#
The localic isotropy group of a topos

##
Simon Henry

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.

Published 2018-12-16.

http://www.tac.mta.ca/tac/volumes/33/41/33-41.pdf

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