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 Galo
is 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.
We show that Connes B-operator on a cyclic differential graded k-module M is a model for the canonical circle action on the geometric realization of M. This implies that the negative cyclic homology and the periodic cyclic homology of a differential
graded category can be identified with the homotopy fixed points and the Tate fixed points of the circle action on its Hochschild complex.
We prove a trace formula in stable motivic homotopy theory over a general base scheme, equating the trace of an endomorphism of a smooth proper scheme with the Euler characteristic integral of a certain cohomotopy class over its scheme of fixed point
s. When the base is a field and the fixed points are etale, we compute this integral in terms of Morels identification of the ring of endomorphisms of the motivic sphere spectrum with the Grothendieck-Witt ring. In particular, we show that the Euler characteristic of an etale algebra corresponds to the class of its trace form in the Grothendieck-Witt ring.
