Homotopy category

In mathematics, the homotopy category is the category whose objects are topological spaces and whose morphisms are the homotopy classes of continuous maps. It is denoted by htop.

Explanation

Homotopy defines an equivalence relation on the set of continuous maps between two topological spaces. The equivalence classes are called homotopy classes. With one notes the set of homotopy classes between topological spaces and.

While top is the classical category of topological spaces and continuous functions, the morphisms of the category htop is precisely the homotopy classes. The objects of both categories are the same.

In other words, it is

And for any two objects

The morphisms are linked rep wise, that is, for topological spaces and continuous mappings apply:

This is well defined, since the homotopy relation is compatible with the sequential execution of functions.

It follows that for an area of ​​Identitätsmorphismus always is the class of all homotopic to the identity map pictures:

Properties

The homotopy category is a symmetric monoidale category with the smash product as a product and the 0 - sphere as a neutral element.

The isomorphisms of the homotopy category are the homotopy equivalences of the category Top.

Independent importance given the category htop, since they do not consists of quantities with an additional structure, and this structure compatible functions. It can not be considered as such. This means that the homotopy category is not determinable for, there is no faithful functor in the category Set of sets.

Generalizations

The homotopy category of an ( arbitrary) model category obtained by localizing with respect to the amount of weak equivalences.

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