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^{-1} x, x \in X are small (they are sets and not proper classes).[1]

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