Category theory Supplemental notes 1 Justin Campbell February 10, 2017 1 Universal properties Informally speaking, a universal property of a mathematical object is a uniform description of morphisms either into or out of the object. One often writes down a universal property and asks if an object with that property exists, which is equivalent to writing down a functor and asking if it is representable. It was understood before the development of category theory that a universal property characterizes an object with that property, if it exists, uniquely up to unique isomorphism. Yoneda's lemma can be understood as the most general form of this assertion. Let C be any category. By way of reminder, a functor F : C op ! Set is called representable if there exists an object X of C and an isomorphism of functors η : hX !~ F . Yoneda's lemma provides a bijection HomFun(C op;Set)(hX ;F )−! ~ F (X); and the element u 2 F (X) corresponding to η is called the universal element. More explicitly, we have u = ηX (idX ) (this is the construction from the proof of Yoneda's lemma). By the Yoneda lemma, the universal element u 2 F (X) determines the isomorphism of functors η : hX ! F . This is important because a priori a morphism of functors contains a great deal of data, but in this case it can be specified by the single element u. Example 1.1. Consider the Cartesian product S × T of two sets S and T , with pS : S × T ! S and pT : S × T ! T being the projection maps defined by pS(s; t) = s and pT (s; t) = t. The universal property of S × T says that maps U ! S × T for any set U correspond bijectively via f 7! (pS ◦ f; pT ◦ f) to pairs of maps U ! S and U ! T . More concisely, S × T represents the functor F : Setop ! Set defined on objects by the formula F (U) := HomSet(U; S) × HomSet(U; T ); with the universal element being the pair (pS; pT ) 2 HomSet(S × T;S) × HomSet(S × T;T ) = F (S × T ) consisting of the two projection maps. We leave the action of F on maps for the reader to discover. More precisely, the isomorphism of functors η : hS×T !~ F consists of bijections ηU : HomSet(U; S × T ) = hS×T (U)−! ~ F (U) = HomSet(U; S) × HomSet(U; T ) for all sets U, which are given by the formula ηU (f) := (pS ◦ f; pT ◦ f): To see that these are bijections, define the inverse as follows: to any pair f : U ! S, g : U ! T assign the map U ! S × T which sends x 7! (f(x); g(x)). To find the universal element, we take U = S × T (the representing object), and observe that ηS×T (idS×T ) = (pS ◦ idS×T ; pT ◦ idS×T ) = (pS; pT ): 1 Now we state and prove \the universal property of a universal element," which includes all classical universal properties as special cases. Proposition 1.2. Let F : C op ! Set be a representable functor, so there is an isomorphism of functors η : hX !~ F for some object X in C , with universal element u := ηX (idX ) 2 F (X). For any object Y in C and element s 2 F (Y ), there exists a unique morphism f : Y ! X in C with the property that F (f): F (X) ! F (Y ) sends u to s. −1 Proof. The desired morphism is f := ηY (s), i.e. f corresponds to s under the bijection ηY : hX (Y )! ~ F (Y ). Since η is a morphism of functors hX ! F , our f : Y ! X gives rise to a commutative square ηX hX (X) F (X) hX (f) F (f) ηY hX (Y ) F (Y ): Therefore we have as needed F (f)(u) = F (f)(ηX (idX )) = ηY (hX (f)(idX )) = ηY (idX ◦f) = ηY (f) = s: As for the uniqueness of f, suppose f 0 : Y ! X is any morphism with the property that F (f)(u) = s. 0 From this equation we obtain, using the square above, that ηY (f ) = s. Thus 0 −1 0 −1 f = ηY (ηY (f )) = ηY (s) = f: From this we can deduce the general uniqueness result. Corollary 1.2.1. If F is represented by X and X0 with universal elements u 2 F (X) and u0 2 F (X0), there exists a unique isomorphism α : X!