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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.

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  • $\begingroup$ I don't think I know of any interesting full subcategories of Top that are locally cartesian closed. (The boring ones I can think of are the discrete spaces and the indiscrete spaces, both being equivalent to Set.) Do you? $\endgroup$
    – Mike Shulman
    Commented Dec 7, 2015 at 1:42
  • $\begingroup$ @Mike Probably that's the better question! Really, I've just seen discussion of cartesian closure, but little discussion of local cartesian closure, and I'm wondering why. From the nlab, I've now seen this paper which has an argument (after Prop 9.3) that [under some hypotheses?] a full subcategory of Top which contains both the discrete and codiscrete topologies on 2 can't be locally cartesian closed. But I don't quite understand how the construction involved goes using just the lccc structure. $\endgroup$
    – Tim Campion
    Commented Dec 7, 2015 at 21:42
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    $\begingroup$ My best guess for why you haven't seen any discussion of local cartesian closure is that no one knows of any convenient lccc subcategories of Top. The closest thing I know of is the fact that the category of k-spaces is "almost" lccc, if you look only at slices over base spaces that are compactly generated. (See May-Sigurdsson Parametrized homotopy theory for references.) $\endgroup$
    – Mike Shulman
    Commented Dec 7, 2015 at 23:45
  • $\begingroup$ @TimCampion have you by any chance found the precise statement of "almost" lccc in May-Sigurdsson? $\endgroup$
    – Arrow
    Commented Jan 30, 2017 at 15:20
  • $\begingroup$ @Arrow I couldn't find the statement there, but they refer to Booth, P.I., 1971. The section problem and the lifting problem. Mathematische Zeitschrift 121, 273–287 (section 3, I think). This uses quasitopological spaces as a tool, which I've successfully avoided learning about -- I also found Booth, P.I., Brown, R., 1978. Spaces of partial maps, fibred mapping spaces and the compact-open topology. General Topology and its Applications 8, 181–195, Theorem 7.3, which seems to use more standard tools. $\endgroup$
    – Tim Campion
    Commented Feb 1, 2017 at 22:57

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