Weak Topologies on Toposes

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 $\mathcal{E}$.