Does essential entail locally connected for hyperconnected geometric morphisms?

Does essential entail locally connected for hyperconnected geometric morphisms?

[streicher]

Consider the functor F from the site for the topos of graphs to the site
for the Sierpinski topos such that the object part of F is a bijection.
Let f be the geometric morphism whose inverse image part is given by
change of base along F. It is the inclusion of graphs with loops only into
the category of graphs.
Of course, p is essential and one easily sees that it is hyperconnected.
One can show that p is not locally connected. However, p is not local
since p_* does not preserve coequalizers.

My attempts to come up with an example of a hyperconnected local geometric
morphism which is is essential but not locally connected have failed so
far.
But all my instincts tell me that there should be a counterexample!

The question came up in discussions with Matias Menni. He told me that one
can prove that essential entails locally connected for hyperconnected
local geometric morphisms. But his argument uses (ii) => (i) of Lemma 3.2
of Peter Johnstone's paper "Calibrated Toposes" whose proof I, however,
find very cryptic. In any case, it would entail a result which I,
personally, would find very surprising...

I would be grateful for any clarification of this puzzling question. My
hope is that someone comes up (with an idea for) a counterexample. But, of
course, I also would highly appreciate any argument that such a
counterexample cannot exist.

Thomas







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Re: Does essential entail locally connected for hyperconnected geometric morphisms?

[Jens Hemelaer]

Morgan Rogers and I were able to construct a counterexample of the form PSh(M) --> PSh(N) where M and N are monoids. It arises from our joint work-in-progress in which we study exactly these kind of geometric morphisms, in a systematic way. After talking about it with Thomas Streicher, we have now written up the counterexample in more detail. You can find it here:
https://arxiv.org/abs/2009.12241 (3 pages).

Jens

________________________________

From: <streicher@mathematik.tu-darmstadt.de<mailto:streicher@mathematik.tu-darmstadt.de>>
Date: Sat, 19 Sep 2020 at 14:28
Subject: categories: Does essential entail locally connected for hyperconnected geometric morphisms?
To: <categories@mta.ca<mailto:categories@mta.ca>>


Consider the functor F from the site for the topos of graphs to the site
for the Sierpinski topos such that the object part of F is a bijection.
Let f be the geometric morphism whose inverse image part is given by
change of base along F. It is the inclusion of graphs with loops only into
the category of graphs.
Of course, p is essential and one easily sees that it is hyperconnected.
One can show that p is not locally connected. However, p is not local
since p_* does not preserve coequalizers.

My attempts to come up with an example of a hyperconnected local geometric
morphism which is is essential but not locally connected have failed so
far.
But all my instincts tell me that there should be a counterexample!

The question came up in discussions with Matias Menni. He told me that one
can prove that essential entails locally connected for hyperconnected
local geometric morphisms. But his argument uses (ii) => (i) of Lemma 3.2
of Peter Johnstone's paper "Calibrated Toposes" whose proof I, however,
find very cryptic. In any case, it would entail a result which I,
personally, would find very surprising...

I would be grateful for any clarification of this puzzling question. My
hope is that someone comes up (with an idea for) a counterexample. But, of
course, I also would highly appreciate any argument that such a
counterexample cannot exist.

Thomas



[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Does essential entail locally connected for hyperconnected geometric morphisms?

[Thomas Streicher]

> Morgan Rogers and I were able to construct a counterexample of the form PSh(M) --> PSh(N) where M and N are monoids. It arises from our joint work-in-progress in which we study exactly these kind of geometric morphisms, in a systematic way. After talking about it with Thomas Streicher, we have now written up the counterexample in more detail. You can find it here:
> https://arxiv.org/abs/2009.12241 (3 pages).

Dear Jens and Morgan,

thanks a lot for your very nice counterexample which I never would
have found on my own!

