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Stream: deprecated: topos theory

Topic: loc conn : conn :: tidy : ?

Tim Campion (May 07 2022 at 18:46):

Connected geometric morphisms bear some sort of relationship to locally connected geometric morphisms which I don't understand too well. In trying to understand it better, I am led to the following vague question:

There is a sort of duality between locally connected geometric morphisms and tidy geometric morphisms -- the former are the stable left Beck-Chevalley geometric morphisms (Elephant C 3.3.16) while the latter are the stable right Beck-Chevalley geometric morphisms (C 3.4.11). Does this analogy include some class of geometric morphisms which stands in relation to the tidy geometric morphisms just as connected geometric morphisms stand in relation to locally connected geometric morphisms?

James Deikun (May 08 2022 at 01:19):

I'd say that the answer to this is probably also "connected". There are slight differences in the relationship, but I think this has to do with the asymmetry where inverse images of geometric morphisms preserve finite limits while direct images don't preserve anything special for a right adjoint. This asymmetry strikes particularly hard sometimes with the locally connected/tidy duality because the inverse image of a locally connected morphism $f$ preserves ($f_{*}$-indexed) cofiltered limits which is just barely enough to combine with the finite limits it already has to preserve and summon a further ($f_{*}$-indexed) left adjoint into existence.

Morgan Rogers (he/him) (May 08 2022 at 10:33):

There are several relationships between locally connected and connected; you might have to be more specific.
Any locally connected (LC) gm factors as a connected LC morphism followed by an étale morphism. This is kind of a coincidence; we can extend this construction to essential geometric morphisms and get a factorization into terminal connected (TC) essential followed by étale, and the orthogonality still holds; TC is strictly weaker than connected in general.
I'm fairly sure we can characterize a locally connected topos $p:\mathcal{E} \to \mathcal{S}$ over a base $\mathcal{S}$ in terms of having "enough connected slices", in the sense that there is a family of objects $\{X_i\}_{i \in I}$ of $\mathcal{E}$ such that the morphisms $\mathcal{E}/X_i \to \mathcal{E}$ are jointly surjective and the composites $\mathcal{E}/X_i \to \mathcal{E} \to \mathcal{S}$ are connected morphisms. (Don't quote me on that, I need to check that I'm not being misled by Set-specific details). If what you're looking for is a corresponding characterization of tidy morphisms, then that might be harder, because the dual of étale morphisms are "complete spreads", studied by Bunge and Funk, which have a looser relationship with tidy morphisms than étale morphisms have with locally connected morphisms.

James Deikun (May 08 2022 at 12:22):

Although ... a bounded tidy morphism, at least, seems to factor as a connected tidy morphism followed by an entire morphism, so maybe 'entire' is dual to 'etale' in a different way ...

James Deikun (May 08 2022 at 12:26):

(Or maybe it's the same way ... I don't know much about spreads yet, but it seems at least that entire morphisms are a kind of spread.)

James Deikun (May 08 2022 at 12:41):

Complete spreads are described in https://doi.org/10.1007/3-540-36359-9 ...

Tim Campion (May 08 2022 at 16:07):

Morgan Rogers (he/him) said:

There are several relationships between locally connected and connected; you might have to be more specific.
Any locally connected (LC) gm factors as a connected LC morphism followed by an étale morphism. This is kind of a coincidence; we can extend this construction to essential geometric morphisms and get a factorization into terminal connected (TC) essential followed by étale, and the orthogonality still holds; TC is strictly weaker than connected in general.
I'm fairly sure we can characterize a locally connected topos $p:\mathcal{E} \to \mathcal{S}$ over a base $\mathcal{S}$ in terms of having "enough connected slices", in the sense that there is a family of objects $\{X_i\}_{i \in I}$ of $\mathcal{E}$ such that the morphisms $\mathcal{E}/X_i \to \mathcal{E}$ are jointly surjective and the composites $\mathcal{E}/X_i \to \mathcal{E} \to \mathcal{S}$ are connected morphisms. (Don't quote me on that, I need to check that I'm not being misled by Set-specific details). If what you're looking for is a corresponding characterization of tidy morphisms, then that might be harder, because the dual of étale morphisms are "complete spreads", studied by Bunge and Funk, which have a looser relationship with tidy morphisms than étale morphisms have with locally connected morphisms.

Here's a guess at a precise statement along these lines:

Conjecture: Let $p : \mathcal E \to \mathcal F$ be a geometric morphism. Then the following are equivalent:

1. $p$ is locally connected.
2. For every etale map $\mathcal F' \to \mathcal F$, there exists an etale cover $\mathcal F'' \twoheadrightarrow \mathcal F'$ such that the pulback $p'' : \mathcal E'' \to \mathcal F''$ is connected.

I'll have to read some more about spreads, thanks!

Tim Campion (May 08 2022 at 16:09):

I've never heard of terminal connected, what's that?

Morgan Rogers (he/him) (May 08 2022 at 16:13):

For essential geometric morphisms it's the property of the extra left adjoint preserving the terminal object, but there's an equivalent formulation in a paper of Osmond that works for arbitrary geometric morphisms, along the lines of the inverse image functor commuting with global sections morphisms (I'll dm you a link to that paper when I'm at my computer next week)

Morgan Rogers (he/him) (May 10 2022 at 10:47):

See section 5.3 here:
https://arxiv.org/pdf/2108.12697.pdf