1. Overview
Let \(X\) be a topological space. Associated to any \(R\)-module \(M\) is a presheaf on \(X\), which is given by mapping every open subset to \(M\) and setting restrictions to be the identity map. The sheafification of this presheaf is called a constant sheaf on \(X\) and is denoted \(\underline{M}\). Consequently, a sheaf \(F\) on \(X\) of \(R\)-modules is said to be locally constant if for every point \(x \in X\), there exists a neighborhood \(U\) such that \(F \mid_{U}\) is isomorphic to a constant sheaf. This is equivalent to giving an open cover \(\{U_{\alpha}\}\) and \(R\)-modules \(\{M_{\alpha}\}\) which satisfy \(F \mid_{U_{\alpha}} \cong \underline{M_{\alpha}}\). We denote the category of locally constant sheaves of \(R\)-modules on \(X\) by \(\LC(X, R)\); it is an abelian category.
Now let \(\Pi(X)\) be the fundamental groupoid of \(X\) with the operator \(\star\) of path composition. We call a functor \(L \colon \Pi(X) \to R\text{-Mod}\) an \(R\)-local system on \(X\). Morphisms between local systems are given by natural transformations of functors. Write \(\LS(X, R)\) for the category of \(R\)-local systems on \(X\); it is again an abelian category. The purpose of this note is to provide a proof of the following theorem.
Theorem. Let \(X\) be a locally path-connected and locally simply connected space. Then the categories \(\LC(X, R)\) and \(\LS(X, R)\) are equivalent.
2. Proof
Again assume \(X\) to be locally path-connected and locally simply connected. We are going to construct a pair of functors
\[\begin{aligned} \CA &\colon \LC(X, R) \to \LS(X, R), \\ \CB &\colon \LS(X, R) \to \LC(X, R) \end{aligned}\]which induce a categorical equivalence. The statement can be generalized to suitable semilocally simply connected spaces, cf. [1]. In the sequel, by an \(R\)-module we mean a left \(R\)-module equipped with the discrete topology, unless specified otherwise. We do not assume covering spaces to be connected.
A continuous map \(f \colon Y \to X\) is called étale if it is a local homeomorphism. For any sheaf \(F\) we define the étale space of \(F\) as the bundle
\[p \colon \coprod_{x \in X} F_x \to X,\]where elements of \(F_x\) are mapped to \(x\). If \(s\in F(U)\) is a section, set
\[\widetilde{s}(U)=\{s_x\in F_x\mid x\in U\}.\]These sets form a basis for the topology on the étale space, and \(p\mid_{\widetilde{s}(U)}\) is a homeomorphism onto \(U\). The utility of this construction is that the sheaf of continuous sections of \(p\) is canonically isomorphic to \(F\). Hence, the categories of étale maps with codomain \(X\) and of sheaves of sets on \(X\) are equivalent, cf. [5, ch.2, §6].
In the case that \(F\) is locally constant, its étale space is a covering. Indeed, let \(U\) be a neighborhood of \(x\) such that \(F \mid_U \cong \underline{M}\). Then the étale space restricted to \(U\) takes the form \(p \colon U \times M \to U\), which is a covering since the topology on \(M\) is discrete. If we assume that \(X\) is locally connected, then this condition is also sufficient.
Lemma 1. Let \(X\) be a locally connected space. A sheaf \(F\) of \(R\)-modules on \(X\) is locally constant if and only if its étale space is a covering.
Proof. Suppose the étale space of \(F\) is a covering. By local connectedness we may choose a connected evenly covered neighborhood \(U\) of any point. Every sheet over \(U\) is determined by its value in one fiber, so the restriction of the sheaf of sections is the constant sheaf associated to that fiber. The fiberwise module operations agree with these identifications because the sum of the two sheets through \(v,w\) is the unique sheet through \(v+w\). Thus \(F\) is constant on \(U\). The converse was proved above. \(\blacksquare\)
Lemma 2. A locally constant sheaf on a simply connected, locally path-connected space is constant.
