# The localic Istropy group of a topos

## TL;DR

This paper introduces the localic isotropy group of a topos, a refined version of the isotropy group with improved properties, and explores its implications for geometric morphism factorizations.

## Contribution

It establishes that the isotropy group is a localic group object, enhancing functoriality and stability, and applies this to factor geometric morphisms uniquely.

## Key findings

- The isotropy group is a localic group object.
- Connected atomic morphisms are quotients by open isotropy actions.
- Any geometric morphism factors into a connected atomic part and an essentially anisotropic part.

## Abstract

It has been shown by J.Funk, P.Hofstra and B.Steinberg that any Grothendieck topos T is endowed with a canonical group object, called its isotropy group, which acts functorially on every object of T. We show that this group is in fact the group of points of a localic group object, called the localic isotropy group, which also acts on every object, and in fact also on every internal locales and on every T-topos. This new localic isotropy group has better functoriality and stability property than the original version and shed some lights on the phenomenon of higher isotropy observed for the ordinary isotropy group. We prove in particular using a localic version of the isotropy quotient that any geometric morphism can be factored uniquely as a connected atomic geometric morphism followed by a so called "essentially anisotropic" geometric morphism, and that connected atomic morphism are exactly the quotient by an open isotropy action.