Higher Galois theory

We generalize toposic Galois theory to higher topoi. We show that locally constant sheaves in a locally (n-1)-connected n-topos are equivalent to representations of its fundamental pro-n-groupoid, and that the latter can be described in terms of Galois torsors. We also show that finite locally constant sheaves in an arbitrary infinity-topos are equivalent to finite representations of its fundamental pro-infinity-groupoid. Finally, we relate the fundamental pro-infinity-groupoid of 1-topoi to the construction of Artin and Mazur and, in the case of the etale topos of a scheme, to its refinement by Friedlander.