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 ($$\infty$$,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 $$C$$ along with a forgetful functor $$T: C \to \mathbf{Set}$$ 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.
 * $$C$$ admits initial (also known as weak) structures with respect to $$T$$
 * Constant functions in $$\mathbf{Set}$$ lift to $$C$$-morphisms
 * Fibers $$T^{-1} x, x \in \mathbf{Set}$$ are small (they are sets and not proper classes).