### Topological category

In category theory, a discipline in mathematics, 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.(Lurie 2009)

A topological category is a category $A$ along with a grounding functor $T: A \to X$ that maps to the category of sets $X$ such that $A$ admits weak or initial structures with respect to $T$. This means that constant functions in $X$ lift to $A$-morphisms and that fibers $T^\left\{-1\right\} x, x \in X$ are small (they are sets and not proper classes).[1]