TITLE: Comparisons of convenient categories for algebraic topology
QUESTION [11 upvotes]: 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?

REPLY [4 votes]: 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.