Title: 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
$\widehat{\mathcal{C}}$.