Abstract
Just as every topos contains a canonical locale, its subobject classifier, every Grothendieck topos contains a canonical group, its isotropy group. In this talk, I will first explain the nature of this group, and develop some of the basic theory of isotropy quotients of toposes. After that, I will explain how isotropy groups give rise to interesting (balanced, closed) monoidal structures on certain toposes, generalizing ideas from Freyd-Yetter.