# Action preserving (weak) topologies on the category of presheaves

## TL;DR

This paper constructs and analyzes action-preserving weak topologies on the category of presheaves over a finitely complete small category, introducing new methods using subfunctors and admissible classes.

## Contribution

It introduces two new weak Lawvere-Tierney topologies on presheaf categories and studies their action-preserving properties, including conditions related to the double negation topology.

## Key findings

- Constructed two weak topologies using subfunctors and admissible classes.
- Identified conditions for these topologies to be action preserving.
- Established an action preserving weak topology on presheaf categories.

## 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 $\widehat{\mathcal{C}}$.