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Let $\mathcal{C}$ be a finitely complete small category. In this paper, firstwe construct two weak (Lawvere-Tierney) topologies on the category ofpresheaves. One of them is established by means of a subfunctor of the Yonedafunctor 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 forthat the two weak topologies and also the double negation topology $\neg\neg$on $\widehat{\mathcal{C}}$ to be action preserving maps. Finally, among otherthings, we constitute an action preserving weak topology on$\widehat{\mathcal{C}}$.

Action preserving (weak) topologies on the category of presheaves