DOCSLIB.ORG

Abelian Categories We Take Only the First Two Axioms: AB1

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Abelian Categories We Take Only the First Two Axioms: AB1 MATH 131B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 1 1. Additive categories From Lang III 3: Abelian categories we take only the first two axioms: AB1, AB2. § Definition 1.1. An additive category is a category for which every hom set Hom (X, Y )isanadditivegroupand C C AB1a composition is biadditive,i.e.,(f + f ) g = f g + f g and f (g + g )= 1 2 ◦ 1 ◦ 2 ◦ ◦ 1 2 f g1 + f g2. AB1b ◦has a zero◦ object 0 having the property that Hom (X, 0) = 0 = Hom (0,X) forC all objects X. C C AB2 has finite products and coproducts (sums). C Theorem 1.2. In an additive category, finite products and finite sums are isomorphic, i.e., the canonical morphism A A is an isomorphism. i ! i Proof. By induction it sufficesL to considerQ sums and products of pairs. We will show A B = A B.Considerthefollowingdiagram. ⊕ ⇠ ⇥ idA (1.1) A / A ; jA pA jA # f g # A B / A B / A B ; ⊕ ⇥ ; ⊕ jB pB jB idB # B / B where f : A B A B is the unique morphism so that pA f jA = idA,pB f jB = id and the⊕ other! two⇥ compositions are zero: p f j ◦=0=◦ p f ◦j ◦.Let B B ◦ ◦ A A ◦ ◦ B g : A B A B be the sum g = jA pA + jB pB.Then,itiseasytoseethatf,g are inverse⇥ ! morphisms:⊕ ◦ ◦ (1) g f j = j p f j + j p f j = j id +0 = j and, ◦ ◦ A A ◦ A ◦ ◦ A B ◦ B ◦ ◦ A A ◦ A A similarly, g f jB = jB.Bydefinitionofcoproduct,thereisauniquemorphism h : A B ◦ A◦ B satisfying h j = j and h j = j .Sinceh = g f and ⊕ ! ⊕ ◦ A A ◦ B B ◦ h = idA B are two solutions, we have g f = idA B. ⊕ ◦ ⊕ (2) pA f g = pA f jA pA + pA f jB pB = pA and, similarly, pB f g = pB. So,◦f ◦g is the◦ identity◦ ◦ on A B◦ . ◦ ◦ ◦ ◦ ◦ ⇥ Therefore, A B and A B are canonically isomorphic in any additive category. ⊕ ⇥ ⇤ Example 1.3. Some examples of additive categories: (1) The category Mod-R of all right R-modules. (2) The category mod-R of finitely generated right R-modules and R-linear homo- morphisms. E.g., mod-Z is the category of finitely generated abelian groups. (3) The category tor-Z of finite abelian groups. (4) The category (Z) of finitely generated free abelian groups (groups isomorphic P to Zn for some finite n). The first and third categories are “abelian” but the other two are not. I don’t want to spend time formally defining what is an abelian category. But I will point out the problems with the second and fourth categories and how to fix them. 2MATH131B:ALGEBRAIIPARTA:HOMOLOGICALALGEBRA 2. Kernels and cokernels In a general additive category, a morphism might not have a kernel or cokernel. However, if they do exist, they are unique. Definition 2.1. The kernel of a morphism f : A B is an object K with a morphism j : K A so that ! ! (1) f j =0:K B (2) For◦ any other! ob ject X and morphism g : X A so that f g =0thereexists aunique˜g : X K so that g = j g˜ (such a!g ˜ is called a lifting◦ of g to K). ! ◦ Since this is a universal property, the kernel is unique (up to isomorphism) if it exists. X !˜g 0 9 g ~ j ✏ f K / A / B Theorem 2.2. K is the kernel of f : A B if and only if ! j f 0 Hom (X, K) ] Hom (X, A) ] Hom (X, B) ! C ! C ! C is exact for any object X. Here f] is left composition with f. Recall that a morphism f : A B is a monomorphism if, for any two maps ! g = h : C A, f g = f h,i,e,f] : Hom (X, A) Hom (X, B) is a monomorphism of6 groups. ! ◦ 6 ◦ C ! C Corollary 2.3. A morphism f : A B is a monomorphism i↵ ker f =0. ! f Notation 2.4. We say that 0 K A B is exact if K =kerf. ! ! ! Remark 2.5. A non-example:Thecategorymod-R of f.g. right