Action preserving (weak) topologies on the category of presheaves

Abstract: Let $\mathcal{C}$ be a finitely complete small category. In this paper, first we construct two weak (Lawvere-Tierney) topologies on the category of presheaves. One of them is established by means of a subfunctor of the Yoneda functor and the other one, is constructed by an admissible class on $\mathcal{C}$ and the internal existential quantifier in the presheaf topos $\widehat{\mathcal{C}}$. Moreover, by using an admissible class on $\mathcal{C},$ we are able to define an action on the subobject classifier $\Omega$ of $\widehat{\mathcal{C}}$. Then we find some necessary conditions for that the two weak topologies and also the double negation topology $\neg\neg$ on $\widehat{\mathcal{C}}$ to be action preserving maps. Finally, among other things, we constitute an action preserving weak topology on $\widehat{\mathcal{C}}$.