CAUTION: The Sender of this email is not from within Dalhousie. 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/ ]