\(\DeclareMathOperator{\LC}{LC} \DeclareMathOperator{\LS}{LS} \newcommand{\CA}{\mathcal{A}} \newcommand{\CB}{\mathcal{B}} \DeclareMathOperator{\Aut}{Aut}\)
Abstract. Let \(X\) be a locally simply connected space. We prove that the categories of locally constant sheaves and local systems on \(X\) are equivalent.
Table of Contents
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 a constant sheaf. This is equivalent to giving an open cover \(\{U_{\alpha}\}\) and \(R\)-modules \(\{M_{\alpha}\}\) which satisfy \(F \mid_{U_{\alpha}} = \underline{M_{\alpha}}\). 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 simply-connected space. Then the categories \(\LC(X, R)\) and \(\LS(X, R)\) are equivalent.
2. Proof
Again assume \(X\) to be locally simply-connected. We are going to construct a pair of functors
\[\begin{aligned} \CA & = \LC(X, R) \to \LS(X, R) \\ \CB & = \LS(X, R) \to \LC(X, R) \end{aligned}\]which induce a categorical equivalence. Our proof can be generalized semi-locally 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 elsewise. We are not assuming coverings to be path-connected. In particular, a cover is universal if every other path-connected cover factors through it.
A continuous map \(f \colon Y \to X\) is called étale if it is a local homeomorphism. For any sheaf \(F\) we define 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 \(x\). This map is indeed étale: for any germ \((U, m)\) the set \(\coprod_{x \in U} \{m\}\) is homeomorphic to \(U\). The utility of this construction is that the sheaf of sections of \(p\) is isomorphic to \(F\). Hence, it is not hard to show that 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 \(S \mid_U = \underline{M}\). Then the étale space restricted to \(U\) takes the form \(p \colon U \times M \to U\), which is a cover by the assumption that 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. Then every point \(x \in X\) has a neighborhood \(U\) such that the restriction is \(p \colon U \times M \to U\). Assuming \(U\) to be connected implies that the sections of this map are constant since the topology on \(M\) is discrete, and thus our sheaf is constant on \(U\). \(\blacksquare\)
Lemma 2. A locally constant sheaf on a simply connected space is constant.
Proof. Let \(X\) be a simply connected space with \(F\) locally constant. Then by lemma 1 the étale space of \(F\) covers \(X\). The only covers of a simply connected space are a discretely indexed copies of itself, and hence we can write our étale space in the form \(p \colon X \times M \to X\). Since \(X\) is connected, the sections of this map are constant, meaning our sheaf is constant. \(\blacksquare\)
Now we are ready to construct are functors and start with constructing \(\CA\). Fix an element \(F \in \LC(X, R)\) and define \(\CA(F) = L\) as follows. For each \(x \in X\) set \(L(x) = F_x\). Next take a homotopy class of paths \([\gamma]\) from \(x\) to \(y\) and choose a representative \(\gamma\). Since \(\gamma\) has compact image, there exists a finite open cover \(U_1, \dots, U_n\) and a partition \(t_0, \dots, t_n\) of \([0, 1]\) such that
- \(\gamma([t_{j-1}, t_j])\) is contained in \(U_j\), and
- \(F \mid_{U_j}\) is the constant sheaf \(\underline{M}\) on each \(U_j\) for some fixed \(M\).
Observe that \(F \mid_{U_j}\) being constant implies that for any \(a, b \in U_j\), there exists a canonical isomorphism \(F_a \cong F(U_j) \cong F_b\). Applying this to the sequence \(\gamma(t_j)\) yields a chain of isomorphisms, which compose to a map \(L([\gamma]) \colon F_x \to F_y\).
It is not hard to see that this isomorphism does not depend on the choice of partition. We do need to show any other representative \(\omega \in [\gamma]\) induces the same map. Indeed, let \(h \colon [0, 1]^2 \to X\) be a homotopy with \(h(0, t) = \gamma(t)\) and \(h(1, t) = \omega(t)\). Since the image of \(h\) is compact, we can choose a partition \(s_0, \dots, s_n\) of \([0, 1]\) such that \(F\) is constant on \([s_j, s_{j+1}] \times [s_k, s_{k+1}]\) for \(0 \leq j, k \leq n-1\). Set \(\gamma_j = h(s_j, t)\) so that \(\gamma_0 = \gamma\) and \(\gamma_n = \omega\). Then \(\gamma_j\) and \(\gamma_{j+1}\) induce the same isomorphism, because they are contained in the same sequence of locally constant neighborhoods. We conclude that our isomorphism is homotopy invariant.
Now let \(f \colon F \to G\) be a map of locally constant sheaves on \(X\). We define \(\CA(f)\) to be the induced map on stalks \(f_x \colon F_x \to G_x\). To show this is a natural transformation, we can restrict ourselves to open subsets \(U\) where \(F\) and \(G\) are both constant sheaves \(\underline{M}\) and \(\underline{N}\), respectively. Our morphism \(f\) is induced by an \(R\)-module homomorphism \(f_0 \colon M \to N\). Then our result follows from the following diagram commuting:
\[\begin{CD} F_x @>\sim>> F(U) @>\sim>> F_y \\ @VVf_0V @VVf_0V @VVf_0V \\ G_x @>\sim>> F(U) @>\sim>> G_y \end{CD}\]Next fix an element \(L \in \LS(X, R)\) and define \(\CA(L) = F\) as follows. Since \(X\) is locally simply connected it has a universal cover \(\varphi \colon \widetilde{X} \to X\), cf. 4, [ch. 3, §8]. The functor \(L\) pulls back to \(\widetilde{X}\) by \(L^{\ast} = L \circ \varphi\), giving us the bundle
\[\nu \colon \coprod_{x \in \widetilde{X}} L^{\ast}(x) \to \widetilde{X}.\]The fundamental groupoid \(\Pi(X)\) acts on \(\widetilde{X}\) by considering the orbits of \(\pi_1(X, x)\) for differing \(x\). In particular, \(\widetilde{X}/\Pi(X) = X\). Likewise, \(\Pi(X)\) acts on each \(L^{\ast}(x)\) via the induced map \(L \colon \pi_1(X, x) \to \Aut(L^{\ast}(x))\). This allows us to form the quotient bundle
\[\overline{\nu} \colon [\coprod_{x \in \widetilde{X}} L^{\ast}(x)] /\Pi(X) \to X.\]We set \(F\) to be its sheaf of sections. Explicitly, elements of \(F(U)\) are sections \(s \colon U \to \coprod_{x \in U} L(x)\), such that for any class of paths \([\gamma]\) from \(x\) to \(y\), we get that \(s(y) = L([\gamma])(s(x))\). Our sheaf being locally constant follows from \(X\) being locally simply connected (lemma 2).
Finally suppose we are given a natural transformation \(\eta \colon L \to L'\) between local systems. Then \(\eta\) induces a morphism \(\CB(\eta) \colon \CB(L) \to \CB(L')\) by pulling back sections, i.e. for each open subset \(U\), we map \(s \in \CB(L)(U)\) to \(\eta \circ s\). This is indeed a morphism of sheaves since \(\eta\) is functorial.
Proof (Theorem). Tracing through our definitions, it is not hard to see that \((\CA \circ \CB)(L) = L\) and \((\CB \circ \CA)(F) = F\). \(\blacksquare\)
3. References
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- bavajee, Why are local systems and representations of the fundamental group equivalent. Link. (version: 2016-02-07).
- Alexandru Dimca, Sheaves in topology, Springer, 2004.
- 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.