Category of topological spaces with continuous maps
This article describes how the collection of all topological spaces can be given the structure of a category
View other category structures on topological spaces
Definition
The category of topological spaces is defined as follows:
- Its objects are topological spaces
- Its morphisms are continuous maps
This is the default category structure on the collection of topological spaces.
Notion of isomorphism
The isomorphisms in the category of topological spaces are precisely the homeomorphisms.
Functors to and from the category
Forgetful functor to sets
This functor sends a topological space to its underlying set, and sends a continuous map of topological spaces, to the corresponding set map. The functor is faithful, hence the category of topological spaces is a concrete category. Some further facts:
- The functor is not conservative: In other words, if a continuous map of topological spaces is set-theoretically a bijection, it need not be a homeomorphism. Thus, the inverse image of an isomorphism under the functor, need not be an isomorphism.
Functor to homotopy category
The [[[homotopy category of topological spaces]] is the homotopy category obtained from the 2-category of topological spaces with continuous maps and homotopies. There is an obvious functor from the category of all topological spaces, to the homotopy category of topological spaces.
- The functor is not faithful: There could be different continuous maps that are homotopy-equivalent
- The functor is full
- The functor is not conservative: A continuous map of topological spaces could be a homotopy-equivalence without being a homeomorphism.