nLab nice category of spaces

For more see at convenient category of topological spaces.

By a “nice” or “convenient” category of space one means a category that subsumes at least most of the given kinds of spaces of interest (typically: topological space) but possibly also less well-behaved spaces, such that the category itself becomes well behaved.

The point is that the default category Top of topological spaces lacks many good category-theoretic properties: It is complete, cocomplete and extensive, but:

The lack of cartesian closure and, to a lesser extent, local presentability, is especially problematic for homotopy theory. Many different solutions for repairing lack of cartesian closure have been proposed, generally involving either restricting to a subcategory of Top (usually reflective or coreflective, so that it inherits completeness and cocompleteness), enlarging it to a supercategory, or some combination thereof. Most involve restricting the topologies to those that can be specified on “small” (and in particular, compact) subsets.

In particular, a convenient category of topological spaces is, in the technical sense of the nLab, a cartesian-closed category of spaces together with some other useful properties (q.v.).



  • Peter May, A Concise Course in Algebraic Topology (Chapter 5, for compactly generated spaces)

  • O. Wyler, Convenient categories for topology

  • L. Fajstrup and J. Rosicky, A convenient category for directed homotopy (for Delta-generated spaces)

  • E. Spanier, Quasi-topologies (for quasi-topological spaces)

  • O. Wyler, Lecture notes on topoi and quasitopoi (for pseudotopological spaces)

  • Peter Johnstone, On a topological topos

  • Peter Johnstone, Stone Spaces

  • J. Milnor, On spaces having the homotopy type of a CW-complex, Trans. Amer. Math. Soc. 90, no. 2 (1959), 272-280.

Last revised on February 6, 2024 at 16:28:44. See the history of this page for a list of all contributions to it.