LOCALLY ANISOTROPIC TOPOSES II

Every Grothendieck topos has internal to it a canonical group object, called its isotropy group [Funk et al., 2012]. We continue our investigation of this group, focusing again on locally anisotropic toposes [Funk and Hofstra, 2018]. Such a topos is one admitting an etale cover by an anisotropic topos. We present a structural analysis of this class of toposes by showing that a topos is locally anisotropic if and only if it is equivalent to the topos of actions of a connected groupoid internal to an anisotropic topos. In particular we may conclude that a locally anisotropic topos, whence an etendue, has isotropy rank at most one, meaning that its isotropy quotient has trivial isotropy [Funk et al., 2018].