An essential, hyperconnected, local geometric morphism that is not locally connected

TL;DR

This paper presents a specific example of a geometric morphism that is essential, hyperconnected, and local but not locally connected, highlighting a nuanced distinction in the theory of geometric morphisms.

Contribution

It provides the first known example of such a morphism, derived from geometric morphisms between presheaf categories on monoids.

Findings

Constructs an explicit example of the morphism

Shows the morphism is hyperconnected but not locally connected

Advances understanding of geometric morphism properties

Abstract

We give an example of an essential, hyperconnected, local geometric morphism that is not locally connected, arising from our work-in-progress on geometric morphisms $\mathbf{PSh}(M) \to \mathbf{PSh}(N)$, where $M$ and $N$ are monoids.