A Novel proof of Cayley’s theorem

Let X be a nonempty connected topological space, and denote the category of sheaves of sets on X with the notation \mathsf{Sh}(X), and the category of étale spaces over X as \acute{\mathrm{e}}\mathrm{t}(X), then we see that we have an equivalence of categories with the functors \mathrm{Sp}\acute{\mathrm{e}}, the functor taking a sheaf to it’s étale space with the natural projection map, and \Sigma the functor taking an étale space over X to the sheaf of continuous sections of the space over X. It is a well-known fact that the subcategory of locally constant sheaves of sets on X is equivalent(via \mathrm{Sp}\acute{\mathrm{e}}(-) and \Sigma(-)) to covering spaces of X which form a full subcategory of \acute{\mathrm{e}}\mathrm{t}(X).

Now consider the constant sheaf on X with stalks G nonempty, which we denote by C_X(G), and assume G has a group structure compatible with the restriction morphisms of C_X(G) in the sense that they are all group homomorphisms. Then we see that if p: \mathrm{Sp}\acute{\mathrm{e}}(C_X(G)) \rightarrow X is the natural projection map, then each g \in C_X(G)(X) \cong G induces an automorphism \hat{g} on \mathrm{Sp}\acute{\mathrm{e}}(C_X(G)), defined pointwise for f \in p^{-1}(x) by \hat{g}(f) := g_x f \in p^{-1}(x). By looking at the basis of the topology on \mathrm{Sp}\acute{\mathrm{e}}(C_X(G)), one sees that \hat{g} is an open map, with inverse \hat{g^{-1}} thus an automorphism of p. Further as C_X(G) is a sheaf, for distinct g,h \in C_X(G)(X) we have \hat{g} \neq \hat{h} since they must differ on the stalk of at least one point(hence the requirement of X being non-empty). Thus we see that G must map injectively into \textrm{Aut}_{\textsf{Set}}(p), known commonly as Deck Transformations.

But \mathrm{Sp}\acute{\mathrm{e}}(C_X(G)) = \coprod_G X so that \textrm{Aut}_{\textsf{Set}}(p) = \mathfrak{S}_G. Thus we see that any group G embeds into \mathfrak{S}_G, recovering Cayley’s theorem.