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Stream: learning: reading & references

Topic: 2-local homeomorphism of toposes

James Deikun (Mar 18 2025 at 14:34):

Let $\pi_{/X}$ denote the projection from the slice over $X$. Given a geometric morphism $f : \mathcal{E \to F}$, call an object $X$ of $\mathcal E$ immune to $f$ if $f_* \circ \pi_{/X}$ is equivalent over $\mathcal F$ to $\pi_{/f_*X}$. If the objects immune to $f$ generate $\mathcal E$ under colimits, call $f$ a 2-local homeomorphism.

Has anyone seen this or a similar definition in the literature? Is there a known way to simplify the whole definition, or specifically the definition of "immune"? Would it be better to have the requirement be that the inclusion of the $f$-immune objects is dense?

Mike Shulman (Mar 18 2025 at 16:21):

By "is equivalent over $\mathcal{F}$ to $\pi_{/f_*X}$" do you mean that the canonical functor $\mathcal{E}_{/X} \to \mathcal{F}_{/f_*X}$ (which forms a commutative square with $\pi_{/X}$, $\pi_{/f_*X}$, and $f_*$) is an equivalence?

James Deikun (Mar 18 2025 at 16:23):

Yes.

Mike Shulman (Mar 18 2025 at 16:34):

Ok, so that functor has a left adjoint given by applying $f^*$ and composing with the counit, whose unit and counit are the same as those of $f$. So $X$ is immune just when $f^*f_*Y \to Y$ is an isomorphism whenever $Y$ admits a map to $X$, and $Z \to f_* f^* Z$ is an isomorphism whenever $Z$ admits a map to $f_*X$. So, for instance, if any object with a global element is immune, then $f$ is an equivalence. Is that right?

Mike Shulman (Mar 18 2025 at 16:35):

More generally, the immune objects are a sieve in $\mathcal{E}$.

Mike Shulman (Mar 18 2025 at 16:35):

What examples do you have in mind?

James Deikun (Mar 18 2025 at 17:00):

It definitely sounds right that if any object with a global element is immune then $f$ is an equivalence. Immunity is intended to be a generalization to the topos context of a locale map $f$ being a homeomorphism on an open subspace, where the same condition applies. And the immune objects being a sieve makes sense as well, since "the restriction of $f$ is a homeomorphism onto its image" is a down-closed property on open sets, hence also a sieve in the (0,1)-categorical sense.

James Deikun (Mar 18 2025 at 17:16):

As for examples, I hope that examples can be obtained from stacks on $\mathcal F$ and maybe vice versa, categorifying the story with locales, sheaves, and local homeomorphisms.

James Deikun (Mar 18 2025 at 17:35):

In particular, $\pi_{/Y}$ for $Y \in \mathcal F$ should be the direct image of a 2-local homeomorphism where the inclusion of immune objects in $\mathcal F/Y$ is the identity.

James Deikun (Mar 18 2025 at 17:51):

Okay, wait, I see where I messed up the definition. I totally forgot that $\pi_{/Y}$ is not the direct image of a geometric morphism, but instead the essentiality of one! So composing slice projections with direct images doesn't really make a lot of sense.

James Deikun (Mar 18 2025 at 19:06):

It looks like the slides at https://www.oliviacaramello.com/Talks/RelativeToposTheoryViaStacks.pdf and the paper at https://arxiv.org/abs/2107.04417 are working on pretty much the same idea, but missing a characterization of the fixed point of the adjunction. That material seems to claim that all the geometric morphisms that would be 2-local homeomorphisms are essential geometric morphisms, so maybe it would be possible to fix up the definition of immune by replacing the direct image $f_*$ with the essentiality $f_!$.

Mike Shulman (Mar 18 2025 at 21:47):

A geometric morphism is a local homeomorphism if it is essential and the induced map $\mathcal{E} \to \mathcal{F}_{/f_!1}$ is an equivalence. That seems like the most direct generalization of "being an homeomorphism onto an open subspace".

James Deikun (Mar 18 2025 at 22:03):

So "immune" should probably be: the induced map $\mathcal E/X \to \mathcal F/f_!X$, given by the action of $f_!$ on arrows, is an equivalence. This should still be a sieve in $\mathcal E$ but what it does in $\mathcal F$ makes more sense, and the corresponding version of 2-local homeomorphism, whether with simple generation under colimits or density, is indeed is a generalization of a (1-)local homeomorphism of toposes.

I'll go spend a little effort off on my own on figuring out whether either of these variants captures the fixed point of Caramello and Zanfa's adjunction. Thanks!

Morgan Rogers (he/him) (Mar 19 2025 at 16:57):

Ah I'm glad you came across the Caramello Zanfa work, that seemed like it would be relevant. :grinning_face_with_smiling_eyes: