# 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.