0 ! HomR(M =M ;G) −! HomR(M ;G) −! HomR(M ;G) −! 0 1 1 1 ∆ 1 λ (λ) p 1 λ i 1 (λ) ∆ −! ExtR(M =M ;G) −! ExtR(M ;G) −! ExtR(M ;G) −! 0 is also exact. From Lemma 2.3, the natural homomorphism i1 is an isomorphism, so that Im(p1) = Ker(i1) = 0. It follows that 0 (λ) ∆ 1 λ (λ) HomR(M ;G) −! ExtR(M =M ;G) ! 0 is exact, hence we have the inequality λ 1 λ (λ) jHomR(M; G)j ≥ ExtR(M =M ;G) : 1 λ (λ) By Lemma 2.4 we know that ExtR(M =M ;G) 6= 0, hence HomR(M; G) 6= 0. @0 By Lemma 2.1, there is a family (Aα)α≤λ@0 of λ countable almost disjoint sub- Aα λ Aα λ (λ) sets of λ.
Load more

Recommended publications
  • Nearly Locally Presentable Categories Are Locally Presentable Is Equivalent to Vopˇenka’S Principle
    NEARLY LOCALLY PRESENTABLE CATEGORIES L. POSITSELSKI AND J. ROSICKY´ Abstract. We introduce a new class of categories generalizing locally presentable ones. The distinction does not manifest in the abelian case and, assuming Vopˇenka’s principle, the same happens in the regular case. The category of complete partial orders is the natural example of a nearly locally finitely presentable category which is not locally presentable. 1. Introduction Locally presentable categories were introduced by P. Gabriel and F. Ulmer in [6]. A category K is locally λ-presentable if it is cocomplete and has a strong generator consisting of λ-presentable objects. Here, λ is a regular cardinal and an object A is λ-presentable if its hom-functor K(A, −): K → Set preserves λ-directed colimits. A category is locally presentable if it is locally λ-presentable for some λ. This con- cept of presentability formalizes the usual practice – for instance, finitely presentable groups are precisely groups given by finitely many generators and finitely many re- lations. Locally presentable categories have many nice properties, in particular they are complete and co-wellpowered. Gabriel and Ulmer [6] also showed that one can define locally presentable categories by using just monomorphisms instead all morphisms. They defined λ-generated ob- jects as those whose hom-functor K(A, −) preserves λ-directed colimits of monomor- phisms. Again, this concept formalizes the usual practice – finitely generated groups are precisely groups admitting a finite set of generators. This leads to locally gener- ated categories, where a cocomplete category K is locally λ-generated if it has a strong arXiv:1710.10476v2 [math.CT] 2 Apr 2018 generator consisting of λ-generated objects and every object of K has only a set of strong quotients.
    [Show full text]
  • AN INTRODUCTION to CATEGORY THEORY and the YONEDA LEMMA Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transfo
    AN INTRODUCTION TO CATEGORY THEORY AND THE YONEDA LEMMA SHU-NAN JUSTIN CHANG Abstract. We begin this introduction to category theory with definitions of categories, functors, and natural transformations. We provide many examples of each construct and discuss interesting relations between them. We proceed to prove the Yoneda Lemma, a central concept in category theory, and motivate its significance. We conclude with some results and applications of the Yoneda Lemma. Contents Introduction 1 1. Categories 2 2. Functors 3 3. Natural Transformations 6 4. The Yoneda Lemma 9 5. Corollaries and Applications 10 Acknowledgments 12 References 13 Introduction Category theory is an interdisciplinary field of mathematics which takes on a new perspective to understanding mathematical phenomena. Unlike most other branches of mathematics, category theory is rather uninterested in the objects be- ing considered themselves. Instead, it focuses on the relations between objects of the same type and objects of different types. Its abstract and broad nature allows it to reach into and connect several different branches of mathematics: algebra, geometry, topology, analysis, etc. A central theme of category theory is abstraction, understanding objects by gen- eralizing rather than focusing on them individually. Similar to taxonomy, category theory offers a way for mathematical concepts to be abstracted and unified. What makes category theory more than just an organizational system, however, is its abil- ity to generate information about these abstract objects by studying their relations to each other. This ability comes from what Emily Riehl calls \arguably the most important result in category theory"[4], the Yoneda Lemma. The Yoneda Lemma allows us to formally define an object by its relations to other objects, which is central to the relation-oriented perspective taken by category theory.