Alas, it leaves open Lawvere and Menni's question whether precohesive
geometric morphisms are always also locally connected. Does your
toolbox also provide a counterexample to this implication. The current
one does not do this job since you show the leftmost adjoint does not
preserve binary products. (By Lemma 2.7 of Johnstone's 2011 TAC paper
preservation of binary products by the leftmost adjoint is equivalent
to preservation of exponentials by the inverse image part for hyperconnected
and local geometric morphisms.)

BTW I think that locally connected, hyperconnected and local is the
correct generalization of essential, 2-valued and local from base Set
to arbitrary base toposes from the point of view of fibered categories.

Best, Thomas


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Re: Does essential entail locally connected for hyperconnected geometric morphisms?

[Jens Hemelaer]

Dear Thomas,


At the moment, we do not have a counterexample to the statement that pre-cohesive geometric morphisms are locally connected. We think we can construct  one of the form PSh(M) --> PSh(N) with M and N monoids, but it will take some more work.

All the best,
Jens

________________________________
From: Thomas Streicher <streicher@mathematik.tu-darmstadt.de>
Sent: Monday, September 28, 2020 1:36 PM
To: Jens Hemelaer
Cc: categories@mta.ca; Morgan Rogers
Subject: Re: categories: Does essential entail locally connected for hyperconnected geometric morphisms?

> Morgan Rogers and I were able to construct a counterexample of the form PSh(M) --> PSh(N) where M and N are monoids. It arises from our joint work-in-progress in which we study exactly these kind of geometric morphisms, in a systematic way. After talking about it with Thomas Streicher, we have now  written up the counterexample in more detail. You can find it here:
> <https://eur01.safelinks.protection.outlook.com/?url=https%3A%2F%2Farxiv.org%2Fabs%2F2009.12241&amp;data=02%7C01%7CJens.Hemelaer%40uantwerpen.be%7C4e384645e45b483168dc08d863a2af9d%7C792e08fb2d544a8eaf72202548136ef6%7C0%7C0%7C637368898108545753&amp;sdata=o5jz9Hahe4VdYGGKQObOulsZH6HIdoh7IXqCAlV9uew%3D&amp;reserved=0> https://arxiv.org/abs/2009.12241 (3 pages).

Dear Jens and Morgan,

thanks a lot for your very nice counterexample which I never would
have found on my own!

Alas, it leaves open Lawvere and Menni's question whether precohesive
geometric morphisms are always also locally connected. Does your
toolbox also provide a counterexample to this implication. The current
one does not do this job since you show the leftmost adjoint does not
preserve binary products. (By Lemma 2.7 of Johnstone's 2011 TAC paper
preservation of binary products by the leftmost adjoint is equivalent
to preservation of exponentials by the inverse image part for hyperconnected
and local geometric morphisms.)

BTW I think that locally connected, hyperconnected and local is the
correct generalization of essential, 2-valued and local from base Set
to arbitrary base toposes from the point of view of fibered categories.

Best, Thomas


[For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Re: Does essential entail locally connected for hyperconnected geometric morphisms?

[Thomas Streicher]

Dear Jens,

> At the moment, we do not have a counterexample to the statement that pre-cohesive geometric morphisms are locally connected. We think we can construct  one of the form PSh(M) --> PSh(N) with M and N monoids, but it will take some more work.

I sincerely hope you will succeed!

BTW it is not known (to me) whether there exists a geometric morphism
f : EE-->SS such that f^* preserves ordinary function spaces (exponentials)
but not dependent functions spaces (i.e. f^*\Pi_u \cong \Pi_{f^*u}
fuer maps u in SS) which is equivalent to f being locally connected.

As follows from Peter Johnstone's Proposition 2.7 in his 2011 TAC paper
"Remarks on Punctual Local Connectedness" for essential hyperconnected local
gm's f : EE-->SS the inverse image part f^* preserves exponentials iff the
left adjoint f_! (to f^*) preserves finite products.
But as shown in Cor.3.11 of Matias Menni's constribution to the
Freyd-Lawvere Festschrift (Georgian Math. Journal 2017) for a hyperconnected
local gm f : EE-->SS, f^* preserving exponentials is equivalent to having
a finite product preserving left adjoint.

Thomas


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