Topological category

Short description: Categorical treatment of topological spaces

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 ([math]\displaystyle{ \infty }[/math],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 [math]\displaystyle{ C }[/math] along with a forgetful functor [math]\displaystyle{ T: C \to \mathbf{Set} }[/math] that maps to the category of sets and has the following three properties:

• [math]\displaystyle{ C }[/math] admits initial (also known as weak) structures with respect to [math]\displaystyle{ T }[/math]
• Constant functions in [math]\displaystyle{ \mathbf{Set} }[/math] lift to [math]\displaystyle{ C }[/math]-morphisms
• Fibers [math]\displaystyle{ T^{-1} x, x \in \mathbf{Set} }[/math] 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.

See also

• Infinity category
• Simplicial category

References

• Lurie, Jacob (2009), Higher topos theory, Annals of Mathematics Studies, 170, Princeton University Press, ISBN 978-0-691-14049-0