Proof. Let \(X\) be a simply connected, locally path-connected space with \(F\) locally constant. Then by lemma 1 the étale space of \(F\) covers \(X\). Every covering of such a space is a discretely indexed union of copies of \(X\), and hence we can write our étale space in the form \(p \colon X \times M \to X\). Its sheaf of sections is the constant sheaf \(\underline M\): over an open set, a section is exactly a locally constant map to the discrete set \(M\). Thus \(F\) is constant. \(\blacksquare\)
We now construct the functor \(\CA\). Fix \(F \in \LC(X, R)\) and let \(p_F\colon E_F\to X\) be its étale space. By lemma 1, \(p_F\) is a covering. Define
\[\CA(F)(x)=F_x.\]If \([\gamma]\colon x\to y\) is a morphism in \(\Pi(X)\) and \(v\in F_x\), lift \(\gamma\) uniquely to a path in \(E_F\) beginning at the point \(v\) over \(x\). Define \(\CA(F)([\gamma])(v)\) to be the endpoint of this lift in \(F_y\). The homotopy-lifting property for coverings shows that this map depends only on the endpoint-preserving homotopy class of \(\gamma\). Uniqueness of lifts gives
\[\CA(F)([\omega\star\gamma]) =\CA(F)([\omega])\circ\CA(F)([\gamma]),\]so \(\CA(F)\) is a functor. The fiberwise addition and scalar multiplication in the étale space show that each transport map is an \(R\)-module isomorphism.
If \(f\colon F\to G\) is a morphism of locally constant sheaves, it induces a continuous map of étale spaces over \(X\) whose restriction to the fiber over \(x\) is the stalk map \(f_x\colon F_x\to G_x\). This map carries lifts to lifts, so
\[f_y\circ\CA(F)([\gamma]) =\CA(G)([\gamma])\circ f_x.\]Thus the collection \((f_x)_{x\in X}\) is a natural transformation, which we define to be \(\CA(f)\).
We next construct \(\CB\). Fix \(L\in\LS(X,R)\) and begin with the set
\[E_L=\coprod_{x\in X}L(x), \qquad p_L\colon E_L\to X.\]Choose a basis of path-connected, simply connected open subsets of \(X\). Let \(U\) be such an open subset, let \(x\in U\), and let \(v\in L(x)\). For \(y\in U\), any two paths in \(U\) from \(x\) to \(y\) are homotopic relative to their endpoints. We may therefore define
\[s_{U,x,v}(y)=L([\gamma_{x,y}])(v),\]where \(\gamma_{x,y}\) is any path in \(U\) from \(x\) to \(y\). Declare the subsets
\[\widetilde{s}_{U,x,v}(U) =\{s_{U,x,v}(y)\mid y\in U\}\]to form a basis for the topology on \(E_L\). Functoriality of \(L\) shows that these sets agree after restriction to smaller simply connected neighborhoods, so the basis is well-defined. Moreover,
\[p_L\mid_{\widetilde{s}_{U,x,v}(U)} \colon \widetilde{s}_{U,x,v}(U)\longrightarrow U\]is a homeomorphism. Thus \(p_L\) is a covering, locally isomorphic over \(U\) to the projection \(U\times L(x)\to U\). Addition and scalar multiplication are continuous in these local trivializations.
Define \(\CB(L)\) to be the sheaf of continuous sections of \(p_L\). The preceding local trivialization shows
\[\CB(L)\mid_U\cong\underline{L(x)},\]so \(\CB(L)\) is locally constant. Explicitly, a section over an open subset \(V\) is a choice \(s(y)\in L(y)\) which is locally compatible with parallel transport; along every path contained in \(V\),
\[s(y)=L([\gamma])(s(x)).\]If \(\eta\colon L\to L'\) is a natural transformation, the fiber maps \(\eta_x\colon L(x)\to L'(x)\) assemble to a continuous map \(E_L\to E_{L'}\) over \(X\). Postcomposition sends a section \(s\) to \(\eta\circ s\) and defines a sheaf morphism \(\CB(\eta)\colon\CB(L)\to\CB(L')\).
Proof (Theorem). Evaluation of the germ of a section at \(x\) gives a canonical isomorphism
\[(\CB(L))_x\cong L(x).\]By the definition of the topology on \(E_L\), monodromy along a path is exactly the map assigned by \(L\). These fiberwise isomorphisms therefore give a natural isomorphism
\[\CA\circ\CB\cong\operatorname{Id}_{\LS(X,R)}.\]Conversely, when \(L=\CA(F)\), lifting paths in the étale space of \(F\) gives precisely the basic local sections used to topologize \(E_L\). Hence \(E_{\CA(F)}\) is canonically isomorphic, over \(X\) and fiberwise as an \(R\)-module, to the original étale space \(E_F\). Passing to sheaves of sections gives a natural isomorphism
\[\CB\circ\CA\cong\operatorname{Id}_{\LC(X,R)}.\]Thus \(\CA\) and \(\CB\) are inverse equivalences. \(\blacksquare\)
3. References
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- J.P. May, A concise course in algebraic topology, Chicago Lectures in Mathematics, 1999.
- Saunders Mac Lane and Ieke Moerdijk, Sheaves in geometry and logic: A first introduction to topos theory, Springer-Verlag, 1992.