Functors and natural transformations functors ; category morphisms natural transformations ; functor morphisms Andrzej Tarlecki: Category Theory, 2018 - 90 - Functors A functor F: K ! K0 from a category K to a category K0 consists of: • a function F: jKj ! jK0j, and • for all A; B 2 jKj, a function F: K(A; B) ! K0(F(A); F(B)) such that: Make explicit categories in which we work at various places here • F preserves identities, i.e., F(idA) = idF(A) for all A 2 jKj, and • F preserves composition, i.e., F(f;g) = F(f);F(g) for all f : A ! B and g : B ! C in K. We really should differentiate between various components of F Andrzej Tarlecki: Category Theory, 2018 - 91 - Examples • identity functors: IdK : K ! K, for any category K 0 0 • inclusions: IK,!K0 : K ! K , for any subcategory K of K 0 0 0 • constant functors: CA : K ! K , for any categories K; K and A 2 jK j, with CA(f) = idA for all morphisms f in K • powerset functor: P: Set ! Set given by − P(X) = fY j Y ⊆ Xg, for all X 2 jSetj − P(f): P(X) ! P(X0) for all f : X ! X0 in Set, P(f)(Y ) = ff(y) j y 2 Y g for all Y ⊆ X op • contravariant powerset functor: P−1 : Set ! Set given by − P−1(X) = fY j Y ⊆ Xg, for all X 2 jSetj 0 0 − P−1(f): P(X ) ! P(X) for all f : X ! X in Set, 0 0 0 0 P−1(f)(Y ) = fx 2 X j f(x) 2 Y g for all Y ⊆ X Andrzej Tarlecki: Category Theory, 2018 - 92 - Examples, cont'd. 0 0 0 • projection functors: π1 : K × K ! K, π2 : K × K ! K • list functor: List: Set ! Monoid, where Monoid is the category of monoids (as objects) with monoid homomorphisms as morphisms: ∗ ∗ − List(X) = hX ; b; i, for all X 2 jSetj, where X is the set of all finite lists of elements from X, b is the list concatenation, and is the empty list. − List(f): List(X) ! List(X0) for f : X ! X0 in Set, List(f)(hx1; : : : ; xni) = hf(x1); : : : ; f(xn)i for all x1; : : : ; xn 2 X • totalisation functor: Tot: Pfn ! Set∗, where Set∗ is the subcategory of Set of sets with a distinguished element ∗ and ∗-preserving functions − Tot(X) = X ] f∗g Define Set∗ as the category of algebras 8 < f(x) if it is defined − Tot(f)(x) = : ∗ otherwise Andrzej Tarlecki: Category Theory, 2018 - 93 - Examples, cont'd. • carrier set functors: j j: Alg(Σ) ! SetS, for any algebraic signature Σ = hS; Ωi, yielding the algebra carriers and homomorphisms as functions between them 0 • reduct functors: σ : Alg(Σ ) ! Alg(Σ), for any signature morphism σ :Σ ! Σ0, as defined earlier • term algebra functors: TΣ : Set ! Alg(Σ) for all (single-sorted) algebraic signatures Σ 2 jAlgSigj Generalise to many-sorted signatures − TΣ(X) = TΣ(X) for all X 2 jSetj # 0 0 − TΣ(f) = f : TΣ(X) ! TΣ(X ) for all functions f : X ! X G G • diagonal functors: ∆K : K ! DiagK for any graph G with nodes N = jGjnodes and edges E = jGjedges , and category K G A A A − ∆K(A) = D , where D is the \constant" diagram, with Dn = A for all A n 2 N and De = idA for all e 2 E G f A B f − ∆K(f) = µ : D ! D , for all f : A ! B, where µn = f for all n 2 N Andrzej Tarlecki: Category Theory, 2018 - 94 - Hom-functors Given a locally small category K, define op HomK : K × K ! Set a binary hom-functor, contravariant on the first argument