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 (or 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 categories of all topological spaces with continuous maps, where one uses the standard forgetful functor.

See also

• Infinity category
• Simplicial category