# What is a noncommutative topos?

## TL;DR

This paper extends the concept of noncommutative frames to noncommutative Grothendieck topologies, introducing associated noncommutative toposes of sheaves of sets, thereby broadening the framework of noncommutative topology.

## Contribution

It introduces noncommutative Grothendieck topologies and constructs their corresponding noncommutative toposes, expanding the theoretical landscape of noncommutative topology.

## Key findings

- Definition of noncommutative Grothendieck topologies
- Construction of noncommutative toposes of sheaves
- Generalization of noncommutative frames

## Abstract

In 1702.04949 noncommutative frames were introduced, generalizing the usual notion of frames of open sets of a topological space. In this paper we extend this notion to noncommutative Grothendieck topologies and their associated noncommutative toposes of sheaves of sets.