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 on a topos is no longer idempotent, when seen as a closure operator. For a given topos ε , 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 ε constitutes a complete residuated lattice provided that ε is (co)complete. Furthermore, when the weak Lawvere–Tierney topology on ε preserves binary meets, we give an explicit description of the (restricted) associated sheaf functor on ε.