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Home , Equivariant map, Projection (mathematics), Representable functor, Universal property, Yoneda lemma

theory Supplemental notes 1

Justin Campbell February 10, 2017

1 Universal properties

Informally speaking, a of a mathematical object is a uniform description of 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 and asking if it is representable. It was understood before the development of that a universal property characterizes an object with that property, if it exists, uniquely unique . 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 → is called representable if there exists an object X of C and an isomorphism of η : hX →˜ F . Yoneda’s lemma provides a

HomFun(C op,Set)(hX ,F )−→ ˜ F (X), and the element u ∈ 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 , the universal element u ∈ F (X) determines the isomorphism of functors η : hX → F . This is important because a priori a 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 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 ) ∈ 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

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 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 , so there is an isomorphism of functors η : hX →˜ F for some object X in C , with universal element u := ηX (idX ) ∈ F (X). For any object Y in C and element s ∈ 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 ∈ F (X) and u0 ∈ 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 and N ⊂ G a normal . Then the quotient group G/N, which has a canonical surjective π : G → G/N, is characterized by the following universal property. For any group H and any homomorphism ϕ : G → H with the property that the restriction ϕ|N is the trivial homomorphism N → H, there exists a homomorphism ϕe : G/N → H uniquely characterized by the property that ϕe ◦ π = ϕ. 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) := {ϕ : G → H | ϕ|N is trivial}. 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 ∈ F (G/N). Indeed, the above proposition (applied with C = Grpop) says that for any group H and ϕ ∈ F (H), i.e. homomorphism ϕ : G → H such that ϕ|N is trivial, there exists a unique morphism ϕe : G/N → H such that F (ϕe)(π) = ϕ. But F (ϕe)(π) = π ◦ ϕe, 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 ϕ : G/N→˜ (G/N)0 such that ϕ ◦ 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 ) ∈ 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 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 that it sends 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 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. In particular, if S or T is nonempty, then the inclusion is not a bijection because S ×S or T ×T is nonempty. Thus F does not preserve products, so by Proposition 2.2 it is not corepresentable.

3 3 Cayley’s theorem and the Yoneda lemma

Let G be a group. A G-set is a set S together with a homomorphism α : G → Sym(S), where Sym(S) := AutSet(S) is the group of bijections S→˜ S (also known as of S). In other words, any element g ∈ G determines a bijection α(g): S→˜ S, and α(gh) = α(g) ◦ α(h) (one also says that G acts on S, or that S is a representation of G). A G- to another G-set T is a map of sets f : S → T such that f(α(g)(s)) = β(g)(f(s)), where β : G → Sym(T ) is the given homomorphism. It is not difficult to check that the composition of two G-equivariant maps is G-equivariant (as is the identity map), so G-sets with G-equivariant maps form a category G-Set. Note that G acts on itself via the formula ρG(g)(h) := gh, which is a homomorphism ρG : G → Sym(G) rev by associativity. There is also an action of G , the group with reversed operation, on G by ρG(g)(h) := hg. rev Proposition 3.1 (Cayley). The homomorphism ρG is injective, and the is the subgroup of G - equivariant maps with respect to the action ρG. Similarly, ρG is injective with image the subgroup of G- equivariant maps.

Proof. First we prove injectivity of λG. If g ∈ G lies in the of ρG then gh = h for any h ∈ G. In particular g = g1 = 1. rev One checks that associativity of the operation on G implies that λG lands in G -equivariant maps. Now suppose that f : G → G is a Grev-equivariant map. This means that for any g, h ∈ G, we have

f(hg) = f(ρG(g)(h)) = ρG(g)(f(h)) = f(h)g.

In particular f(g) = f(1)g, meaning f = ρG(f(1)) as desired. One obtains the second statement by replacing G with Grev everywhere (note that (Grev)rev = G).

Suppose we are given a functor F : BG → Set. This determines a set F (•G) and a homomorphism of monoids G → EndSet(F (•G)), whose image is contained in the group Sym(F (•G)) because G is a group and monoid preserve invertibility. Moreover, if F 0 : BG → Set is another functor and we are 0 0 given a morphism of functors η : F → F , we obtain a map η•G : F (•G) → F (•G). For any g ∈ G the square

η •G 0 F (•G) F (•G)

F (g) F 0(g) η •G 0 F (•G) F (•G)

commutes, which precisely says that η•G is G-equivariant. Proposition 3.2. The functor Fun(BG, Set) → G-Set constructed above is an equivalence.

Proof. Given a G-set S with action α : G → Sym(S), define F : BG → Set by F (•G) := S and F (g) := α(g). It is not hard to check that this defines the inverse functor G-Set → Fun(BG, Set).

Now let’s explain why Cayley’s theorem is a special case of the Yoneda lemma. First observe that (BG)op = B(Grev) (really there is a canonical equivalence of categories between the two). The Yoneda lemma applied to BG implies that the functor y : BG → Fun(B(Grev), Set) is fully faithful. By definition

y(•G)(•Grev ) = EndBG(•G) = G,

rev so in terms of the equivalence in Proposition 3.2, y sends •G to the set G with the G -action ρG. Moreover, the functor y determines a homomorphism

G = AutBG(•G) −→ AutFun(B(Grev),Set)(y(•G)) = AutGrev-Set(G), and one checks from the definition of y that the composition of this homomorphism with the inclusion AutGrev-Set(G) ⊂ Sym(G) is just λG. Thus the Yoneda lemma says that λG is an isomorphism onto AutGrev-Set(G), which is Cayley’s theorem.

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