Weak Topologies on Toposes

TL;DR

This paper explores weak Lawvere-Tierney topologies on toposes, revealing their lattice structure and characterizing associated sheaf functors when certain preservation properties hold.

Contribution

It introduces the concept of weak Lawvere-Tierney topologies, analyzes their algebraic structure, and describes sheaf functors under specific conditions.

Findings

Weak Lawvere-Tierney topologies form a complete residuated lattice.

Composition of two such topologies is not necessarily idempotent.

Explicit description of associated sheaf functors when meets are preserved.

Abstract

This paper deals with the notion of weak Lawvere-Tierney topology on a topos. Our motivation to study such a notion is based on the observation that the composition of two Lawvere-Tierney topologies is no longer idempotent, when seen as a closure operator. For a given topos $\mathcal{E}$, in this paper we investigate some properties of this notion. Among other things, it is shown that the set of all weak Lawvere-Tierney topologies on $\mathcal{E}$ constitutes a complete residuated lattice provided that $\mathcal{E}$ is (co)complete. Furthermore, when the weak Lawvere-Tierney topology on $\mathcal{E}$ preserves binary meets we give an explicit description of the (restricted) associated sheaf functor on $\mathcal{E}$.