R-modules does not have kernels if R is not right Noetherian (right ideals are finitely generated). To see this suppose that I is a right ideal which is not finitely generated. Then IR as R-module is not an object in the category. So, the morphism R R/I does not have a kernel in the category mod-R. ! Cokernel is defined analogously and satisfies the following theorem which can be used as the definition. Theorem 2.6. The cokernel of f : A B is an object C with a morphism p : B C so that ! ! p] f ] 0 Hom (C, Y ) Hom (B,Y ) Hom (A, Y ) ! C ! C ! C is exact for any object Y where f ] is right composition with f. The standard diagram defining coker f is the following. f p A / B / C g 0 !g ✏ 9 Y Again, letting C =0wegetthestatement: MATH 131B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 3 Corollary 2.7. A morphism f is an epimorphism (i.e., f ] is a monomorphism) if and only if coker f =0. f Notation 2.8. We say A B C 0isexact if C =cokerf. f g ! ! ! We say 0 A B C 0isexact if A =kerg and C =cokerf. ! ! ! ! These two theorems can be summarized by the following statement. Corollary 2.9. For any additive category , Hom is left exact in each coordinate. C C Exercise 2.10. (1) Show that mod-R has cokernels. (2) In the category of f.g. free abelian groups, show that the morphism f : Z Z 2 ! given by multiplication by 2 is an epimorphism: 0 Z Z 0isexact!! ! ! ! 3. Projective and injective objects We will discuss the definition of projective and injective objects in any additive cat- egory. And it was easy to show that the category of R-modules has sufficiently many projectives. But the category of free R-modules does not. Definition 3.1. An object P of an additive category is called projective if for any epimorphism p : A B and any morphism g : P C B there exists a morphism g˜ : P A so that p !g˜ = g.Themap˜g is called a lifting! of g to A. ! ◦ A ? g˜ 9 p g ✏ P / B Theorem 3.2. P is projective if and only if Hom (P, ) is an exact functor. C − Proof. If 0 A B C 0 is exact then, by left exactness of Hom we get an exact sequence: ! ! ! ! 0 Hom (P, A) Hom (P, B) Hom (P, C) ! C ! C ! C By definition, P is projective if and only if the last map is always an epimorphism, i.e., i↵we get a short exact sequence 0 Hom (P, A) Hom (P, B) Hom (P, C) 0 ! C ! C ! C ! ⇤ Theorem 3.3. Any free R-module is projective. Proof. Suppose that F is free with generators x↵.TheneveryelementofF can be written uniquely as x↵r↵ where the coefficients r↵ R are almost all zero (only finitely many are nonzero). Suppose that g : F B is2 a homomorphism. Then, for P ! every index ↵,theelementg(x↵)comesfromsomeelementy↵ A.I.e.,g(x↵)=f(y↵). Then a liftingg ˜ of g is given by 2 (3.1)g ˜( xar↵)= y↵r↵ The verification that this is a liftingX is straightforward.X ⇤ 4MATH131B:ALGEBRAIIPARTA:HOMOLOGICALALGEBRA To do the last step in full detail, we use the universal property of free modules: A homomorphism from the free module F generated by a set S to another module M is uniquely determined by a set mapping S M which is arbitrary, i.e., (3.1) defines the ! unique homomorphism F A sending x↵ to y↵. For every R-module M !there is a free R-module which maps onto M, namely the free module F generated by symbols [x]forallx M and with projection map p : F M given by 2 ! p r↵[x↵] = r↵x↵ The notation [x]isusedtodistinguishbetweentheelement⇣X ⌘ X x M and the corresponding generator [x] F . Using the universal property of free modules,2 the homomorphism p is defined by the2 equation p[x]=x. For M afinitelygeneratedR-module, take a finite generating set S.Thenthefree R-module generated by S maps onto M..
