Some Types of Lawvere-Tierney Topologies

In this article, we start with a collection M of morphisms of a small category X , that satisfies certain conditions and we construct a subobject M of the subobject classifier Ω in the topos Set X op of presheaves. By classifying this subobject we get a morphism j : Ω → Ω. Finally, we find the equivalent conditions on M rendering j a Lawvere-Tierney topology. In fact, we extend constructing Lawvere-Tierney topologies induced from subpresheaves with principal sieves to subpresheaves with finitely generated sieves