The geometry of Coherent topoi and Ultrastructures

TL;DR

This paper explores the geometric properties of coherent topoi, demonstrating their right Kan injectivity with respect to flat embeddings, and relates this to ultrastructures on their points, with speculations on ultracategories in model theory.

Contribution

It establishes that coherent topoi are right Kan injective regarding flat embeddings and connects this to ultrastructures, offering new insights into their categorical geometry.

Findings

Coherent topoi are right Kan injective with respect to flat embeddings.

Ultrastructures on the category of points are derived from this injectivity.

Speculations on ultracategories in formal model theory are presented.

Abstract

We show that coherent topoi are right Kan injective with respect to flat embeddings of topoi. We recover the ultrastructure on their category of points as a consequence of this result. We speculate on possible notions of ultracategory in various arenas of formal model theory.