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 (${\displaystyle \infty }$,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. (Lurie 2009)

In another approach, a topological category is defined as a category ${\displaystyle C}$ along with a forgetful functor ${\displaystyle T:C\to \mathbf {Set} }$ that maps to the category of sets and has the following three properties:

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

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

• Infinity category
• Simplicial category

• Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, vol. 170, Princeton University Press, arXiv:math.CT/0608040, ISBN 978-0-691-14049-0, MR 2522659