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

[Jens Hemelaer <Jens.Hemelaer@uantwerpen.be>]

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


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