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Stream: learning: questions

Topic: Found a rare geometric morphism

James Deikun (Jan 22 2024 at 13:41):

So I encountered in the wild a kind of geometric morphism which

• is essential, i.e. the inverse image $f^*$ has a further left adjoint $f_!$
• preserves, via the inverse image, the subobject classifier and $\top$ ($f^* \Omega \cong \Omega$)
• preserves, via essentialism, the terminal object ($f_! 1 \cong 1$)

An [[atomic geometric morphism]] can only do this if it is hyperconnected, but the examples I'm interested in are localic (they're even geometric embeddings). The examples are the essential subtoposes of the toposes of simplicial, cubical, and reflexive globular sets (where if the cubes have diagonals they also have connections).

I tried looking for these in the Elephant and the nLab and failed; has anyone heard of them?

Morgan Rogers (he/him) (Jan 22 2024 at 13:45):

There's a typo, so it should be $f^* \Omega \cong \Omega$, right?

James Deikun (Jan 22 2024 at 13:45):

Ah, right, fixed, thanks.

James Deikun (Jan 22 2024 at 13:49):

The corresponding property for a full subcategory is that you can recover all the monomorphisms into any of the objects in it along with their targets by splitting enough idempotents.

James Deikun (Jan 22 2024 at 13:52):

(In other words, that it is closed under incoming monomorphisms up to Morita equivalence.)

Morgan Rogers (he/him) (Jan 22 2024 at 14:17):

Is it locally connected?

James Deikun (Jan 22 2024 at 14:38):

If it is locally connected, then $f^*$ is (locally) Cartesian closed and you have $f^*(\Omega^A) \cong (f^*\Omega)^{f^*A} \cong \Omega^{f^*A}$ and the geometric morphism is atomic, and since it's localic then it's the slice over $f_!1 \cong 1$, hence an equivalence.

James Deikun (Jan 22 2024 at 14:40):

As a rule these are basically essential geometric morphisms that stubbornly refuse to let their extra left adjoint be indexed.

James Deikun (Jan 22 2024 at 15:03):

I think they are all dominant though.

Morgan Rogers (he/him) (Jan 22 2024 at 16:04):

James Deikun said:

If it is locally connected, then $f^*$ is (locally) Cartesian closed and you have $f^*(\Omega^A) \cong (f^*\Omega)^{f^*A} \cong \Omega^{f^*A}$ and the geometric morphism is atomic, and since it's localic then it's the slice over $f_!1 \cong 1$, hence an equivalence.

Right, with preservation of the subobject classifier that would correspond to the inverse image functor being logical oops.
So having $f_!1 = 1$ is called being "terminal connected" in a few places (I have used that name myself before, and I've seen it used by Caramello; @Axel Osmond wrote about a generalization where essential is not required). They're orthogonal to etale morphisms just as connected morphisms are.

Morgan Rogers (he/him) (Jan 22 2024 at 16:06):

No nlab page for that yet.
As for your specific examples, there is certainly material (eg of Lawvere) on essential subtoposes of simplicial sets, for example. There might be some further properties developed for those that you can lift.

Morgan Rogers (he/him) (Jan 22 2024 at 16:22):

What kind of result are you interested in?

James Deikun (Jan 22 2024 at 17:28):

Morgan Rogers (he/him) said:

So having $f_!1 = 1$ is called being "terminal connected" in a few places (I have used that name myself before, and I've seen it used by Caramello; Axel Osmond wrote about a generalization where essential is not required). They're orthogonal to etale morphisms just as connected morphisms are.

Hm, among essential geometric morphisms there's even a factorization system; then all these are terminal-connected just because these toposes are two-valued. Still, I don't think that makes them have to preserve the subobject classifier?

James Deikun (Jan 22 2024 at 17:29):

As for the kind of result I'm interested in, this is part of my project to characterize what "degeneracies" exactly are.

James Deikun (Jan 28 2024 at 04:57):

Seems Proposition A4.5.8 in the Elephant talks a bit about these guys.

Morgan Rogers (he/him) (Jan 29 2024 at 17:17):

So it does, well spotted! Johnstone doesn't name them or mention them elsewhere in the Elephant, and it's not immediately obvious to me that any of the equivalent conditions clarify the relationship with terminal-connectedness (if there is any). Did it help you?

James Deikun (Jan 30 2024 at 01:01):

Well, it's grist for the mill. For now I'm attacking the general problem by other means and hoping the two angles will shed some light on each other.