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Notes on Category Theory (In Progress) Notes on Category Theory (in progress) George Torres Last updated February 28, 2018 Contents 1 Introduction and Motivation 3 2 Definition of a Category 3 2.1 Examples . .4 3 Functors 4 3.1 Natural Transformations . .5 3.2 Adjoint Functors . .5 3.3 Units and Counits . .6 3.4 Initial and Terminal Objects . .7 3.4.1 Comma Categories . .7 4 Representability and Yoneda's Lemma 8 4.1 Representables . .9 4.2 The Yoneda Embedding . 10 4.3 The Yoneda Lemma . 10 4.4 Consequences of Yoneda . 11 5 Limits and Colimits 12 5.1 (Co)Products, (Co)Equalizers, Pullbacks and Pushouts . 13 5.2 Topological limits . 15 5.3 Existence of limits and colimits . 15 5.4 Limits as Representable Objects . 16 5.5 Limits as Adjoints . 16 5.6 Preserving Limits and GAFT . 18 6 Abelian Categories 19 6.1 Homology . 20 6.1.1 Biproducts . 21 6.2 Exact Functors . 23 6.3 Injective and Projective Objects . 26 6.3.1 Projective and Injective Modules . 27 6.4 The Chain Complex Category . 28 6.5 Homological dimension . 30 6.6 Derived Functors . 32 1 CONTENTS CONTENTS 7 Triangulated and Derived Categories 35 ||||||||||| Note to the reader: This is an ongoing collection of notes on introductory category theory that I have kept since my undergraduate years. They are aimed at students with an undergraduate level background in topology and algebra. These notes started as lecture notes for the Fall 2015 Category Theory tutorial led by Danny Shi at Harvard. There is no single textbook that these notes follow, but Categories for the Working Mathematician by Mac Lane and Lang's Algebra are good standard resources. For a more modern take, see Emily Riehl's Category Theory in Context. Please let me know of any typos you may find: [email protected]. Any text in red is TODO and will be done at some point when I find time. 2 2 DEFINITION OF A CATEGORY 1 Introduction and Motivation Categories are a way of abstracting the mechanics of various subjects in mathematics into a coherent theory. It is the modern standard for communicating and formulating many \algebraic" fields of mathematics, including Algebraic Topology and Algebraic Geometry. As a warm-up, consider two abelian groups A; B. The reader knows that the tensor product A ⊗ B is defined to be an abelian group together with a map f : A×B ! A⊗B satisfying certain properties. It is unique in the sense that, if X is any other abelian group with a map g : A × B ! X satisfying the same properties as f, then there is a unique map φ such that the following diagram commutes: A × B f A ⊗ B g φ X In this sense A ⊗ B is the \smallest\ such group, because any other such groups factor through A ⊗ B in the above diagram. This is an example of a universal property construction, and is used in various areas of mathematics. Category theory can characterize how and why this works, and most importantly, it can define it without reference to any axioms of group theory. 2 Definition of a Category \Category theory is the subject where you can leave the definitions as exercises" - John Baez Definition 2.1. A category C consists of the following: 1. A collection of objects Obj(C) 2. For each A; B 2 Obj(C), there is a collection of morphisms HomC(A; B) (\maps" between A and B) 3. For every A 2 Obj(C), there exists an identity morphism 1A 2 HomC(A; A). 4. A composition rule for morphisms: ◦ : HomC(A; B) × HomC(B; C) ! HomC(A; C) Such that the following two axioms are satisfied: • Associativity: for morphisms g; f; h: (h ◦ g) ◦ f = h ◦ (g ◦ f) • Identity: for objects A; B and morphism f 2 HomC(A; B): f idA A B idB ) f ◦ idA = idB ◦ f = f Categories are usually best thought of as dots and arrows between dots. Remark 2.2. Notation for objects and morphisms varies. We often drop the C subscript in Hom and sometimes we write C(A; B) instead of Hom(A; B). We also often drop the \obj" notation and think of C itself as the set of objects (i.e. we write A 2 C instead of A 2 Obj(C)). Additionally, we denote Hom(C) to be the collection of all morphisms in C. Definition 2.3. Let A; B 2 Obj(C) and f 2 Hom(A; B). We say A and B are isomorphic (A =∼ B) if there exists g 2 Hom(B; A) such that g ◦ f = idA and f ◦ g = idB. 3 2.1 Examples 3 FUNCTORS 2.1 Examples Below are some familiar examples of categories. B Sets (Set), where Obj(Set) are sets and morphisms are functions between sets. B Groups (Grp), where Obj(Grp) are groups and morphisms are group homomorphisms. B Modules (Rmod), where Obj(Rmod) are R modules and morphisms are linear homomorphisms B Topological spaces (Top), where Obj(Top) are topological spaces and morphisms are continuous functions. B Pointed topological spaes (Top∗), where Obj(Top∗) are spaces with a specified basepoint and morphisms are continuous maps that send basepoints to basepoints. B The trivial category, 1. This is the category with only one object (often denoted \ · ") and its identity morphism. Definition 2.4. Given a category C, we define the dual (or opposite) category Cop as the category whose objects are those of C and f 2 C(A; B) () f 2 Cop(B; A). It is the same category but with arrows (morphisms) pointed in the reversed direction. 3 Functors Definition 3.1. A (covariant) functor F : A!B between categories is a function taking objects in A to objects in B. Further it maps morphisms to morphisms in the following way: f 2 A(A; A0) ) F (f) 2 B(F (A);F (A0)) Further, F must satisfy two conditions: 1. F (g ◦ f) = F (g) ◦ F (f). 2. F (idA) = idF (a). Remark 3.2. Consider the category of categories (called Cat) : the objects are categories and the morphisms are functors. This allows us to compose functors and declare when two categories are equivalent. Two categories are equivalent if they are isomorphic as elements of Cat. Examples: • Forgetful functors { functors that \forget" certain information about objects in a category: a) F : Grp ! Set sending groups to their underlying groups and homomorphisms to maps of sets. b) F : Ring ! Grp sending rings to their abelian group and homomorphisms to homomorphisms. • F : AbGrp ! Grp sending groups to themselves, forgetting that they are abelian. • The fundamental group { This sends a topological space to its fundamental group: π1 : Top∗ ! Grp • Singular homology { This sends a topological space to its nth singular homology group: Hn(−): Top ! AbGrp Definition 3.3. A contravariant functor is a covariant functor F : Aop !B. Definition 3.4. A functor F : A!B is faithful if, given A; A0 2 Obj(A) then F as a map of A(A0;A) is injective. Definition 3.5. A subcategory C of a category A is a category such that: • Obj(C) ⊂ Obj(A) • C(C; C0) ⊂ A(C; C0) A subcategory is called full if 8A; A0 2 Obj(A) we have C(A; A0) = A(A; A0). 4 3.1 Natural Transformations 3 FUNCTORS 3.1 Natural Transformations Consider two functors F; G : A!B. We would like a way to make these compatible: Definition 3.6. A natural transformation N between two functors is a family of morphisms, each of which associates an object of A to a morphism N(A) (sometimes denoted NA) of B. Any natural transformation must make the following square commute: F (f) F (A) F (A0) N(A) N(A0) G(f) G(A) G(A0) Natural transformations are often denoted as double arrows: F A N B G If NA is an isomorphism for each A, then we call N an isomorphism of functors. Examples: • Let G be a group thought of as a one-object category. A functor F : G ! Set gives a group action on a set S. A natural transformation of group actions is a map of sets that respects the group action. Expand • Let Mn(−): CRing ! Monoid be the functor sending a commutative ring to the monoid of matrices over that ring. Let U : Cring ! Monoid be the forgetful functor that forgets ring addition. A natural transformation between Mn(−) and U is the determinant. Fix Remark 3.7. Natural transformations can compose to form another natural transformation. This allows us to define the functor category: Definition 3.8. For fixed categories A; B the functor category [A; B] is the category whose objects are functors A!B and whose morphisms are natural transformations. Definition 3.9. We say two functors are naturally isomorphic if they are isomorphic as objects in the functor category. Equivalently, F; G : A!B are naturally isomorphic if there is a natural transformation α : F ! G such that αA : F (A) ! G(A) is an isomorphism for all A 2 A (such an α is called a natural isomorphism). 3.2 Adjoint Functors Definition 3.10. Let F : A B : G be functors. We say F is left adjoint to G (and G is right adjoint to F ) if for every A 2 A;B 2 B we have a natural isomorphism: B(F (A);B) =∼ A(A; G(B)) In other words, this is a natural bijection. For g 2 B(F (A);B), we denote the corresponding element of A(A; G(B)) as g.
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