# 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 along with a forgetful functor that maps to the category of sets and has the following three properties:

- admits initial (or weak) structures with respect to
- Constant functions in lift to -morphisms
- Fibers 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.^{[1]}

## See also

## References

- ↑ Brümmer, G. C. L. (September 1984). "Topological categories".
*Topology and its Applications*.**18**(1): 27–41. doi:10.1016/0166-8641(84)90029-4. Retrieved 1 October 2013.

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