nLab isotropy group of a topos

Definition

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

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

Isotropy quotient

Like a free loop space object in any category, $I_T$ acts on all toposes over $T$. In particular, it acts on etale geometric morphisms over $T$, and hence on objects of $T$. The isotropy quotient of $T$ 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 $Z_T$ is trivial, but it may be strictly smaller.

References

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

• Simon Henry, The localic isotropy group of a topos, arxiv