Topologies for intermediate logics

TL;DR

This paper explores how to characterize classes of Grothendieck toposes based on their internal logic satisfying certain intermediate logic assertions, introducing analogues of classical topologies for these logics.

Contribution

It introduces natural analogues of double negation and De Morgan topologies for a broad class of intermediate logics within Grothendieck toposes.

Findings

Characterization of Grothendieck toposes with specific internal logics

Introduction of new topologies for intermediate logics

Extension of classical topologies to a wider logic class

Abstract

We investigate the problem of characterizing the classes of Grothendieck toposes whose internal logic satisfies a given assertion in the theory of Heyting algebras, and introduce natural analogues of the double negation and De Morgan topologies on an elementary topos for a wide class of intermediate logics.