Topological category

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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[edit]


  1. ^ 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.