From coextensive varieties to the Gaeta topos

TL;DR

This paper explores properties of coextensive varieties, showing the functoriality and representability of central elements, and establishing the Gaeta topos as a classifier for certain models.

Contribution

It demonstrates that in coextensive varieties, the central elements form a functorial, representable assignment, and links the Gaeta topos to classifying central-free models.

Findings

Central elements form a functorial, representable assignment in coextensive varieties.

Finitely presented algebras in coextensive varieties are also coextensive.

The Gaeta topos classifies central-free models in (0,1)-dense coextensive varieties.

Abstract

In this paper, we show that in every coextensive variety V, the assignment that maps each algebra to its set of central elements is both functorial and representable. Furthermore, we prove that the full subcategory of finitely presented algebras in V is coextensive. Finally, we establish that if V is additionally (0, 1)-dense, the Gaeta topos classifies central-free V-models.