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 ($\infty$,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.

In another approach, a topological category is defined as a category $C$ along with a forgetful functor $T: C \to \mathbf$ that maps to the category of sets and has the following three properties:
* $C$ admits initial (also known as weak) structures with respect to $T$
* Constant functions in $\mathbf$ lift to $C$-morphisms
* Fibers $T^ x, x \in \mathbf$ 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

*{{Citation , last1=Lurie , first1=Jacob , title=Higher topos theory , arxiv=math.CT/0608040 , publisher=Princeton University Press , series=Annals of Mathematics Studies , isbn=978-0-691-14049-0 , mr=2522659 , year=2009 , volume=170

Category theory