    [Show full text]
  • Direct Products and Homomorphisms
    Commuting properties Direct products and homomorphisms Simion Breaz logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Outline 1 Commuting properties Products and coproducts Contravariant functors Covariant functors logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits. logo Simion Breaz Direct products and homomorphisms Products and coproducts Commuting properties Contravariant functors Covariant functors Introduction Important properties of objects in particular categories (e.g. varieties of universal algebras) can be described using commuting properties of some canonical functors. For instance, in [Ad´amek and Rosicki: Locally presentable categories] there are the following examples: If V is a variety of finitary algebras and A ∈ V then A is finitely generated iff the functor Hom(A, −): V → Set preserves direct unions (i.e. directed colimits of monomorphisms); A is finitely presented (i.e. it is generated by finitely many generators modulo finitely many relations) iff the functor Hom(A, −): V → Set preserves directed colimits.
    [Show full text]
  • Lecture 1.3: Direct Products and Sums
    Lecture 1.3: Direct products and sums Matthew Macauley Department of Mathematical Sciences Clemson University http://www.math.clemson.edu/~macaule/ Math 8530, Advanced Linear Algebra M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 1 / 5 Overview In previous lectures, we learned about vectors spaces and subspaces. We learned about what it meant for a subset to span, to be linearly independent, and to be a basis. In this lecture, we will see how to create new vector spaces from old ones. We will see several ways to \multiply" vector spaces together, and will learn how to construct: the complement of a subspace the direct sum of two subspaces the direct product of two vector spaces M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 2 / 5 Complements and direct sums Theorem 1.5 (a) Every subspace Y of a finite-dimensional vector space X is finite-dimensional. (b) Every subspace Y has a complement in X : another subspace Z such that every vector x 2 X can be written uniquely as x = y + z; y 2 Y ; z 2 Z; dim X = dim Y + dim Z: Proof Definition X is the direct sum of subspaces Y and Z that are complements of each other. More generally, X is the direct sum of subspaces Y1;:::; Ym if every x 2 X can be expressed uniquely as x = y1 + ··· + ym; yi 2 Yi : We denote this as X = Y1 ⊕ · · · ⊕ Ym. M. Macauley (Clemson) Lecture 1.3: Direct products and sums Math 8530, Advanced Linear Algebra 3 / 5 Direct products Definition The direct product of X1 and X2 is the vector space X1 × X2 := (x1; x2) j x1 2 X1; x2 2 X2 ; with addition and multiplication defined component-wise.
    [Show full text]
  • Forcing with Copies of Countable Ordinals
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 4, April 2015, Pages 1771–1784 S 0002-9939(2014)12360-4 Article electronically published on December 4, 2014 FORCING WITH COPIES OF COUNTABLE ORDINALS MILOSˇ S. KURILIC´ (Communicated by Mirna Dˇzamonja) Abstract. Let α be a countable ordinal and P(α) the collection of its subsets isomorphic to α. We show that the separative quotient of the poset P(α), ⊂ is isomorphic to a forcing product of iterated reduced products of Boolean γ algebras of the form P (ω )/Iωγ ,whereγ ∈ Lim ∪{1} and Iωγ is the corre- sponding ordinal ideal. Moreover, the poset P(α), ⊂ is forcing equivalent to + a two-step iteration of the form (P (ω)/Fin) ∗ π,where[ω] “π is an ω1- + closed separative pre-order” and, if h = ω1,to(P (ω)/Fin) . Also we analyze δ I the quotients over ordinal ideals P (ω )/ ωδ and the corresponding cardinal invariants hωδ and tωδ . 1. Introduction The posets of the form P(X), ⊂,whereX is a relational structure and P(X) the set of (the domains of) its isomorphic substructures, were considered in [7], where a classification of the relations on countable sets related to the forcing-related properties of the corresponding posets of copies is described. So, defining two structures to be equivalent if the corresponding posets of copies produce the same generic extensions, we obtain a rough classification of structures which, in general, depends on the properties of the model of set theory in which we work. For example, under CH all countable linear orders are partitioned in only two classes.
