nLab isotropy group of a topos

Isotropy group of a topos


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Category Theory

Isotropy group of a topos


The isotropy group of a topos TT is its free loop space object I TI_T in Topos, regarded as a group object over TT. It is therefore a group topos? in the world of TT-toposes.

Specifically, this means it is the pullback of the diagonal TT×TT \to T\times T against itself, in the sense appropriate to ToposTopos 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 I TTI_T \to T is automatically localic, since it is a pullback of the diagonal TT×TT\to T\times T which is localic, and therefore corresponds to a localic group internal to TT. (In general, if TT is an nn-topos, then I TTI_T\to T will be (n1)(n-1)-localic?.) The group of points of this localic group is then a group object Z TZ_T in TT, the etale isotropy group of TT. The latter captures some but not all of the information about I TI_T.

Isotropy quotient

Like a free loop space object in any category, I TI_T acts on all toposes over TT. In particular, it acts on etale geometric morphisms over TT, and hence on objects of TT. The isotropy quotient of TT 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 Z TZ_T 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

  • Jonathon Funk, Pieter Hofstra, and Benjamin Steinberg, Isotropy and crossed toposes, TAC

It was shown to be the group of points of the localic isotropy group in

Created on June 16, 2017 at 04:36:59. See the history of this page for a list of all contributions to it.