Homotopy category of topological spaces

This article describes a category (in the mathematical sense) whose objects are topological spaces. In other words, it gives a category structure to the collection of all topological spaces.
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Definition

The homotopy category of topological spaces is defined as follows:

• Its objects are topological spaces
• Its morphisms are homotopy classes of continuous maps

This can be viewed as the homotopy category obtained from the 2-category of topological spaces with continuous maps and homotopies.

Notion of isomorphism

The isomorphisms in this category are the homotopy equivalences.

Functors to and from the category

The homotopy category of topological spaces is not a concrete category, because, although its objects are sets, the morphisms are not concrete continuous maps but rather continuous maps upto homotopy.

Functor from the category of topological spaces

This functor sends a topological space to itself, and sends a continuous map to its homotopy class of continuous maps.

This functor is not conservative, because there are homotopy equivalences of topological spaces that are not homeomorphisms.