The Nori fundamental gerbe of a fibered category

We give a condition that ensures that a fibered category over a field admits a universal morphism to a profinite gerbe. This fundamental gerbe generalizes both Nori's fundamental group scheme and Deligne's relative fundamental groupoid. Using a simplified notion of essentially finite bundle, we also give a tannakian construction. As an application, we show how the fundamental gerbe enables to formulate a version of Grothendieck's section conjecture in arbitrary characteristic. We then study various natural quotients of the fundamental gerbe.