Profinite Groupoids and Presheaves

A groupoid is a category in which every morphism is an isomorphism. A group G determines a groupoid \(\mathbb{G}\) with a single formal object ★, and \(\mathbb{G}\) (★,★) = G as set of (iso)morphisms. The Galois theory of rings will use a Galois groupoid, with possibly several objects, instead of a group. A profinite groupoid will be one whose set of objects and set of morphisms are profinite spaces, while all operations are continuous. The notion of profinite presheaf on a profinite groupoid then generalizes the notion of profinite G-space, for a profinite group G. These notions and results in the profinite case are special instances of internal category theory in a category with pullbacks.