Atomic Toposes with Co-Well-Founded Categories of Atoms

TL;DR

The paper provides a criterion for atomic sites that yield atomic topoi with co-well-founded categories of atoms, ensuring local finite presentability whenever the atom category stabilizes on $\omega$-chains. It applies this to derive a general description of atoms as quotients $m/G$ of representable atoms and shows that, under suitable conditions, all sheaves coincide with pullback-preserving presheaves, simplifying the atom structure. The Malitz--Gregory atomic topos is constructed as a counterexample to the conjecture that every locally finitely presentable topos has enough points, demonstrating a pointless, locally finitely presentable atomic topos built from $\mathbb{I}$-trees, and its atoms are again expressible in quotient form. The results illuminate the interplay between atom structure and presentability, and provide tools to identify when atomic topoi have simple atom descriptions and when they may fail to have 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.

Theorems & Definitions (20)

• Remark 1
• Proposition 2
• Remark 3
• Example 4
• Lemma 5
• Lemma 6
• Proposition 7
• Theorem 8
• Lemma 9
• Theorem 10