Topological category explained

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 (

inf ty ,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 Set 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

Set lift to C -morphisms

• Fibers

T ^{-1} x, x \in 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]

See also

• Infinity category
• Simplicial category

Notes and References

1. Brümmer. G. C. L.. Topological categories. Topology and Its Applications. September 1984. 18. 1. 27–41. 10.1016/0166-8641(84)90029-4. free.