Load more

Recommended publications
  • 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.
    [Show full text]
  • 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.......................
    [Show full text]
  • 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.
    [Show full text]
  • Abelian Categories
    Abelian Categories Lemma. In an Ab-enriched category with zero object every finite product is coproduct and conversely. π1 Proof. Suppose A × B //A; B is a product. Define ι1 : A ! A × B and π2 ι2 : B ! A × B by π1ι1 = id; π2ι1 = 0; π1ι2 = 0; π2ι2 = id: It follows that ι1π1+ι2π2 = id (both sides are equal upon applying π1 and π2). To show that ι1; ι2 are a coproduct suppose given ' : A ! C; : B ! C. It φ : A × B ! C has the properties φι1 = ' and φι2 = then we must have φ = φid = φ(ι1π1 + ι2π2) = ϕπ1 + π2: Conversely, the formula ϕπ1 + π2 yields the desired map on A × B. An additive category is an Ab-enriched category with a zero object and finite products (or coproducts). In such a category, a kernel of a morphism f : A ! B is an equalizer k in the diagram k f ker(f) / A / B: 0 Dually, a cokernel of f is a coequalizer c in the diagram f c A / B / coker(f): 0 An Abelian category is an additive category such that 1. every map has a kernel and a cokernel, 2. every mono is a kernel, and every epi is a cokernel. In fact, it then follows immediatly that a mono is the kernel of its cokernel, while an epi is the cokernel of its kernel. 1 Proof of last statement. Suppose f : B ! C is epi and the cokernel of some g : A ! B. Write k : ker(f) ! B for the kernel of f. Since f ◦ g = 0 the map g¯ indicated in the diagram exists.
    [Show full text]
  • Homological Algebra Lecture 1
    Homological Algebra Lecture 1 Richard Crew Richard Crew Homological Algebra Lecture 1 1 / 21 Additive Categories Categories of modules over a ring have many special features that categories in general do not have. For example the Hom sets are actually abelian groups. Products and coproducts are representable, and one can form kernels and cokernels. The notation of an abelian category axiomatizes this structure. This is useful when one wants to perform module-like constructions on categories that are not module categories, but have all the requisite structure. We approach this concept in stages. A preadditive category is one in which one can add morphisms in a way compatible with the category structure. An additive category is a preadditive category in which finite coproducts are representable and have an \identity object." A preabelian category is an additive category in which kernels and cokernels exist, and finally an abelian category is one in which they behave sensibly. Richard Crew Homological Algebra Lecture 1 2 / 21 Definition A preadditive category is a category C for which each Hom set has an abelian group structure satisfying the following conditions: For all morphisms f : X ! X 0, g : Y ! Y 0 in C the maps 0 0 HomC(X ; Y ) ! HomC(X ; Y ); HomC(X ; Y ) ! HomC(X ; Y ) induced by f and g are homomorphisms. The composition maps HomC(Y ; Z) × HomC(X ; Y ) ! HomC(X ; Z)(g; f ) 7! g ◦ f are bilinear. The group law on the Hom sets will always be written additively, so the last condition means that (f + g) ◦ h = (f ◦ h) + (g ◦ h); f ◦ (g + h) = (f ◦ g) + (f ◦ h): Richard Crew Homological Algebra Lecture 1 3 / 21 We denote by 0 the identity of any Hom set, so the bilinearity of composition implies that f ◦ 0 = 0 ◦ f = 0 for any morphism f in C.
