Every Grothendieck topos has internal to it a canonical group object, called its isotropy. We continue our investigation of this group, focusing again on locally anisotropic toposes. Such a topos is one admitting an étale 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 étendue, has isotropy rank at most one, meaning that its isotropy quotient has trivial isotropy.
Keywords: toposes, isotropy, Galois theory, inverse semigroups
2020 MSC: 18B25, 18B40, 18E50, 18F10
Theory and Applications of Categories, Vol. 37, 2021, No. 27, pp 914-939.