# Comparisons of convenient categories for algebraic topology

I've heard that there are many convenient categories for algebraic topology. Such categories often have many nice properties like being cartesian closed, complete, cocomplete, the forgetful functor creates limits, containing all "nice" spaces like CW complexes and topological manifolds, et cetera. But the only convenient category for algebraic topologies that I know are the category of compact generated weakly Hausdorff (CGWH) spaces. Can someone briefly summarize other such categories and their advantages and disadvantages when compared to the category of CGWH spaces?

From the nLab (although I was the author of these words):

A reasonably large class of examples, including the examples of compactly generated spaces and sequential spaces, is given in the article by Escardó, Lawson, and Simpson (ref). These may be outlined as follows. An exponentiable space in $$Top$$ is a space $$X$$ such that $$X \times -: Top \to Top$$ has a right adjoint. These may be described concretely as core-compact spaces (spaces whose topology is a [[continuous lattice]]). Suppose given a collection $$\mathcal{C}$$ of core-compact spaces, with the property that the product of any two spaces in $$\mathcal{C}$$ is a colimit in $$Top$$ of spaces in $$\mathcal{C}$$. Such a collection $$\mathcal{C}$$ is called productive. Spaces which are $$Top$$-colimits of spaces in $$\mathcal{C}$$ are called $$\mathcal{C}$$-generated.

Theorem (Escardó, Lawson, Simpson): If $$\mathcal{C}$$ is a productive class, then the full subcategory of $$Top$$ whose objects are $$\mathcal{C}$$-generated is a coreflective subcategory of $$Top$$ (hence complete and cocomplete) that is cartesian closed.

Other convenience conditions, such as inclusion of CW-complexes and closure under closed subspaces, are in practice usually satisfied as well. For example, if closed subspaces of objects of $$\mathcal{C}$$ are $$\mathcal{C}$$-generated, then closed subspaces of $$\mathcal{C}$$-generated spaces are also $$\mathcal{C}$$-generated. If the unit interval $$I$$ is $$\mathcal{C}$$-generated, then so are all CW-complexes.

A number of examples are scattered throughout the paper.

If you want a convenient category of pointed spaces, then the category $$\mathbf{NG}_0$$ of pointed numerically generated spaces, discussed for instance in

Kazuhisa Shimakawa, Kohei Yoshida, Tadayuki Haraguchi, Homology and cohomology via enriched bifunctors Kyushu Journal of Mathematics, 2018, Volume 72, Issue 2, Pages 239-252, doi:10.2206/kyushujm.72.239

is one. The authors say that numerically generated is equivalent to being $$\Delta$$-generated, as discussed in Dan Dugger's Notes on Delta-generated spaces.

• An important advantage of $\Delta$-generated spaces -- shared with sequential spaces, or any full subcategory of $Top$ which is the colimit-closure of a small subcategory (a la Todd's response) is that they are locally presentable, which simplifies certain infinitary constructions like the small object argument. – Tim Campion Dec 7 at 18:15

One property that one sometimes wants, but which is not satisfied by most of the "usual" convenient categories, is that of being locally cartesian closed, or equivalently that each pullback functor $$f^* : \mathcal{C}/Y \to \mathcal{C}/X$$ has a right adjoint $$f_*$$.

However, I don't know of any nontrivial subcategory of the usual category $$\mathrm{Top}$$ of topological spaces that is locally cartesian closed. The closest locally cartesian closed categories to $$\mathrm{Top}$$ that I know of are quasitoposes, such as the category of subsequential spaces (which contains the category of sequential spaces mentioned by Todd, and is locally presentable) or pseudotopological spaces (which contains the entire category $$\mathrm{Top}$$, but is not locally presentable).

Another approach, used by May and Sigurdsson, is to mix two convenient categories to approximate local cartesian closure. As explained in their book, if $$\mathcal{K}$$ denotes the category of not-necessarily-weak-Hausdorff $$k$$-spaces, then the pullback functor $$f^*:\mathcal{K}/Y\to \mathcal{K}/X$$ has a right adjoint as long as $$X$$ and $$Y$$ are weak Hausdorff.