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: theory: category theory

Topic: geometric morphisms induced by lex functors

Jonas Frey (Jun 08 2025 at 14:13):

Given a lex functor $F : \mathbb{C}\to \mathbb{D}$ between lex categories, we obtain a triple
$F_!\dashv F^*\dashv F_*:\hat{\mathbb{C}}\to\hat{\mathbb{D}}$ of lex adjoints, constituting an adjunction of geometric morphisms, ie the geometric morphism $(F^*\dashv F_*)$ has a right (!) adjoint in the $2$-category of Grothendieck toposes and geometric morphisms.

Does this situation have a name?

I know that the further left adjoint means that the geometric morphism $(F^*\dashv F_*)$ is essential, and if $F$ is [[absolutely dense]] (co-fully-faithful) then the g.m. is [[totally connected]], but is there something to be said about non-connected essential g.m.s with lex leftest adjoint?

Mike Shulman (Jun 08 2025 at 15:38):

Doesn't the lexness of $F_!$ (in particular, preserving 1) imply the g.m. is connected?

Jonas Frey (Jun 08 2025 at 17:50):

Maybe that's only true if the gm is already locally connected? I think eg the essential gm between presheaves induced by the functor 1-->FinSet is a counterexample.

Mike Shulman (Jun 08 2025 at 20:51):

Hmm, I suppose so. I think essential geometric morphisms that are not locally connected aren't studied very much by topos theorists; they're not very "topos-theoretic" since they aren't indexed.

Morgan Rogers (he/him) (Jun 09 2025 at 11:42):

I have seen the geometric morphism in the opposite direction get some attention specifically in the case that $F$ is fully faithful, since then that morphism is local.

Thinking in the direction you indicated now: preserving the terminal object makes the morphism terminally connected -- if it were also locally connected this would imply connectedness by Lemma C3.3.3 of the Elephant, which confirms your counterexample @Jonas Frey . Osmond generalizes the notion to non-essential geometric morphisms $f:\mathcal{F} \to \mathcal{E}$ by observing that the isomorphism $\mathcal{F}(1_{\mathcal{F}},f^*(X)) \cong \mathcal{E}(f_!(1_{\mathcal{F}}),X)$ natural in $X$ characterizes these morphisms and still makes sense after identifying $f_!(1_{\mathcal{F}})$ with $1_\mathcal{E}$.

As far as I can tell, that approach doesn't generalize easily (although I would be happy for someone to try and prove me wrong!) On the other hand, if we keep the extra left adjoint around, we can take a "Beck-Chevalley" approach instead. Consider this diagram:
image.png

Here $D$ is an "external" diagram category (a locally constant one, if you like, obtained from the base topos by pullback). Since $f^*$ has a left adjoint, we have an isomorphism $f^*d_* \cong e_*[D,f]^*$; this is the expression that $f^*$ preserves small limits, with "small" again referring to the base topos of sets. Now we can use the existence of the left adjoint to transpose that isomorphism to a natural transformation $f_!e_* \Rightarrow d_*[D,f]_!$. Its components translate to the comparison morphisms for $D$-indexed limits along $f_!$. So your conditions amounts to this one being an isomorphism for $D$ finite. An important class to compare with are proper and tidy morphisms, whose direct images preserve filtered colimits.

This raises the question of whether one can, in light of finiteness, extend this characterization to a more intrinsic one, for instance by identifying a characteristic, pullback stable property of $d$ and $e$ strong enough to carry the BC condition on its own. While essential geometric morphisms aren't stable under pullback, it's plausible that this class might be, which is all the more reason to expect it to have an intrinsic characterization.

Ivan Di Liberti (Jun 10 2025 at 15:46):

In my paper with Ye we study this construction and hence we needed to give it a name. See the notion of ethereal geometric morphism (Defn 1.2.1) and then Section 1.3.