
Topological category

In category theory, a discipline in mathematics, the notion of topological category has a number of different, inequivalent definitions.
In one approach, a topological category is a category that is enriched over the category of compactly generated Hausdorff spaces. They can be used as a foundation for higher category theory, where they can play the role of (formula_1,1)-categories. An important example of a topological category in this sense is given by the category of CW complexes, where each set Hom("X","Y") of continuous maps from "X" to "Y" is equipped with the compact-open topology. 
In another approach, a topological category is defined as a category formula_2 along with a forgetful functor formula_3 that maps to the category of sets and has the following three properties:
An example of a topological category in this sense is the category of all topological spaces with continuous maps, where one uses the standard forgetful functor.