**Functors**

Definition:

A functor from a category to a category is a function which

– For each object , we have an object .

– For each , we have a morphism

Furthermore, is required to satisfy the two axioms:

– For each object , we have . That is, maps the identity morphism on to the identity morphism on .

– For , we have That is, functors must preserve composition of morphisms.

Definition:

A cofunctor (also called contravariant functor) from a category to a category is a function which

– For each object , we have an object .

– For each we have a morphism satisfying the two axioms:

– For each object we have . That is, preserves identity morphisms.

– For each and we have Note that cofunctors reverse the direction of composition.

**Example**

Given a fixed pointed space , we define a functor as follows: for each we assign . Given in we define by for every .

We can check the two axioms:

– for every .

– For , we have for every .

Similarly, we can define a cofunctor by taking and for in we define for every .

Note that if rel , then and similarly . Therefore (resp.\ ) can also be regarded as defining a functor (resp.\ cofunctor) .

**Homotopy Sets and Groups**

Theorem:

If , , , , Hausdorff and locally compact, then there is a natural equivalence defined by , where if is a map then is given by .

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

Proposition 1:

The exponential function induces a continuous function which is a homeomorphism if and are Hausdorff and is locally compact\footnote{every point of has a compact neighborhood}.

Proposition 2:

If is an equivalence relation on a topological space and is a homotopy such that each stage factors through , i.e.\ , then induces a homotopy such that .

Nice post! Learned a lot from it.

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Thanks!

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