1 Categories, Functors, and Natural Transformations. Discrete categories. A category is discrete when every arrow is an identity.

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1 MacLane: Categories or Working Mathematician 1 Categories, Functors, and Natural Transormations 1.1 Axioms or Categories 1.2 Categories Discrete categories. A category is discrete when every arrow is an identity. 1.3 Functors ull=hom-unction is surjective (or every pair o objects), aithull=injective 1.4 Natural Transormations Given two unctors S, T : C B a natural transormation τ : S T is a unction which assigns to each object c o C an arrow τ c = τc: Sc T c o B in such a way that every arrow : c c in C yields a diagram c Sc τc T c S c Sc τc T c which is commutative. A natural transormation τ with every component τ c invertible in B is called natural equivalence or better a natural isomorphism. An equivalence between categories C and D is deined to be a pair o unctors S : C D, T : D C together with natural isomorphisms I c = T S, ID = S T. 1.5 Monics, Epis and Zeros monic=monomorphism, epic=epimorphism right inverse=section, let inverse=retraction An object t is terminal in C i to each object a in C there is exactly one arrow a t. Dual: initial object. A null object z in C is an object which is both initial and terminal. For any two objects a and b o C there is a unique arrow a z b, celled the zero arrow rom a to b. A groupoid is a category in which every arrow is invertible. A typical groupoid is the undamental groupoid π(x) o a topological space X. (objects=points, arrow x x =homotopy classes o paths rom x to x ) connected groupoid: there is an arrow joining any two o its objects. (It is determined up to isomorphism by a group.) 1.6 Foundations 1.7 Large Categories Ab-categories? (hom-sets are abelian groups) T 1

2 2 Construction on categories 2.1 Duality 2.2 Contravariance and Opposites 2.3 Products o categories Functors S : B C D rom a product category are called biunctors. Proposition Let B, C and D be categories. For all objects c C and b B, let L c : B D, M b : C D be unctors such that M b (c) = L c (b) or all b and c. Then there exists a biunctor S : B C D with S(, c) = L c or all c and S(b, ) = M b or all b i and only i or every pair o arrows : b b and g : c c one has M b g L c = L c M b g. These equal arrow in D are then the value S(, g) o the arrow unction o S. Proposition For biunctors S, S, the unction α which assigns to each pair o objects b B, c C an arrow α(b, c): S(b, c) S (b, c) is a natural transormation α : S S (i.e., o biunctors) i and only i α(b, c) is natural in b or each c C and natural in c or each b B. Such natural transormations appear in the undamental deinition o adjoint unctor (Chapter IV.) A unctor F : X C is the let adjoint o a unctor G: C X (opposite direction) when there is a bijection natural in x X and c C. 2.4 Functor categories hom C (F x, c) = hom x (x, Gc) B C F unct(c, B) with objects the unctors T : C B and morphism the natural transormations. Nat(S, T ) := B C (S, T ) = {τ τ : S T natural}. 2.5 The Category o All Categories We have deined a vertical composite τ σ, C σ τ B Given unctors and natural transormations, C S τ T B S τ T A (2.1) 2

3 one may construct natural transormation τ τ as (τ τ)c = T τc τ Sc = τ T C S τc. (2.2) This composition is readily shown to be associative. It moreover has identities. It is convenient to let the symbols S or a unctor also denote the identity transormation 1 S : S S. The deinition (2.2) can then be rewritten using also the vertical composition, as τ τ = (T τ) (τ S) = (τ T ) (S τ) (2.3) There is a more general rule. Given three categories and our transormations C σ τ B σ τ A (2.4) the vertical composites under and the horizontal composites under are related by the identity (interchange law) (τ σ ) (τ σ) = (τ τ) (σ σ) (2.5) Exercise 4: Let G be a topological group with identity element e, while σ, σ, τ, τ are continuous paths in G starting and ending at e. - skladanie ciest, pointwise product. Then interchange law holds. Exercise 5: (Hilton=Eckmann). Let S be a set with two (everywhere deined) binary operations : S S S, : S S S which both have the same (two-sided) unit element e and which satisy the interchange identity (2.5). Prove that and are equal, and that each is commutative. Exercise 6: Combine Exercises 5 and 6 to prove that the undamental group o a topological group is abelian. 2.6 Comma categories Category o objects under b is the category (b C) with objects all pairs, c, where c is an object o C and : b c is and arrow o c. Arrows are arrows o C (resp. commutative triangles). I a is an object o C, the category (C a) o objects over a has objects : c a. I b is an object o C and S : D C a unctor, the category (b S) o objects S-under b has an objects all pairs, d with : b Sd. (T a) o objects T -over a. Here is the general construction. Given categories and unctors E T C S D the comma category (T S), also written (T, S) has objects all triples e, d,, with d ObjD, e ObjE, and : T e Sd and as arrows e, d, e, d, all pairs k, h o arrows k : e e, h: d d such that T k = Sh. In pictures Objects e, d, T e ; arrows k, h T e T k T e (2.6) Sd Sd Sh Sd with the square commutative. The composite k, h k, h = k k, h h, when deined. 3

