Functors, Homotopy Sets and Groups

Functors
Definition:
A functor F from a category \mathscr{C} to a category \mathscr{D} is a function which
– For each object X\in\mathscr{C}, we have an object F(X)\in\mathscr{D}.
– For each f\in\hom_\mathscr{C}(X,Y), we have a morphism \displaystyle F(f)\in\hom_\mathscr{D}(F(X),F(Y)).

Furthermore, F is required to satisfy the two axioms:
– For each object X\in\mathscr{C}, we have F(1_X)=1_{F(X)}. That is, F maps the identity morphism on X to the identity morphism on F(X).

– For f\in\hom_{\mathscr{C}}(X,Y), g\in\hom_\mathscr{C}(Y,Z) we have \displaystyle F(g\circ f)=F(g)\circ F(f)\in\hom_\mathscr{D}(F(X),F(Z)). That is, functors must preserve composition of morphisms.

Definition:
A cofunctor (also called contravariant functor) F^* from a category \mathscr{C} to a category \mathscr{D} is a function which
– For each object X\in\mathscr{C}, we have an object F^*(X)\in\mathscr{D}.
– For each f\in\hom_\mathscr{C}(X,Y) we have a morphism \displaystyle F^*(f)\in\hom_\mathscr{D}(F^*(Y),F^*(X)) satisfying the two axioms:
– For each object X\in\mathscr{C} we have F^*(1_X)=1_{F^*(X)}. That is, F^* preserves identity morphisms.
– For each f\in\hom_\mathscr{C}(X,Y) and g\in\hom_\mathscr{C}(Y,Z) we have \displaystyle F^*(g\circ f)=F^*(f)\circ F^*(g)\in\hom_\mathscr{D}(F^*(Z),F^*(X)). Note that cofunctors reverse the direction of composition.

Example

Given a fixed pointed space (K,k_0)\in\mathscr{PT}, we define a functor \displaystyle F_K:\mathscr{PT}\to\mathscr{PS} as follows: for each (X,x_0)\in\mathscr{PT} we assign F_K(X,x_0)=[K,k_0; X,x_0]\in\mathscr{PS}. Given f: (X,x_0)\to (Y,y_0) in \hom((X,x_0),(Y,y_0)) we define F_K(f)\in\hom([K,k_0; X,x_0],[K,k_0;Y,y_0]) by \displaystyle F_k(f)[g]=[f\circ g]\in[K,k_0; Y,y_0] for every [g]\in [K,k_0; X,x_0].

We can check the two axioms:
– F_k(1_X)[g]=[1_X\circ g]=[g] for every [g]\in[K,k_0; X, x_0].
– For f\in\hom((X,x_0),(Y,y_0)), h\in\hom((Y,y_0),(Z,z_0)) we have \displaystyle F_K(h\circ f)[g]=[h\circ f\circ g]=F_K(h)\circ F_K(f)[g]\in[K,k_0; Z,z_0] for every [g]\in[K,k_0; X,x_0].

Similarly, we can define a cofunctor F_K^* by taking F_K^*(X,x_0)=[X,x_0; K,k_0] and for f:(X,x_0)\to (Y,y_0) in \hom((X,x_0),(Y,y_0)) we define \displaystyle F_K(f)[g]=[g\circ f]\in[X,x_0; K,k_0] for every [g]\in[Y,y_0; K,k_0].

Note that if f\simeq f' rel x_0, then F_K(f)=F_K(f') and similarly F_K^*(f)=F_K^*(f'). Therefore F_K (resp.\ F_K^*) can also be regarded as defining a functor (resp.\ cofunctor) \mathscr{PT}'\to\mathscr{PS}.

Homotopy Sets and Groups
Theorem:
If (X,x_0), (Y,y_0), (Z,z_0)\in\mathscr{PT}, X, Z Hausdorff and Z locally compact, then there is a natural equivalence \displaystyle A: [Z\wedge X, *; Y,y_0]\to [X, x_0; (Y,y_0)^{(Z,z_0)}, f_0] defined by A[f]=[\hat{f}], where if f:Z\wedge X\to Y is a map then \hat{f}: X\to Y^Z is given by (\hat{f}(x))(z)=f[z,x].

We need the following two propositions in order to prove the theorem.

Proposition 1:
The exponential function E: Y^{Z\times X}\to (Y^Z)^X induces a continuous function \displaystyle E: (Y,y_0)^{(Z\times X, Z\vee X)}\to ((Y,y_0)^{(Z,z_0)}, f_0)^{(X,x_0)} which is a homeomorphism if Z and X are Hausdorff and Z is locally compact\footnote{every point of Z has a compact neighborhood}.

Proposition 2:
If \alpha is an equivalence relation on a topological space X and F:X\times I\to Y is a homotopy such that each stage F_t factors through X/\alpha, i.e.\ x\alpha x'\implies F_t(x)=F_t(x'), then F induces a homotopy F':(X/\alpha)\times I\to Y such that F'\circ (p_\alpha\times 1)=F.

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2 Responses to Functors, Homotopy Sets and Groups

  1. Olcay O. says:

    Nice post! Learned a lot from it.

    Liked by 1 person

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