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 the topos. 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 locale and on every $T$-topos. This new localic isotropy group has better functoriality and stability property than the original version and sheds some light 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 morphisms are exactly the quotients by open isotropy actions, hence providing a form of Galois theory for general (unpointed) connected atomic geometric morphisms.
Keywords: Topos, Isotropy, localic groups
2010 MSC: 18B25, 03G30
Theory and Applications of Categories, Vol. 33, 2018, No. 41, pp 1318-1345.