and covariant on the second argument, as follows: op • HomK(hA; Bi) = K(A; B), for all hA; Bi 2 jK × Kj, i.e., A; B 2 jKj 0 0 0 0 • HomK(hf; gi): K(A; B) ! K(A ;B ), for hf; gi: hA; Bi ! hA ;B i in Kop × K, i.e., f : A0 ! A and g : B ! B0 in K, as a function given by Hom (hf; gi)(h) = f;h;g. f K A 0 A K(A; B) h K(A0;B0) ? g ? B -B0 Also: HomK(A; ): K ! Set op HomK(f; g) 6 HomK( ;B): K ! Set Andrzej Tarlecki: Category Theory, 2018 - 95 - Functors preserve. • Check whether functors preserve: − monomorphisms − epimorphisms − (co)retractions − isomorphisms − (co)cones − (co)limits − ... • A functor is (finitely) continuous if it preserves all existing (finite) limits. Which of the above functors are (finitely) continuous? Dualise! Andrzej Tarlecki: Category Theory, 2018 - 96 - Functors compose. Given two functors F: K ! K0 and G: K0 ! K00, their composition F;G: K ! K00 is defined as expected: • (F;G)(A) = G(F(A)) for all A 2 jKj • (F;G)(f) = G(F(f)) for all f : A ! B in K Cat, the category of (sm)all categories − objects: (sm)all categories − morphisms: functors between them − composition: as above Characterise isomorphisms in Cat Define products, terminal objects, equalisers and pullback in Cat Try to define their duals Andrzej Tarlecki: Category Theory, 2018 - 97 - Comma categories Given two functors with a common target, F: K1 ! K and G: K2 ! K, define their comma category (F; G) − objects: triples hA1; f : F(A1) ! G(A2);A2i, where A1 2 jK1j, A2 2 jK2j, and f : F(A1) ! G(A2) in K − morphisms: a morphism in (F; G) is any pair hh1; h2i: hA1; f : F(A1) ! G(A2);A2i ! hB1; g : F(B1) ! G(B2);B2i, where h1 : A1 ! B1 in K1, h2 : A2 ! B2 in K2, and F(h1);g = f;G(h2) in K. K1: K: K2: f - − composition: component-wise A1 F(A1) G(A2) A2 h1 F(h1) G(h2) h2 ? ? g ? ? B1 F(B1) - G(B2) B2 Andrzej Tarlecki: Category Theory, 2018 - 98 - Examples • The category of graphs as a comma category: Graph = (IdSet; CP) where CP: Set ! Set is the (Cartesian) product functor (CP(X) = X × X and CP(f)(hx; x0i) = hf(x); f(x0)i). Hint: write objects of this category as hE; hsource; targeti: E ! N × N;Ni • The category of algebraic signatures as a comma category: + AlgSig = (IdSet; ( ) ) where ( )+ : Set ! Set is the non-empty list functor ((X)+ is the set of all + non-empty lists of elements from X, (f) (hx1; : : : ; xni) = hf(x1); : : : ; f(xn)i). Hint: write objects of this category as hΩ; harity; sorti:Ω ! S+;Si Define K!, K#A as comma categories. The same for Alg(Σ). Andrzej Tarlecki: Category Theory, 2018 - 99 - Cocompleteness of comma categories Fact: If K1 and K2 are (finitely) cocomplete categories, F: K1 ! K is a (finitely) cocontinuous functor, and G: K2 ! K is a functor then the comma category (F; G) is (finitely) cocomplete. Proof (idea): Construct coproducts and coequalisers in (F; G), using the corresponding constructions in K1 and K2, and cocontinuity of F. State and prove the dual fact, concerning completeness of comma categories Andrzej Tarlecki: Category Theory, 2018 - 100 - Coproducts: f A1 F(A1) - G(A2) A2 ιA1 ¡ A ¡ A ιA2 F(ιA1 ) G(ιA2 ) ¡ ¡ AAU ¡ ¡ AAU A1 + B1 F(A1 + B1) - G(A2 + B2) A2 + B2 AK ¡ AK ¡ A