nLab
nice category of spaces

The category Top of topological spaces lacks many good categorical properties. It is complete and cocomplete, but:

• not cartesian closed or locally cartesian closed,
• not locally presentable, and
• not a topos or even a quasitopos.

The lack of cartesian closure and, to a lesser extent, local presentability, is especially problematic for homotopy theory. Many different solutions 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.

A convenient category of topological spaces is, in particular, a cartesian-closed category of spaces.

I'm not sure that we really want to use the terminology that way, but Ronnie already created that page, so I'm linking these together. —Toby

Examples

• The most common approach among algebraic topologists today is to use the subcategory of compactly generated spaces, which is cartesian closed, but not locally cartesian closed. It is a coreflective subcategory of a reflective subcategory of $\mathrm{Top}$.

• The subcategory of Delta-generated spaces, recently proposed by J. H. Smith, is both cartesian closed and locally presentable.

• An approach of mainly historical interest is to use quasitopological spaces?, an enlargement of $\mathrm{Top}$ which is cartesian closed.

• The category $\mathrm{PsTop}$ of pseudotopological spaces (also called Choquet spaces) is a quasitopos containing $\mathrm{Top}$ as a full reflective subcategory. In particular, $\mathrm{PsTop}$ is locally cartesian closed (but not locally presentable).

• In his paper On a topological topos, Peter Johnstone described a Grothendieck topos $E$ which contains the category of sequential topological spaces as a full reflective subcategory which is closed under many colimits (including all those used to define CW complexes). Since $E$ is a Grothendieck topos, it is locally presentable and locally cartesian closed. Moreover, the geometric realization and singular complex? functors form a geometric morphism between $E$ and the category of simplicial sets. The “underlying set” functor $E\to \mathrm{Set}$ is not faithful, but it is faithful on the full subcategory of subsequential spaces, which contain the sequential spaces and form a quasitopos.

• One can just forget topological spaces and use the category of simplicial sets as the subject of homotopy theory. The fact that every topological space has a simplicial set as its singularization? then becomes an application of the homotopy theory of simplicial sets to the study of topological spaces, rather than a way to use simplicial sets to study the homotopy theory of topological spaces.

  For more on this see Top, homotopy theory and infinity-groupoid.

References

• 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)

• P. Johnstone, On a topological topos