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If topology were invented for algebraic geometry or logic, in ignorance of Euclidean space, we might reasonably regard connected compact Hausdorff spaces as pathological, or even doubt their existence. After all, Hausdorffness is a "separation" condition whereas connectedndess is a "nonseparation" condition -- it would seem strange that they should be compatible if you'd only seen spectral spaces or Stone spaces but not Euclidean spaces.

Luckily though, we know about Euclidean space and spaces built from it, and their central importance to mathematics. But they are very special, being as they are balanced between separatedness and nonseparatedness. For instance, every connected compact abelian group is a torus. Every number field has finitely many infinite places -- which are mysterious in a way that finite places are not. And so forth.

When it comes to $\infty$-topos theory as a generalization of topology, I wonder what the infinite places are. All the generalized concepts are there:

  • A proper geometric morphism (HTT 7.3.1.4) $\mathcal Y \to \mathcal X$ is one which satisfies a Beck-Chevalley condition, stably under pullback.

    • So a compact $\infty$-topos $\mathcal X$ is one such that the global sections geometric morphism $t_\ast: \mathcal X \to Spaces$ is proper.

    • And a Hausdorff $\infty$-topos $\mathcal X$ is one such that the diagonal $\mathcal X \to \mathcal X \otimes \mathcal X$ is proper (here $\mathcal X \otimes \mathcal X$ is the Lurie tensor product, which is the product in the category of toposes by HTT 7.3.3.9).

  • Generalizing from ordinary topos theory, an $\infty$-connected geometric morphism is one such that the left adjoint $t^\ast: Spaces \to \mathcal X$ to global sections is fully faithful.

  • Generalizing again from ordinary topos theory, a locally $\infty$-connected topos is one such that $t^\ast$ has a further left adjoint $t_!: \mathcal X \to Spaces$.

Question:

What's an example of a compact Hausdorff, $\infty$-connected, locally $\infty$-connected $\infty$-topos other than $Spaces$ itself?

There are various ways to relax these conditions:

  • If we don't require $\infty$-connectedness, then any étale topos (i.e. a slice $Spaces/X$) satisfies the remaining conditions; I'd be interested in examples other than these.

  • If we don't require Hausdorffness, then we have a weakening of the notion of a cohesive topos (i.e. a topos which is local (a strengthening of compactness) and strongly connected (a strenghtening of connected + locally connected)). My sense is that there are lots of cohesive toposes, so I think that Hausdorffness may be what sets this question apart from questions about cohesion.

  • We could relax compactness to local compactness or exponentiability or even do away with it altogether. I'd be interested in such examples.

  • ...

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    $\begingroup$ To be honest, I find the opening sentence bizarre. Both connectedness and compactness (properness) are important in locale and topos theory. "Hausdorff" means the diagonal embedding is proper. $\endgroup$
    – Todd Trimble ♦
    Oct 8, 2018 at 23:39
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    $\begingroup$ I'm trying to convey two things: 1.) There is an inherent tension between Hausdorffness (a separation condition) and connectedness (a nonseparation condition). 2.) Basically every construction of a connected Hausdorff space is built from the central example of Euclidean space. So if you didn't know about Euclidean space (say you primarily know spectral spaces, or Stone spaces) , you might regard the tension between connectedness and Hausdorffness as exotic and bizarre. In $\infty$-topos theory, I don't know what "seeds" like Euclidean space to start from to get such exotic behavior. $\endgroup$
    – Tim Campion ♦
    Oct 8, 2018 at 23:43
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    $\begingroup$ Great! Thanks for asking about it, I've added some of that explanation to the question body to clarify. $\endgroup$
    – Tim Campion ♦
    Oct 8, 2018 at 23:53
  • $\begingroup$ I would be tempted to rather say that connectedness is a cohesiveness condition (perhaps in the presence of local connectedness, else one would want pro-objects or similar). $\endgroup$
    – David Roberts ♦
    Oct 8, 2018 at 23:56
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    $\begingroup$ @ToddTrimble In defence of the opening sentence, in noncommutative spaces, where the algebra comes first, people did think "connected spaces" (C$^*$-algebras whose only projections are $0$ and $1$) were pathological, and Kaplansky doubted that a simple C$^*$-algebra with no nontrivial projections could exist (they do). $\endgroup$
    – Robert Furber
    Oct 9, 2018 at 7:00

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Every contractible finite CW complex $X$ satisfies these conditions. This follows from results in Section 7.3 of HTT and Appendix A of HA: we have $Shv(X) \otimes Shv(X)=Shv(X\times X)$ since $X$ is locally compact, proper morphisms between locally compact Hausdorff spaces give proper morphisms of $\infty$-topoi (so $Shv(X)$ is compact and Hausdorff), contractible topological spaces have trivial shape (so $t^*$ is fully faithful), and CW complexes are locally of constant shape (so $t_!$ exists).

The Hilbert cube is a non-CW example.

A non-$0$-localic example is presheaves on an $\infty$-category with finite sums, or more generally any weakly contractible $\infty$-category $C$ such that $C$ and $C_{(x,y)/}$ (for all $x,y\in C$) admit coinitial functors from finite $\infty$-categories (these guarantee compactness and Hausdorfness, respectively).

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  • $\begingroup$ Thanks, this is greatI The last class of examples seems to show that these conditions are actually pretty mild. So maybe there's a missing condition somewhere. I'm not sure if compact Hausdorff implies "locally compact" in some sense, or at least exponentiable, so maybe I should ask for it explicitly. But every presheaf topos is exponentiable, so the last class of examples meets this criterion too (for that matter, the localic examples presumably do as well). I'm not sure if Hausdorff toposes are closed under etale covers -- maybe I should ask for all etale covers to be Hausdorff as well. $\endgroup$
    – Tim Campion ♦
    Oct 9, 2018 at 12:52
  • $\begingroup$ Maybe here's a way of sharpening the question: what's an example of a coherent, locally coherent, $\infty$-connected, locally $\infty$-connected, Hausdorff $\infty$-topos other than $Spaces$? $\endgroup$
    – Tim Campion ♦
    Oct 9, 2018 at 13:40
  • $\begingroup$ The Hilbert cube example shows that Hausdorff toposes need not be hypercomplete. Alternatively, the $\infty$-topos of parameterized spectra, or more generally of $n$-excisive functors, is Hausdorff. For it is a localization (i.e. embedded subtopos) of the pointed object classifier $Spaces^{Spaces^{fin}_\ast}$, which is Hausdorff by the criterion you give, and Hausdorff $\infty$-toposes are closed under embedded subtoposes (for the diagonal of an embedded subtopos is an equivalence, and hence proper, and "separated" (= proper diagonal) geometric morphisms are closed under composition). $\endgroup$
    – Tim Campion ♦
    Oct 9, 2018 at 14:12
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    $\begingroup$ I agree there might be some missing conditions (such as properness of diagonals for any finite ∞-category, not just discrete ones), though I don't have any intuition for what a non-0-localic continuum should be... $X$ locally compact Hausdorff does imply $Shv(X)$ exponentiable (SAG 21.1.7). It's also easy for presheaf ∞-topoi to be coherent and locally coherent, you only need finite limits in the ∞-category. I think the ∞-topoi of $n$-excisive functors are also proper so they give another family of examples. $\endgroup$
    – Marc Hoyois
    Oct 9, 2018 at 15:49

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