One topos, many sites

We give characterizations, for various fragments of geometric logic, of the class of theories classified by a locally connected (resp. connected and locally connected, atomic, compact, presheaf) topos, and exploit the existence of multiple sites of definition for a given topos to establish some properties of quotients of theories of presheaf type.

💡 Research Summary

The paper “One topos, many sites” investigates the relationship between fragments of geometric logic and the class of theories classified by various kinds of Grothendieck toposes. The authors focus on five principal classes of toposes—locally connected, connected and locally connected, atomic, compact, and presheaf—and give precise logical characterisations of the theories each class classifies.

For locally connected toposes, the authors show that the classified theories are exactly those whose models decompose into a coproduct of connected components; this mirrors the categorical property that every object is a sum of connected subobjects. When a topos is both connected and locally connected, an additional global connectedness condition is imposed, forcing every model to be a single connected structure rather than a disjoint union.

Atomic toposes are treated by translating the categorical notion of an atom (a non‑zero object with no proper subobjects) into logical terms. The paper proves that a theory classified by an atomic topos admits a “atomic decomposition”: every subterminal object in the classifying topos splits into a family of atoms, and consequently every model can be expressed as a disjoint union of indecomposable pieces that carry independent logical information. This leads to the notion of a “completely separable theory,” distinct from ordinary complete theories.

Compact toposes are characterised by a finitary generation property. The authors demonstrate that a theory classified by a compact topos has a presentation in which all axioms and inference rules are derivable from a finite subset. This logical compactness aligns with the categorical compactness of objects and yields model‑theoretic consequences such as ultrahomogeneity: any finite partial isomorphism between substructures extends to a global automorphism.

Presheaf toposes constitute the most general case considered. The central technical tool is the “multiple‑site phenomenon”: a single presheaf topos Sh(C) can be presented by many different small categories C equipped with various Grothendieck topologies J. By exploiting this flexibility, the authors analyse quotients (or “quotient theories”) of a theory of presheaf type. They prove that adding or weakening axioms corresponds, at the topos level, to passing to a subsite (C′,J′) and to the inverse‑image part of a geometric morphism Sh(C′,J′) → Sh(C,J). Consequently, any quotient of a presheaf‑type theory remains classified by the same underlying presheaf topos, though its site of definition changes. This result provides a clean categorical explanation for why quotients preserve the presheaf nature while altering the logical presentation. Concrete examples—such as the passage from the theory of groups to its abelian quotient—illustrate the mechanism.

The paper concludes with several research directions. First, it suggests studying the intersection of locally connected and atomic properties, potentially defining a “partial atomicity” class of theories. Second, it proposes investigating “compact presheaf theories,” i.e., theories that are both compact and of presheaf type, to understand their model‑theoretic behaviour. Third, the authors advocate a systematic exploration of the landscape of sites that present a given topos, aiming to uncover new logical completeness or decidability criteria. Overall, the work deepens the bridge between categorical topology (toposes) and logical syntax (geometric theories), offering new classification theorems and a powerful method for handling quotients of presheaf‑type theories.