    [Show full text]
  • Combinatorial Categorical Equivalences of Dold-Kan Type 3
    COMBINATORIAL CATEGORICAL EQUIVALENCES OF DOLD-KAN TYPE STEPHEN LACK AND ROSS STREET Abstract. We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let X denote an additive category with finite direct sums and split idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in X are equivalent to chain complexes in X ; (b) the observation of Church, Ellenberg and Farb [9] that X -valued species are equivalent to X -valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of “Dold-Kan type”; and so on. When X is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourn. Contents 1. Introduction 1 2. The setting 4 3. Basic examples 6 4. The kernel module 8 5. Reduction of the problem 11 6. The case P = K 12 7. Examples of Theorem 6.7 14 8. When X is semiabelian 16 Appendix A. A general result from enriched category theory 19 Appendix B. Remarks on idempotents 21 References 22 2010 Mathematics Subject Classification: 18E05; 20C30; 18A32; 18A25; 18G35; 18G30 arXiv:1402.7151v5 [math.CT] 29 Mar 2019 Key words and phrases: additive category; Dold-Kan theorem; partial map; semi-abelian category; comonadic; Joyal species. 1. Introduction The intention of this paper is to prove a class of equivalences of categories that seem of interest in enumerative combinatorics as per [22], representation theory as per [9], and homotopy theory as per [2].
    [Show full text]
  • On Ideals and Homology in Additive Categories
    IJMMS 29:8 (2002) 439–451 PII. S0161171202011675 http://ijmms.hindawi.com © Hindawi Publishing Corp. ON IDEALS AND HOMOLOGY IN ADDITIVE CATEGORIES LUCIAN M. IONESCU Received 28 January 2001 and in revised form 26 July 2001 Ideals are used to define homological functors in additive categories. In abelian categories the ideals corresponding to the usual universal objects are principal, and the construction reduces, in a choice dependent way, to homology groups. The applications considered in this paper are: derived categories and functors. 2000 Mathematics Subject Classification: 18G50, 18A05. 1. Introduction. Categorification is by now a commonly used procedure [1, 6, 9]. The concept of an additive category generalizes that of a ring in the same way group- oids generalize the notion of groups. Additive categories were called “rings with sev- eral objects” in [14], and were studied by imitating results and proofs from noncom- mutative homological ring theory, to additive category theory. Alternatively, the addi- tive category theory may be applied, as in [15], to the ring theory. Subsequent related papers adopted the “ideal theory” point of view, for example, [5], and in [17] the prob- lem of lifting algebraic geometry to the category theory level was considered and a notion of prime spectrum of a category was defined. In this paper, we consider the Dedekind’s original aim for introducing ideals [7], and leading to the study of general rings, not only principal ideal rings (PIR). In the context of categories, we relax the requirements of an exact category for the existence of kernels and cokernels, and define homological objects in an intrinsic way, using ideals.
    [Show full text]
  • Introduction to Categories
    6 Introduction to categories 6.1 The definition of a category We have now seen many examples of representation theories and of operations with representations (direct sum, tensor product, induction, restriction, reflection functors, etc.) A context in which one can systematically talk about this is provided by Category Theory. Category theory was founded by Saunders MacLane and Samuel Eilenberg around 1940. It is a fairly abstract theory which seemingly has no content, for which reason it was christened “abstract nonsense”. Nevertheless, it is a very flexible and powerful language, which has become totally indispensable in many areas of mathematics, such as algebraic geometry, topology, representation theory, and many others. We will now give a very short introduction to Category theory, highlighting its relevance to the topics in representation theory we have discussed. For a serious acquaintance with category theory, the reader should use the classical book [McL]. Definition 6.1. A category is the following data: C (i) a class of objects Ob( ); C (ii) for every objects X; Y Ob( ), the class Hom (X; Y ) = Hom(X; Y ) of morphisms (or 2 C C arrows) from X; Y (for f Hom(X; Y ), one may write f : X Y ); 2 ! (iii) For any objects X; Y; Z Ob( ), a composition map Hom(Y; Z) Hom(X; Y ) Hom(X; Z), 2 C × ! (f; g) f g, 7! ∞ which satisfy the following axioms: 1. The composition is associative, i.e., (f g) h = f (g h); ∞ ∞ ∞ ∞ 2. For each X Ob( ), there is a morphism 1 Hom(X; X), called the unit morphism, such 2 C X 2 that 1 f = f and g 1 = g for any f; g for which compositions make sense.
