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Stream: deprecated: topos theory

Topic: (in)stability of geometric surjections

James Deikun (May 03 2022 at 13:23):

Geometric surjections are known to be unstable under pullbacks along general geometric morphisms and in particular along geometric embeddings, see e.g. https://math.stackexchange.com/questions/3992838/geometric-surjections-are-not-stable-under-pullback .

Is it known whether they are stable under pullbacks along tidy geometric morphisms? (This would allow a particular notion of 'cofunctorial geometric morphisms' to exist ...)

Tim Campion (May 03 2022 at 15:14):

James Deikun said:

Is it known whether they are stable under pullbacks along tidy geometric morphisms? (This would allow a particular notion of 'cofunctorial geometric morphisms' to exist ...)

Not sure about your question, but I'd be interested to hear more about this...

Jens Hemelaer (May 03 2022 at 20:28):

They are at least stable under pullback along tidy inclusions (these are exactly the closed inclusions). If $f$ is a surjection and $p$ is tidy, then the Beck–Chevalley condition $f^* p_* \simeq q_*g^*$ holds, with $g$ the pullback of $f$ along $p$, and $q$ the pullback of $p$ along $f$. To show that $g$ is surjective, take a morphism $\phi$ such that $g^*(\phi)$ is an isomorphism. Then $q_*(g^*(\phi)) \simeq f^*(p_*(\phi))$ is an isomorphism as well. Because both $p_*$ and $f^*$ are conservative, it then follows that $\phi$ is an isomorphism. So $g^*$ is conservative as well, in other words $g$ is surjective.

James Deikun (May 03 2022 at 20:56):

That at least eliminates the class of instability examples that I know of! It seems like the remaining cases would need a different approach though ...

Tim Campion (May 04 2022 at 19:43):

Jens Hemelaer said:

They are at least stable under pullback along tidy inclusions (these are exactly the closed inclusions). If $f$ is a surjection and $p$ is tidy, then the Beck–Chevalley condition $f^* p_* \simeq q_*g^*$ holds, with $g$ the pullback of $f$ along $p$, and $q$ the pullback of $p$ along $f$. To show that $g$ is surjective, take a morphism $\phi$ such that $g^*(\phi)$ is an isomorphism. Then $q_*(g^*(\phi)) \simeq f^*(p_*(\phi))$ is an isomorphism as well. Because both $p_*$ and $f^*$ are conservative, it then follows that $\phi$ is an isomorphism. So $g^*$ is conservative as well, in other words $g$ is surjective.

By this observation, it now suffices to show that surjections are stable under base change along proper surjections (since any proper map may be factored as a proper surjection followed by a proper inclusion -- see Elephant C.3.2.17). The "converse to this statement" is included in the Elephant C.5.1.7 or C.3.2.23. I strongly suspect this statement is true, and to be found somewhere in the Elephant...

Tim Campion (May 04 2022 at 19:47):

Er... I said "proper" there when you asked about "tidy" -- is there a counterexample to the statement when pulling back along some proper, non-tidy morphism?

Tim Campion (May 04 2022 at 19:57):

Oh nevermind, the Elephant statements I cited are less general than I initially thought

James Deikun (May 04 2022 at 22:29):

As for the "cofunctorial geometric morphisms" ... geometric surjections are a close analogue to identity-on-objects functors (including the way they are induced by a monad of proarrows) and tidy morphisms share a characterizing property of discrete opfibrations (the Beck-Chevalley condition above) which means that the assignment of $p_{*}f^{*}$ to $f : \mathcal{E} \larr \mathcal{\bar{E}} \to \mathcal{F} : p$ is a pseudofunctor from the category (and hopefully bicategory) of spans of geometric morphisms with right leg tidy to the bicategory of lex functors.

The stability property would confirm that this makes sense as a topos theory analogue of cofunctors.

James Deikun (May 05 2022 at 12:09):

Perhaps a path to this result would be factorizing the finitary lex functors into a lex comonadic left adjoint and a finitary lex functor with a lex left adjoint ...

James Deikun (May 05 2022 at 13:29):

James Deikun said:

... the assignment of $p_{*}f^{*}$ to $f : \mathcal{E} \larr \mathcal{\bar{E}} \to \mathcal{F} : p$ is a pseudofunctor from the category (and hopefully bicategory) of spans of geometric morphisms with right leg tidy to the bicategory of lex functors.

