Let be a nonempty connected topological space, and denote the category of sheaves of sets on with the notation , and the category of étale spaces over as , then we see that we have an equivalence of categories with the functors , the functor taking a sheaf to it’s étale space with the natural projection map, and the functor taking an étale space over to the sheaf of continuous sections of the space over . It is a well-known fact that the subcategory of locally constant sheaves of sets on is equivalent(via and ) to covering spaces of which form a full subcategory of .
Now consider the constant sheaf on with stalks nonempty, which we denote by , and assume has a group structure compatible with the restriction morphisms of in the sense that they are all group homomorphisms. Then we see that if is the natural projection map, then each induces an automorphism on , defined pointwise for by . By looking at the basis of the topology on , one sees that is an open map, with inverse thus an automorphism of . Further as is a sheaf, for distinct we have since they must differ on the stalk of at least one point(hence the requirement of being non-empty). Thus we see that must map injectively into , known commonly as Deck Transformations.
But so that . Thus we see that any group embeds into , recovering Cayley’s theorem.