    [Show full text]
  • Categories of Mackey Functors
    Categories of Mackey Functors Elango Panchadcharam M ∗(p1s1) M (t2p2) M(U) / M(P) ∗ / M(W ) 8 C 8 C 88 ÖÖ 88 ÖÖ 88 ÖÖ 88 ÖÖ 8 M ∗(p1) Ö 8 M (p2) Ö 88 ÖÖ 88 ∗ ÖÖ M ∗(s1) 8 Ö 8 Ö M (t2) 8 Ö 8 Ö ∗ 88 ÖÖ 88 ÖÖ 8 ÖÖ 8 ÖÖ M(S) M(T ) 88 Mackey ÖC 88 ÖÖ 88 ÖÖ 88 ÖÖ M (s2) 8 Ö M ∗(t1) ∗ 8 Ö 88 ÖÖ 8 ÖÖ M(V ) This thesis is presented for the degree of Doctor of Philosophy. Department of Mathematics Division of Information and Communication Sciences Macquarie University New South Wales, Australia December 2006 (Revised March 2007) ii This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration except where specifically indicated in the text. This work has not been submitted for a higher degree to any other university or institution. Elango Panchadcharam In memory of my Father, T. Panchadcharam 1939 - 1991. iii iv Summary The thesis studies the theory of Mackey functors as an application of enriched category theory and highlights the notions of lax braiding and lax centre for monoidal categories and more generally for promonoidal categories. The notion of Mackey functor was first defined by Dress [Dr1] and Green [Gr] in the early 1970’s as a tool for studying representations of finite groups. The first contribution of this thesis is the study of Mackey functors on a com- pact closed category T . We define the Mackey functors on a compact closed category T and investigate the properties of the category Mky of Mackey func- tors on T .
    [Show full text]
  • Lectures on Homological Algebra
    Lectures on Homological Algebra Weizhe Zheng Morningside Center of Mathematics Academy of Mathematics and Systems Science, Chinese Academy of Sciences Beijing 100190, China University of the Chinese Academy of Sciences, Beijing 100049, China Email: [email protected] Contents 1 Categories and functors 1 1.1 Categories . 1 1.2 Functors . 3 1.3 Universal constructions . 7 1.4 Adjunction . 11 1.5 Additive categories . 16 1.6 Abelian categories . 21 1.7 Projective and injective objects . 30 1.8 Projective and injective modules . 32 2 Derived categories and derived functors 41 2.1 Complexes . 41 2.2 Homotopy category, triangulated categories . 47 2.3 Localization of categories . 56 2.4 Derived categories . 61 2.5 Extensions . 70 2.6 Derived functors . 78 2.7 Double complexes, derived Hom ..................... 83 2.8 Flat modules, derived tensor product . 88 2.9 Homology and cohomology of groups . 98 2.10 Spectral objects and spectral sequences . 101 Summary of properties of rings and modules 105 iii iv CONTENTS Chapter 1 Categories and functors Very rough historical sketch Homological algebra studies derived functors between • categories of modules (since the 1940s, culminating in the 1956 book by Cartan and Eilenberg [CE]); • abelian categories (Grothendieck’s 1957 T¯ohokuarticle [G]); and • derived categories (Verdier’s 1963 notes [V1] and 1967 thesis of doctorat d’État [V2] following ideas of Grothendieck). 1.1 Categories Definition 1.1.1. A category C consists of a set of objects Ob(C), a set of morphisms Hom(X, Y ) for every pair of objects (X, Y ) of C, and a composition law, namely a map Hom(X, Y ) × Hom(Y, Z) → Hom(X, Z), denoted by (f, g) 7→ gf (or g ◦ f), for every triple of objects (X, Y, Z) of C.
