Atomic Toposes with Co-Well-Founded Categories of Atoms

Jérémie Marquès

Theory and applications of categories, Tome 44 (2025), pp. 420-438.

Résumé

The atoms of the Schanuel topos can be described as the formal quotients n/G where n is a finite set and G is a subgroup of Aut(n). We give a general criterion on an atomic site (A,Jₐₜ) ensuring that the atoms of Sh(A,Jₐₜ) can be described in a similar fashion, as the formal quotients n/G where n ∈ A and G ⊆ Aut(n) is a “valid” subgroup. It might happen that every group of automorphisms is valid in this sense, and we show that it is the case if and only if the Jₐₜ-sheaves coincide with the pullback-preserving presheaves. We show that if the criterion is satisfied and the groups Aut(n) are Noetherian, then Sh(A,Jₐₜ) is locally finitely presentable. By applying this to the Malitz-Gregory atomic topos, we obtain a negative answer to a question of Di Liberti and Rogers: Does every locally finitely presentable topos have enough points? We also provide an example of an atomic topos which is not locally finitely presentable.

Détail

Publié le : 2025-05-22

Classification : Primary: 03G30, Secondary: 18B25, 18C35
Keywords: Atomic toposes, toposes without points, nominal sets, locally finitely presentable toposes