The isotropy group of a topos is its free loop space object in Topos, regarded as a group object over . It is therefore a group topos? in the world of -toposes.
Specifically, this means it is the pullback of the diagonal against itself, in the sense appropriate to as a higher category.
This definition makes sense for (Grothendieck) 1-toposes and also for higher toposes. But in the 1-topos case, the map is automatically localic, since it is a pullback of the diagonal which is localic, and therefore corresponds to a localic group internal to . (In general, if is an -topos, then will be -localic?.) The group of points of this localic group is then a group object in , the etale isotropy group of . The latter captures some but not all of the information about .
Like a free loop space object in any category, acts on all toposes over . In particular, it acts on etale geometric morphisms over , and hence on objects of . The isotropy quotient of is the full subcategory of objects for which this action is trivial.
The isotropy quotient is contained in the etale isotropy quotient, namely the full subcategory of objects for which the action of the etale isotropy group is trivial, but it may be strictly smaller.
The etale isotropy group of a 1-topos (then called just the “isotropy group”) was defined in
It was shown to be the group of points of the localic isotropy group in
Created on June 16, 2017 at 08:36:59. See the history of this page for a list of all contributions to it.