Atomic Toposes with Co-Well-Founded Categories of Atoms

TL;DR

This paper characterizes atoms in atomic toposes using co-well-founded categories of atoms, provides criteria for their description, and explores implications for local finite presentability and points in such toposes.

Contribution

It introduces a general criterion for describing atoms in atomic toposes and applies it to specific cases, including a counter-example to a conjecture about locally finitely presentable toposes.

Findings

Atoms in certain atomic toposes can be described via pairs (n,G).

The toposes studied are locally finitely presentable.

Counter-example to the conjecture that all locally finitely presentable toposes have enough points.

Abstract

The atoms of the Schanuel topos can be described as the pairs $(n,G)$ where $n$ is a finite set and $G$ is a subgroup of $\operatorname{Aut}(n)$. We give a general criterion on an atomic site ensuring that the atoms of the topos of sheaves on that site can be described in a similar fashion. We deduce that these toposes are locally finitely presentable. By applying this to the Malitz-Gregory atomic topos, we obtain a counter-example to the conjecture that every locally finitely presentable topos has enough points. We also work out a combinatorial property satisfied exactly when the sheaves for the atomic topology are the pullback-preserving functors. In this case, the category of atoms is particularly simple to describe.