~ X0 such that F (α)(u0) = u. Proof. Proposition 1.2 tells us that there exist unique morphisms α : X ! X0 and β : X0 ! X such that F (α)(u0) = u and F (β)(u) = u0. Note that β ◦ α has the property that F (β ◦ α)(u) = F (α)(F (β)(u)) = F (β)(u0) = u; as does idX , but the proposition tells us that there is only one endomorphism X ! X such that the induced map F (X) ! F (X) sends u 7! u. Therefore β ◦ α = idX . On proves similarly that α ◦ β = idX0 , so α is an isomorphism as needed. Example 1.3. Let G be a group and N ⊂ G a normal subgroup. Then the quotient group G=N, which has a canonical surjective homomorphism π : G ! G=N, is characterized by the following universal property. For any group H and any homomorphism ' : G ! H with the property that the restriction 'jN is the trivial homomorphism N ! H, there exists a homomorphism 'e : G=N ! H uniquely characterized by the property that 'e ◦ π = '. Let's say this in the language of representable functors. Having fixed G and N, let F : Grp ! Set be the functor defined on objects by F (H) := f' : G ! H j 'jN is trivialg: We leave it to the reader to define the action of F on morphisms. The universal property of π : G ! G=N is encapsulated by the fact that F is corepresented by G=N, with universal element π 2 F (G=N). Indeed, the above proposition (applied with C = Grpop) says that for any group H and ' 2 F (H), i.e. homomorphism ' : G ! H such that 'jN is trivial, there exists a unique morphism 'e : G=N ! H such that F ('e)(π) = '. But F ('e)(π) = π ◦ 'e, so this is the universal property stated above. We leave it to the reader as a worthwhile exercise to deduce the following uniqueness assertion from the corollary: for any group (G=N)0 and homomorphism π0 : G ! (G=N)0 with the above universal property, there exists a unique isomorphism ' : G=N!~ (G=N)0 such that ' ◦ π = π0. 2 2 Products and representable functors Let's generalize the discussion of Cartesian product of sets in Example 1.1 to products in an arbitrary category C . Definition 2.1. For any objects X and Y in C , their product, if it exists, is an object X × Y equipped with maps pX : X × Y ! X and pY : X × Y ! Y , satisfying the following universal property: for any object Z in C and any morphisms f : Z ! X and g : Z ! Y , there exists a unique morphism (f; g): Z ! X × Y satisfying pX ◦ (f; g) = f and pY ◦ (f; g) = g. This can be restated succinctly by saying that X × Y represents the functor F : C op ! Set given on objects by F (Z) := hX (Z) × hY (Z) = HomC (Z; X) × HomC (Z; Y ) and morphisms by F (f) := hX (f) × hY (f), and that (pX ; pY ) 2 F (X × Y ) is a universal element (i.e. the corresponding morphism of functors hX×Y ! F is an isomorphism). By Corollary 1.2.1, this characterizes the product uniquely up to unique isomorphism, and in particular the notation X × Y is unambiguous. Now let F : C ! D be an arbitrary functor. Note that for any objects X and Y in C whose product exists, there is a canonical morphism (F (pX );F (pY )) : F (X × Y ) −! F (X) × F (Y ) in D. We say that F preserves products if (F (pX );F (pY )) is an isomorphism whenever X × Y exists. If X and Y are objects of C and their product in C op exists, we call it the coproduct X t Y in C . Proposition 2.2. A (co)representable functor preserves products. Proof. Suppose F : C ! Set is corepresented by an object X and η : hX !~ F . By replacing C with C op this proof applies to representable functors as well. Notice that hX preserves products: for any objects Y and Z whose product exists, the map X X X X X (h (pY ); h (pZ )) : h (Y × Z) = HomC (X; Y × Z) −! HomC (X; Y ) × HomC (X; Z) = h (Y ) × h (Z) is bijective by the definition of the product. We have a commutative square η hX (Y × Z) Y ×Z F (Y × Z) (hX (pY );hX (pZ )) (F (pY );F (pZ )) η ×η hX (Y ) × hX (Z) Y Z F (Y ) × F (Z); and the right vertical map is bijective because the other three maps are bijective. In the case of a representable functor F : C op ! Set, preservation of products means that it sends coproducts in C to products of sets. Example 2.3. Let's use Proposition 2.2 to prove that a functor is not corepresentable. Namely, consider F : Set ! Set defined on objects as F (S) := S t S (here t is disjoint union of sets) and morphismss by F (f) := f t f. Observe that for any two sets S and T , we have F (S) × F (T ) = (S t S) × (T t T ) =∼ (S × S) t (S × T ) t (S × T ) t (T × T ): On other hand, F (S × T ) = (S × T ) t (S × T ); and (F (pS);F (pT )) is the inclusion of the latter set into the former.