  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.
