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Concept
Homotopy category
Summary
In mathematics, the homotopy category is a built from the category of topological spaces which in a sense identifies two spaces that have the same shape. The phrase is in fact used for two different (but related) categories, as discussed below.
More generally, instead of starting with the category of topological spaces, one may start with any and define its associated homotopy category, with a construction introduced by Quillen in 1967. In this way, homotopy theory can be applied to many other categories in geometry and algebra.
The Top has objects the topological spaces and morphisms the continuous maps between them. The older definition of the homotopy category hTop, called the naive homotopy category for clarity in this article, has the same objects, and a morphism is a homotopy class of continuous maps. That is, two continuous maps f: X → Y are considered the same in the naive homotopy category if one can be continuously deformed to the other. There is a functor from Top to hTop that sends spaces to themselves and morphisms to their homotopy classes. A map f: X → Y is called a homotopy equivalence if it becomes an isomorphism in the naive homotopy category.
Example: The circle S1, the plane R2 minus the origin, and the Möbius strip are all homotopy equivalent, although these topological spaces are not homeomorphic.
The notation [X,Y] is often used for the set of morphisms from a space X to a space Y in the naive homotopy category (but it is also used for the related categories discussed below).
Quillen (1967) emphasized another category which further simplifies the category of topological spaces. Homotopy theorists have to work with both categories from time to time, but the consensus is that Quillen's version is more important, and so it is often called simply the "homotopy category".
One first defines a weak homotopy equivalence: a continuous map is called a weak homotopy equivalence if it induces a bijection on sets of path components and a bijection on homotopy groups with arbitrary base points.
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