The idea here is that just as cofunctors in $\bold{Cat}$, these "cogeometric morphisms" are really a subclass of proarrows that can be exhibited as spans of ordinary morphisms, and since the spans are formed in an analogous way it should hopefully be "the same" class of proarrows. A lot remains to be verified; a nice intrinsic characterization of cofunctors-as-proarrows would also help with this!

Bryce Clarke (May 05 2022 at 16:54):

James Deikun said:

The idea here is that just as cofunctors in $\bold{Cat}$, these "cogeometric morphisms" are really a subclass of proarrows that can be exhibited as spans of ordinary morphisms, and since the spans are formed in an analogous way it should hopefully be "the same" class of proarrows. A lot remains to be verified; a nice intrinsic characterization of cofunctors-as-proarrows would also help with this!

Possibly silly question, but does an intrinsic characterisation of functors-as-profunctors exist? Assuming that the codomain is not Cauchy complete?

John Baez (May 05 2022 at 16:59):

I've never heard of one if one doesn't assume Cauchy completeness. That used to bug me a lot; then like everyone else (apparently) I got used to thinking of Cauchy completion as a very innocuous maneuver, sort of like "neatening up" a category.

John Baez (May 05 2022 at 17:00):

But I like your question, because I can imagine not wanting to accept working only with Cauchy complete categories.

Nathanael Arkor (May 05 2022 at 17:03):

I don't think it's possible to recover a functor $A \to B$ from a profunctor $A \not\rightarrow B$ in general. You can only recover a functor between the Cauchy completions of $A$ and $B$. This is essentially one of the reasons for the introductions of formalisms for formal category theory like proarrow equipments, because there was the realisation that remembering the profunctors that arise from functors is a very useful.

Mike Shulman (May 05 2022 at 17:06):

I don't think of Cauchy completion as innocuous!! As Nathanael said, this is why proarrow equipments are better than just bicategories of profunctors.

Mike Shulman (May 05 2022 at 17:07):

It's definitely impossible to detect which profunctors are functors between non-Cauchy complete categories in the bicategory Prof, because in Prof a category is equivalent to its Cauchy completion, so they have exactly the same profunctors into them.

Mike Shulman (May 05 2022 at 17:09):

Cauchy completion of ordinary categories, which is just idempotent-splitting, might be innocuous, although I can imagine arguments even there. But for enriched categories, it becomes a much more nontrivial operation. E.g. for $V=[0,\infty]$ there are lots of important metric spaces, like $\mathbb{Q}$, that aren't Cauchy-complete. And for $V=\rm SupLat$, the Cauchy completion of a small category is nearly always a large category!

James Deikun (May 05 2022 at 17:33):

Indeed. I meant an intinsic characterization in the proarrow equipment $\mathbb{P}\bold{rof}$; such a characterization could make use of (co)representability of profunctors as well.

James Deikun (May 05 2022 at 17:38):

(In $\mathbb{P}\bold{rof}$ in particular I think profunctors that merely have a right adjoint generally correspond to semifunctors ...)

Mike Shulman (May 05 2022 at 17:39):

A semifunctor being another name for "a functor into the Cauchy completion".

Nathanael Arkor (May 05 2022 at 17:59):

The usual identification of left-adjoint profunctors with functors between Cauchy complete categories uses the axiom of choice. Is there a nice constructive analogue? Is it appropriate simply to replace "functor" with "anafunctor"?

John Baez (May 05 2022 at 18:45):

When I said Cauchy completion was "innocuous" I was talking about the Set-enriched case. In my work with Todd Trimble and Joe Moeller on Vect-enriched categories, I've been feeling that Cauchy completion is fairly innocuous though technically quite crucial: splitting idempotents and throwing in biproducts.

John Baez (May 05 2022 at 18:46):

But anyway, "innocuous" is the sort of vague emotive remark of the sort I try to avoid, since they're destined to start arguments I'd rather not get into.

John Baez (May 05 2022 at 18:54):

I guess what's really true is that the guy who tried to talk me into accepting "functors are just left adjoint profunctors... well, at least for Cauchy complete categories" was trying to act like Cauchy completion was innocuous.

Graham Manuell (May 05 2022 at 18:58):

Nathanael Arkor said:

The usual identification of left-adjoint profunctors with functors between Cauchy complete categories uses the axiom of choice. Is there a nice constructive analogue? Is it appropriate simply to replace "functor" with "anafunctor"?

I'm pretty sure this is correct. Certainly anafunctors are the same thing as 'representable' profunctors.