    [Show full text]
  • Ring (Mathematics) 1 Ring (Mathematics)
    Ring (mathematics) 1 Ring (mathematics) In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition (called the additive group of the ring) and a monoid under multiplication such that multiplication distributes over addition.a[›] In other words the ring axioms require that addition is commutative, addition and multiplication are associative, multiplication distributes over addition, each element in the set has an additive inverse, and there exists an additive identity. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article. Polynomials, represented here by curves, form a ring under addition The branch of mathematics that studies rings is known and multiplication. as ring theory. Ring theorists study properties common to both familiar mathematical structures such as integers and polynomials, and to the many less well-known mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.[1] Ring theory may be used to understand fundamental physical laws, such as those underlying special relativity and symmetry phenomena in molecular chemistry. The concept of a ring first arose from attempts to prove Fermat's last theorem, starting with Richard Dedekind in the 1880s. After contributions from other fields, mainly number theory, the ring notion was generalized and firmly established during the 1920s by Emmy Noether and Wolfgang Krull.[2] Modern ring theory—a very active mathematical discipline—studies rings in their own right.
    [Show full text]
  • Friday September 20 Lecture Notes
    Friday September 20 Lecture Notes 1 Functors Definition Let C and D be categories. A functor (or covariant) F is a function that assigns each C 2 Obj(C) an object F (C) 2 Obj(D) and to each f : A ! B in C, a morphism F (f): F (A) ! F (B) in D, satisfying: For all A 2 Obj(C), F (1A) = 1FA. Whenever fg is defined, F (fg) = F (f)F (g). e.g. If C is a category, then there exists an identity functor 1C s.t. 1C(C) = C for C 2 Obj(C) and for every morphism f of C, 1C(f) = f. For any category from universal algebra we have \forgetful" functors. e.g. Take F : Grp ! Cat of monoids (·; 1). Then F (G) is a group viewed as a monoid and F (f) is a group homomorphism f viewed as a monoid homomor- phism. e.g. If C is any universal algebra category, then F : C! Sets F (C) is the underlying sets of C F (f) is a morphism e.g. Let C be a category. Take A 2 Obj(C). Then if we define a covariant Hom functor, Hom(A; ): C! Sets, defined by Hom(A; )(B) = Hom(A; B) for all B 2 Obj(C) and f : B ! C, then Hom(A; )(f) : Hom(A; B) ! Hom(A; C) with g 7! fg (we denote Hom(A; ) by f∗). Let us check if f∗ is a functor: Take B 2 Obj(C). Then Hom(A; )(1B) = (1B)∗ : Hom(A; B) ! Hom(A; B) and for g 2 Hom(A; B), (1B)∗(g) = 1Bg = g.
    [Show full text]
  • The Yoneda Ext and Arbitrary Coproducts in Abelian Categories
    THE YONEDA EXT AND ARBITRARY COPRODUCTS IN ABELIAN CATEGORIES ALEJANDRO ARGUD´IN MONROY Abstract. There are well known identities involving the Ext bifunctor, co- products, and products in Ab4 abelian categories with enough projectives. Namely, for every such category A, given an object X and a set of ob- n ∼ jects {Ai}i∈I , the following isomorphism can be built ExtA i∈I Ai,X = n n L i∈I ExtA (Ai,X), where Ext is the n-th derived functor of the Hom func- tor.Q The goal of this paper is to show a similar isomorphism for the n-th Yoneda Ext, which is a functor equivalent to Extn that can be defined in more general contexts. The desired isomorphism is constructed explicitly by using colimits, in Ab4 abelian categories with not necessarily enough projec- tives nor injectives, answering a question made by R. Colpi and K R. Fuller in [8]. Furthermore, the isomorphisms constructed are used to characterize Ab4 categories. A dual result is also stated. 1. Introduction The study of extensions is a theory that has developed from multiplicative groups [21, 15], with applications ranging from representations of central simple algebras [4, 13] to topology [10]. In this article we will focus on extensions in an abelian category C. In this context, an extension of an object A by an object C is a short exact sequence 0 → A → M → C → 0 up to equivalence, where two exact sequences are equivalent if there is a morphism arXiv:1904.12182v2 [math.CT] 8 Sep 2020 from one to another with identity morphisms at the ends.
