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 (∞,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. (Lurie 2009)

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:

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).

An example of a topological category in this sense is the category of all topological spac es with continuous maps, where one uses the standard forgetful functor.[1]