For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff $f^{-1}(U)$ is open for every continuous $f: J \to X$ for $J \in \mathcal{J}$. Then $\mathsf{Top}_\mathcal{J}$ is a coreflective subcategory of $\mathsf{Top}$.

Then Dugger Prop 1.15, referring to Vogt, section 3, asserts that if every $J \in \mathcal{J}$ is exponentiable in $\mathsf{Top}$ (in particular, if every such $J$ is locally compact Hausdorff), and if $\mathcal{J}\times \mathcal{J}$ maps to $\mathsf{Top}_\mathcal{J}$ under binary product in $\mathsf{Top}$, then $\mathsf{Top}_\mathcal{J}$ is cartesian closed.

In particular, if $\mathcal{J}$ is compact Hausdorff spaces, we see that the $k$-spaces are cartesian closed. If $\mathcal{J}$ is the singleton consisting of the one-point compactification of the natural numbers, we see that the sequential spaces are cartesian closed. If $\mathcal{J}$ is the set of simplices, we see that the $\Delta$-generated spaces are cartesian closed.

My question is: under what conditions is $\mathsf{Top}_\mathcal{J}$ locally cartesian closed? Of course, I would be interested in answers that apply to more general subcategories of $\mathsf{Top}$, or even to more general cartesian closed categories.

For a class $\mathcal{J}$ of topological spaces, let $\mathsf{Top}_\mathcal{J}$ denote the category of $\mathcal{J}$-generated spaces, i.e. those spaces $X$ such that $U\subseteq X$ is open iff $f^{-1}(U)$ is open for every continuous $f: J \to X$ for $J \in \mathcal{J}$. Then $\mathsf{Top}_\mathcal{J}$ is a coreflective subcategory of $\mathsf{Top}$.

Then Dugger Prop 1.15, referring to Vogt, section 3, asserts that if every $J \in \mathcal{J}$ is exponentiable in $\mathsf{Top}$ (in particular, if every such $J$ is locally compact Hausdorff), and if $\mathcal{J}\times \mathcal{J}$ maps to $\mathsf{Top}_\mathcal{J}$ under binary product in $\mathsf{Top}$, then $\mathsf{Top}_\mathcal{J}$ is cartesian closed.

In particular, if $\mathcal{J}$ is compact Hausdorff spaces, we see that the $k$-spaces are cartesian closed. If $\mathcal{J}$ is the singleton consisting of the one-point compactification of the natural numbers, we see that the sequential spaces are cartesian closed. If $\mathcal{J}$ is the set of simplices, we see that the $\Delta$-generated spaces are cartesian closed.

My question is: under what conditions is $\mathsf{Top}_\mathcal{J}$ locally cartesian closed? In particular, which of the above spaces are locally cartesian closed? In fact, I don't even know if the countably-generated spaces are cartesian closed. Of course, I would be interested in answers that apply to more general subcategories of $\mathsf{Top}$, or even to more general cartesian closed categories.