Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits.

On the other hand, the category of topological spaces has a competing notion of compactness. Not every compact object in $\operatorname{\underline{Top}}$ is a compact topological space, as is explained here. Todd Trimble asked (in the $n$-category café) if the situation is any better if $X$ is assumed compact Hausdorff.

More generally, is there some sort of classification of compact objects in $\operatorname{\underline{Top}}$?

Let $\mathscr C$ be a locally small category that has filtered colimits. Then an object $X$ in $\mathscr C$ is compact if $\operatorname{Hom}(X,-)$ commutes with filtered colimits.

On the other hand, the category of topological spaces has a competing notion of compactness. Not every compact topological space is a compact object in $\operatorname{\underline{Top}}$, as is explained here. Todd Trimble asked (in the $n$-category café) if the situation is any better if $X$ is assumed compact Hausdorff.

More generally, is there some sort of classification of compact objects in $\operatorname{\underline{Top}}$?