F(ιB )¡ AG(ιB ) ¡ ιB1 1 2 ιB2 A ¡ g A ¡ B1 F(B1) - G(B2) B2 Coequalisers: f A1 F(A1) - G(A2) A2 0 0 0 0 h1 h1 F(h1) F(h1) G(h2) G(h2) h2 h2 ?? ?? g ?? ?? B1 F(B1) - G(B2) B2 c1 F(c1) G(c2) c2 ? ? ? ? C1 F(C1) - G(C2) C2 Andrzej Tarlecki: Category Theory, 2018 - 101 - Indexed categories An indexed category is a functor C : Indop ! Cat . Standard example: Alg: AlgSigop ! Cat The Grothendieck construction: Given C : Indop ! Cat, define a category Flat(C): − objects: hi; Ai for all i 2 jIndj, A 2 jC(i)j − morphisms: a morphism from hi; Ai to hj; Bi, hσ; fi: hi; Ai ! hj; Bi, consists of a morphism σ : i ! j in Ind and a morphism f : A !C(σ)(B) in C(i) − composition: given hσ; fi: hi; Ai ! hi0;A0i and hσ0; f 0i: hi0;A0i ! hi00;A00i, their composition in Flat(C), hσ; fi;hσ0; f 0i: hi; Ai ! hi00;A00i, is given by hσ; fi;hσ0; f 0i = hσ;σ0; f;C(σ)(f 0)i Fact: If Ind is complete, C(i) are complete for all i 2 jIndj, and C(σ) are continuous for all σ : i ! j in Ind, then Flat(C) is complete. Try to formulate and prove a theorem concerning cocompleteness of Flat(C) Andrzej Tarlecki: Category Theory, 2018 - 102 - Natural transformations Given two parallel functors F; G: K ! K0, a natural transformation from F to G τ : F ! G 0 is a family τ = hτA : F(A) ! G(A)iA2jKj of K -morphisms such that for all f : A ! B in K (with A; B 2 jKj), τA;G(f) = F(f);τB K: K0: τ A F(A) A - G(A) f F(f) G(f) Then, τ is a natural isomorphism if for ? ? ? τB all A 2 jKj, τA is an isomorphism. B F(B) - G(B) Andrzej Tarlecki: Category Theory, 2018 - 103 - Examples 0 • identity transformations: idF : F ! F, where F: K ! K , for all objects A 2 jKj, (idF)A = idA : F(A) ! F(A) • singleton functions: sing : IdSet ! P (: Set ! Set), where for all X 2 jSetj, singX : X ! P(X) is a function defined by singX (x) = fxg for x 2 X List • singleton-list functions: sing : IdSet ! jListj (: Set ! Set), where jListj = List;j j: Set(! Monoid) ! Set, and for all X 2 jSetj, List ∗ List singX : X ! X is a function defined by singX (x) = hxi for x 2 X • append functions: append : jListj;CP ! jListj (: Set ! Set), where for all ∗ ∗ ∗ X 2 jSetj, append X :(X × X ) ! X is the usual append function (list concatenation) polymorphic functions between algebraic types Andrzej Tarlecki: Category Theory, 2018 - 104 - Polymorphic functions Work out the following generalisation of the last two examples: − for each algebraic type scheme 8α1 : : : αn · T , built in Standard ML using at least products and algebraic data types (no function types though), define the corresponding functor [[T ]]: Setn ! Set − argue that in a representative subset of Standard ML, for each polymorphic 0 expression E : 8α1 : : : αn · T ! T its semantics is a natural transformation [[E]]: [[T ]] ! [[T 0]] Theorems for free! (see Wadler 89) Andrzej Tarlecki: Category Theory, 2018 - 105 - Yoneda lemma Given a locally small category K, functor F: K ! Set and object A 2 jKj: ∼ Nat(HomK(A; ); F) = F(A) natural transformations from HomK(A; ) to F, between functors from K to Set, are given exactly by the elements of the set F(A) EXERCISES: • Dualise: for G: Kop ! Set, ∼ Nat(HomK( ;A); G) = G(A) .