Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: reading & references

Topic: Special properties of ∞-toposes of the form ∞Grpd/X

Oscar Cunningham (Apr 09 2024 at 16:09):

Related to the other thread about special properties of presheaf toposes, I'm wondering about the $\infty$-toposes you get by taking an $\infty$-groupoid $X$ and looking at the over-category $\infty\mathrm{Grpd}/X$ (equivalently $\infty\mathrm{Grpd}^X$). This is slightly more specific than 'presheaf $\infty$-topos', since I'm only interested in $X$ being an $\infty$-groupoid rather than an $(\infty, 1)$-category. Again I'm mostly intereted in the properties these have from an internal logic point of view.

Brendan Murphy (Apr 09 2024 at 20:12):

Interesting question! I'll just reiterate my point from the other question, again not internal, that ooGrpd/X will always be lfp while general grothendieck topoi are not

Brendan Murphy (Apr 09 2024 at 20:15):

If we break up X into its connected components we get that ooGrpd^X is a product of topoi of G-equivariant oo groupoids for G an infinity group (eg topological group)

Brendan Murphy (Apr 09 2024 at 20:20):

A fancy way to say nothing is that a grothendieck oo topos E is of the form you're looking at iff the terminal geometric morphism E -> ooGrpd is étale

Mike Shulman (Apr 09 2024 at 22:06):

This is even easier: they're always Boolean and satisfy the axiom of choice!

Graham Manuell (Apr 10 2024 at 17:13):

Mike Shulman said:

This is even easier: they're always Boolean and satisfy the axiom of choice!

Really? That's interesting, because the 1-dimensional analogue doesn't satisfy choice. What leads to the difference?

Brendan Murphy (Apr 10 2024 at 18:22):

Brendan Murphy said:

A fancy way to say nothing is that a grothendieck oo topos E is of the form you're looking at iff the terminal geometric morphism E -> ooGrpd is étale

Oh, this is slightly less useless than I thought. Rezk uses an intrinsic characterization of being étale to give an answer to this question https://golem.ph.utexas.edu/category/2018/10/topoi_of_gsets.html#c054844

Oscar Cunningham (Apr 10 2024 at 18:32):

@Brendan Murphy Thanks! That's the kind of thing I was looking for.

Oscar Cunningham (Apr 10 2024 at 18:33):

I'm interested in Graham's question too. Could it be that they satisfy internal choice but not external choice?

Mike Shulman (Apr 10 2024 at 18:45):

Yes, I mean internal choice, since you said you were interested in properties of the internal logic. (Although if by "1-dimensional version" you mean a 1-topos $\mathrm{Set}/X$ when $X$ is a set, I think those do also satisfy "external choice" -- isn't it only when $X$ starts to be a groupoid that external choice fails?)

David Corfield (Apr 11 2024 at 08:06):

You might find something of interest in Lurie's answer to my MO question. There was some nForum discussion here.

Reid Barton (Apr 11 2024 at 16:16):

Slicing always preserves internal properties.

Oscar Cunningham (Apr 14 2024 at 07:35):

@David Corfield One interesting thing about CABAs is that they can also be characterised as the complete boolean algebras in which limits and colimits distribute over each other. I find this algebraic definition nicer than demanding atomicity. I wonder if something similar works in the $\infty$-topos case.

Oscar Cunningham (Apr 14 2024 at 07:36):

@Reid Barton Is it possible that these are precisely the $\infty$-toposes with the same internal properties as $\infty\mathrm{Grpd}$?

Reid Barton (Apr 14 2024 at 17:04):

I doubt it, but I don't really know.