4 2.7 Graphs and Free Categories skipped 2.8 Quotient Categories skipped 3 Universals and limits Deinition I S : D C is a unctor and c an object o C, a universal arrow rom c to S is a pair r, u consisting o an object r od D and an arrow u: c Sr o C, such that to every pair d, with d an object o D and : c Sd an arrow o C, there is a unique arrow : r d o D with S u =. In other words, every arrow to S actors uniquely through the universal arrow u, as in the commutative diagram u c Sr Sd S (3.1) Examples: Bases o vector spaces, ree categories orm graphs, ields o quotients, completion o metric space (universal or orgetul unctor rom complete metric spaces). I D is a category and H : D Set unctor, a universal element o the unctor H is a pair r, e consisting o an object r D and an element e Hr such that or every pair d, x with x Hd there is a unique arrow : r d o D with (H)e = x. Diagonal unctor: : C C C, c = c, c. 3.1 The Yoneda Lemma Proposition For a unctor S : D C a pair ru: c Sr is universal rom orm c to S i and only i the unction sending each : r d into S u: c Sd is a bijection o hom-sets D(r, d) = C(c, Sd) (3.2) This bijection is natural in d. Conversely, given r and c, any natural isomorphism (3.2) is determined in this way by a unique arrow u: c Sr such that r, u is universal rom C to s. Deinition Let D have small hom-sets. A representation o a unctor K : D Set is a pair r, ψ with r an object o D and ψ : D(r, ) = K (3.3) a natural isomorphism. The object r is called the representing object. The unctor K is said to be representable when such a representation exists. Proposition Let denote any one-point set and let D have small hom-sets. I r, u: Kr is a universal arrow rom to k : D Set, then the unction ψ which or each object d o D send the arrow : r d to K( )(u ) Kd is a representation o K. Every representation o K is obtained in this way rom exactly one such universal arrow. 4

5 The argument or Proposition rested on the observation that each natural transormation ϕ : D(r, ) K is completely determined by the image under ϕ r o the identity 1: r r. This act may be stated as ollows: Lemma (Yoneda). I k : D Set is a unctor rom D and r an object in D (or D a category with small hom-sets), there is a bijection y : Nat(D(r, ), K) = Kr (3.4) which sends each natural transormation α : D(r, ) K to α r 1 r, the image o the identity r r. Corollary For objects r, s D each natural transormation D(r, ) D(s, ) has the orm D(h, ) or a unique arrow h: s r. Lemma The bijection o 3.4 is a natural isomorphism y : N E between the unctors E, N : Set D D Set. 3.2 Coproducts and Colimits Cokernels. Suppose that C has zero object z, so that or any two objects b, c C there is a zero arrow 0: b z c. The cokernel o : a b is then an arrow u: b e such that (i) u = 0: ae; (ii) i h: b c has h = 0, then h = h u or a unique arrow h : e c. The picture is 3.3 Products and limits a b 3.4 Categories with inite products u h e u = 0, h c h = 0. Proposition I a category C has a terminal object t and a product diagram a a b b or any two o its objects, then C has all inite products. The product object provide, by a, b a b, a biunctor C C C. For any three objects there is an isomorphism α = α a,b,c : a (b c) = (a b) c (3.5) natural in a, b and c. For any object a there are isomorphisms λ = λ a : t a = a ϱ = ϱ a : a t = a (3.6) which are natural in a, where t is the terminal object o C. 3.5 Groups in categories Let C be a category with inite products products and a terminal object t. Then a monoid in C is a triple c, µ: c c c, η : t c, such that the ollowing diagrams commute c (c c) α (c c) c η 1 c c (3.7) 1 µ c c µ c µ 5

6 η 1 t c c c λ µ 1 η ϱ c c t (3.8) µ=multiplication, α=asocitivity isomorphism o (3.5). We now deine a group in C to be a monoid c, µ, η together with an arrow ξ : c c which makes the diagram (with δ c the diagonal) c δ c c c 1 η c c (3.9) t η c µ commute (ξ=right inverse). Proposition I C is a category with inite products, then an object c is a group (or, a monoid) in C i and only i the hom unctor C(, c) is a qroup (respectively, a monoid) in the unctor category Set Cop. 4 Adjoints 4.1 Adjunctions Deinition Let A and X be categories. F, G, ϕ : XA, where F and G are unctors An adjunction rom X to A is a triple F G F G while ϕ is a unction which assigns to each pair o objects x X, a A a bijection which is natural in x and a. ϕ = ϕ x,a : A(F x, a) = X(x, Ga) (4.1) An adjunction may also be described without hom-sets directly in terms o arrows. It is a bijection which assigns to each arrow : F x a an arrow ϕ = rad : x Ga, the right adjunct o, in such a way that the naturality conditions ϕ( F h) = ϕ h, ϕ(k ) = Gk ϕ, (4.2) hold or all and all arrows h: x x and k : a a. It is equivalent to require that ϕ 1 be natural; i.e., that or every h, k and g : x Ga one has ϕ 1 (gh) = ϕ 1 g F h, ϕ 1 (Gk g) = k ϕ 1 g. (4.3) Given such an adjunctions, the unctor F is said to be a let-adjoint or G, while G is called a right adjoint or F. (TODO Find TEXequivalent!!!) Every adjunction yields a universal arrow. Speciically, set a = F x in (4.1). The let hand hom-set o (4.1) then contains the identity 1: F x F x; call its ϕ-image η x. By Yoneda s 6