    [Show full text]
  • Category Theory Course
    Category Theory Course John Baez September 3, 2019 1 Contents 1 Category Theory: 4 1.1 Definition of a Category....................... 5 1.1.1 Categories of mathematical objects............. 5 1.1.2 Categories as mathematical objects............ 6 1.2 Doing Mathematics inside a Category............... 10 1.3 Limits and Colimits.......................... 11 1.3.1 Products............................ 11 1.3.2 Coproducts.......................... 14 1.4 General Limits and Colimits..................... 15 2 Equalizers, Coequalizers, Pullbacks, and Pushouts (Week 3) 16 2.1 Equalizers............................... 16 2.2 Coequalizers.............................. 18 2.3 Pullbacks................................ 19 2.4 Pullbacks and Pushouts....................... 20 2.5 Limits for all finite diagrams.................... 21 3 Week 4 22 3.1 Mathematics Between Categories.................. 22 3.2 Natural Transformations....................... 25 4 Maps Between Categories 28 4.1 Natural Transformations....................... 28 4.1.1 Examples of natural transformations........... 28 4.2 Equivalence of Categories...................... 28 4.3 Adjunctions.............................. 29 4.3.1 What are adjunctions?.................... 29 4.3.2 Examples of Adjunctions.................. 30 4.3.3 Diagonal Functor....................... 31 5 Diagrams in a Category as Functors 33 5.1 Units and Counits of Adjunctions................. 39 6 Cartesian Closed Categories 40 6.1 Evaluation and Coevaluation in Cartesian Closed Categories. 41 6.1.1 Internalizing Composition................. 42 6.2 Elements................................ 43 7 Week 9 43 7.1 Subobjects............................... 46 8 Symmetric Monoidal Categories 50 8.1 Guest lecture by Christina Osborne................ 50 8.1.1 What is a Monoidal Category?............... 50 8.1.2 Going back to the definition of a symmetric monoidal category.............................. 53 2 9 Week 10 54 9.1 The subobject classifier in Graph.................
    [Show full text]
  • Chapter 9. Derived Categories
    HOMOLOGICAL ALGEBRA Contents 9. Derived categories of abelian categories 1 9.1. Summary 1 9.2. Localization of categories 3 9.3. The derived category D(A) of an abelian category A 4 9.4. Truncations 5 9.5. Homotopy category descriptions of derived categories 7 9.6. Derived functors RF : D+(A) → D+(B) 8 9.7. Usefulness of the derived category 10 9.8. Four functors formalism in geometry 11 9.9. The topological context 14 9. Derived categories of abelian categories 9.1. Summary. The course covers three main constructions Hn (1) The cohomology functors C(A) −→ A for an abelian category A. One tool for computing cohomology are spectral sequences. (2) The homotopy category of complexes K(A) for an additive category A. These are used to make injective/projective resolutions canonical and therefore to make sense of the derived functors. If A is abelian with enough injectives then to an additive RF functor F : A → B, we associate its right derived functor K+(A) −−→ K+(B) by RF (A) def= F (I) for any injective resolution I of A. One can say that this construction makes manifest some information hidden in the functor F itself. (3) The derived category of complexes D(A) for an abelian category A. It works RF much the same as one produces derived functors D+(A) −−→ D+(B) by the same formulas RF (A) def= F (I) for any injective resolution I of A. However, the same idea of de4rived functors works technically better in this setting. Date: ? 1 2 The treatment of Homological Algebra in this course is mostly “classical” in the sense that we concentrate on defining functors RnF : A→A and we use spectral sequences to calculate these.
    [Show full text]