    [Show full text]
  • Adjoint and Representable Functors
    What underlies dual-process cognition? Adjoint and representable functors Steven Phillips ([email protected]) Mathematical Neuroinformatics Group, Human Informatics Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8566 JAPAN Abstract & Stanovich, 2013, for the model and responses to the five criticisms). According to this model, cognition defaults to Despite a general recognition that there are two styles of think- ing: fast, reflexive and relatively effortless (Type 1) versus slow, Type 1 processes that can be intervened upon by Type 2 pro- reflective and effortful (Type 2), dual-process theories of cogni- cesses in response to task demands. However, this model is tion remain controversial, in particular, for their vagueness. To still far from the kinds of formal, computational explanations address this lack of formal precision, we take a mathematical category theory approach towards understanding what under- that prevail in cognitive science. lies the relationship between dual cognitive processes. From our category theory perspective, we show that distinguishing ID Criticism of dual-process theories features of Type 1 versus Type 2 processes are exhibited via adjoint and representable functors. These results suggest that C1 Theories are multitudinous and definitions vague category theory provides a useful formal framework for devel- C2 Type 1/2 distinctions do not reliably align oping dual-process theories of cognition. C3 Cognitive styles vary continuously, not discretely Keywords: dual-process; Type 1; Type 2; category theory; C4 Single-process theories provide better explanations category; functor; natural transformation; adjoint C5 Evidence for dual-processing is unconvincing Introduction Table 2: Five criticisms of dual-process theories (Evans & Intuitively, at least, human cognition involves two starkly dif- Stanovich, 2013).
    [Show full text]
  • 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.
    [Show full text]
  • 1 Category Theory
    {-# LANGUAGE RankNTypes #-} module AbstractNonsense where import Control.Monad 1 Category theory Definition 1. A category consists of • a collection of objects • and a collection of morphisms between those objects. We write f : A → B for the morphism f connecting the object A to B. • Morphisms are closed under composition, i.e. for morphisms f : A → B and g : B → C there exists the composed morphism h = f ◦ g : A → C1. Furthermore we require that • Composition is associative, i.e. f ◦ (g ◦ h) = (f ◦ g) ◦ h • and for each object A there exists an identity morphism idA such that for all morphisms f : A → B: f ◦ idB = idA ◦ f = f Many mathematical structures form catgeories and thus the theorems and con- structions of category theory apply to them. As an example consider the cate- gories • Set whose objects are sets and morphisms are functions between those sets. • Grp whose objects are groups and morphisms are group homomorphisms, i.e. structure preserving functions, between those groups. 1.1 Functors Category theory is mainly interested in relationships between different kinds of mathematical structures. Therefore the fundamental notion of a functor is introduced: 1The order of the composition is to be read from left to right in contrast to standard mathematical notation. 1 Definition 2. A functor F : C → D is a transformation between categories C and D. It is defined by its action on objects F (A) and morphisms F (f) and has to preserve the categorical structure, i.e. for any morphism f : A → B: F (f(A)) = F (f)(F (A)) which can also be stated graphically as the commutativity of the following dia- gram: F (f) F (A) F (B) f A B Alternatively we can state that functors preserve the categorical structure by the requirement to respect the composition of morphisms: F (idC) = idD F (f) ◦ F (g) = F (f ◦ g) 1.2 Natural transformations Taking the construction a step further we can ask for transformations between functors.
    [Show full text]
  • 1.7 Categories: Products, Coproducts, and Free Objects
    22 CHAPTER 1. GROUPS 1.7 Categories: Products, Coproducts, and Free Objects Categories deal with objects and morphisms between objects. For examples: objects sets groups rings module morphisms functions homomorphisms homomorphisms homomorphisms Category theory studies some common properties for them. Def. A category is a class C of objects (denoted A, B, C,. ) together with the following things: 1. A set Hom (A; B) for every pair (A; B) 2 C × C. An element of Hom (A; B) is called a morphism from A to B and is denoted f : A ! B. 2. A function Hom (B; C) × Hom (A; B) ! Hom (A; C) for every triple (A; B; C) 2 C × C × C. For morphisms f : A ! B and g : B ! C, this function is written (g; f) 7! g ◦ f and g ◦ f : A ! C is called the composite of f and g. All subject to the two axioms: (I) Associativity: If f : A ! B, g : B ! C and h : C ! D are morphisms of C, then h ◦ (g ◦ f) = (h ◦ g) ◦ f. (II) Identity: For each object B of C there exists a morphism 1B : B ! B such that 1B ◦ f = f for any f : A ! B, and g ◦ 1B = g for any g : B ! C. In a category C a morphism f : A ! B is called an equivalence, and A and B are said to be equivalent, if there is a morphism g : B ! A in C such that g ◦ f = 1A and f ◦ g = 1B. Ex. Objects Morphisms f : A ! B is an equivalence A is equivalent to B sets functions f is a bijection jAj = jBj groups group homomorphisms f is an isomorphism A is isomorphic to B partial order sets f : A ! B such that f is an isomorphism between A is isomorphic to B \x ≤ y in (A; ≤) ) partial order sets A and B f(x) ≤ f(y) in (B; ≤)" Ex.
    [Show full text]