7 Proposition this η x is a universal arrow η x or every objects x. Moreover, the unction x η x is a natural transormation I X F G. The bijection ϕ can be expressed in terms o the arrow η x as ϕ() = G()η x or : F x a. (4.4) Theorem An adjunction F, G, ϕ : X A determines (i) A natural transormation η : I x GF such that or each object x the arrow η x is universal to G rom x, while the right adjunct o each : F x a is ϕ = G η x : x Ga; (4.5) (ii) A natural transormation ε: F G I A such that arrow ε a is universal to a rom F, while each g : x Ga has let adjunct ϕ 1 g = ε a F g : F x a. (4.6) Moreover, both the ollowing composites are the identities (o G, resp. F ). G ηg GF G Gε G, F We call η unit and ε the counit o the adjunction. F η F GF εf F. (4.7) Theorem Each adjunction F, G, ϕ : X A is completely determined by the items in any one o the ollowing lists: (i) Functors, G and a natural transormation η : 1 x GF such that each η x : x GF x is universal rom G to x. The ϕ id deined by (4.5). (ii) The unctor G: A X and or each x X an object F 0 x A and a universal arrow η x : x GF 0 x rom x to G. Then the unctor F has object unction F 0 and is deined on arrow h: x x by GF h η x = η x h. (iii) Functors F, G and a natural transormation ε : F G I A such that each ε a : F Ga a is universal rom F to a. Here ϕ 1 is deined by (4.6). (iv) The unctor F : X A and or each a A an object G 0 a X and an arrow ε a : F G 0 a a universal rom F to a. (v) Functors F, G and natural transormations η : I X GF and ε : GF I A such that both composites (4.7) are the identity transormations. Here ϕ is deined by (4.5) and ϕ 1 by (4.6). Corollary Any two let-adjoints F and F o a unctor G: A X are naturally isomorphic. Corollary A unctor G: A X has a let adjoint i and only i, or each x X, the unctor X(x, Ga) is representable as a unctor o a A. I ϕ: A(F 0 x, a) = X(x, Ga) is representation o this unctor, then F 0 is the object unction o a let-adjoint o G or which the bijection ϕ is natural in a and gives the adjunction. Theorem I the additive unctor G: A M between Ab-categories A and M has a let adjoint F : M A, then F is additive and the adjunction bijections ϕ: A(F m, a) = M(m, Ga) are isomorphisms o abelian groups (or all m M, a A). 7

8 4.2 Examples o Adjoints 4.3 Relective Subcategories Theorem For an adjunction F, G, η, ε : X A: (i) G is aithul i and only i every component ε a o the counit ε is epi, (ii) G is ull i and only i every ε a is a split monic. Hence G is ull and aithul i and only i each ε a is an isomorphism F Ga = a. Lemma Let : A(a, ) A(b, ) be the natural transormation induced by an arrow : b a o A. Then is monic i and only i is epi, while is epi i and only i is a split monic (i.e., i and only i has a let inverse). A subcategory A o B is called relective in B when the inclusion unctor K : A B has a let adjoint F : B A. This unctor F may be called a relector and the adjunction F, K, ϕ = F, ϕ : B A a relection o B in its subcategory A. 4.4 Equivalence o Categories skipped 4.5 Adjoints or Preorders skipped 4.6 Cartesian Closed Categories To assert that a category C has all inite products and coproducts is to assert that the unctors C 1 and : C C C have both let and right adjoints. Indeed, the let adjoints give initial object and coproduct respectively, while the right adjoints give terminal object and product, respectively. A category C with all inite products speciically given is called cartesian closed when each o the ollowing unctors C 1, C C C, C b C, c 0, c c, c, a a b has a speciied right adjoint (with a speciied adjunction). This adjoints are written as ollows t 0, a b a, b, c b c TODO oprav to na otocena mapsto The third required adjoint speciies or each unctor b: C C a right adjoint, with the corresponding bijection hom(a b, c) = hom(a, c b ) natural in a and c. By the parameter theorem (to be proved in the next section), b, c c b is then (the object unction o) a biunctor C op C C. Speciying the adjunction amounts to speciying or each c and b an arrow e e: c b b c 8

9 which is natural in c and universal orm b to c. We call e = e b the evaluation map. Exercise 1: a) I U is any set, show that the preorder P(U) o all subsets o U is a cartesian closed category. b) Show that any Boolean algebra, regarded as a preorder, is cartesian closed. Exercise 3: In any cartesian closed category, prove c t = c and c b b = (c b ) b. Exercise 4: In any cartesian closed category obtain a natural transormation c b b a c a which agrees in Set with composition o unctions. Prove it (like composition) associative. Exercise 5: Show that A cartesian closed need not imply A J cartesian closed. 9