Unramified F-divided objects and the etale fundamental pro-groupoid in positive characteristic

Let X /S be a flat algebraic stack of finite presentation. We define a new & eacute;tale fundamental pro-groupoid pi(1)(X /S), generalizing Grothendieck's enlarged & eacute;tale fundamental group from SGA 3 to the relative situation. When S is of equal positive characteristic p, we prove that pi(1)(X /S) naturally arises as colimit of the system of relative Frobenius morphisms X -> X-p/S -> X-p2/S -> center dot center dot center dot in the pro-category of Deligne Mumford stacks. We give an interpretation of this result as an adjunction between pi(1) and the stack Fdiv of F -divided objects. In order to obtain these results, we study the existence and properties of relative perfection for algebras in characteristic p.