### Locally Constant Sheaves

#### Posted by Mike Shulman

Urs and I recently got into a discussion about the correct definition of “locally constant ∞-sheaf.” In trying to sort it out, I realized that I don’t even know the right definition of a locally constant 1-sheaf, in general! Specifically, there are two definitions of “locally constant sheaf” which are the same when the base space (or, more generally, topos) is *locally connected*—but otherwise they’re not the same. Can anyone tell me which is the right definition in general, and why?

Here they are. Let $X$ be a topological space, for maximum familiarity and concreteness.

**Definition 1.** A sheaf $F$ on $X$ is **locally constant** if there exists an open cover $(U_i)$ of $X$ such that $F|_{U_i}$ is (isomorphic to) a constant sheaf for each $i$.

(Recall that for a set $A$, the constant sheaf $\Delta A$ on a space $U$ is the sheafification of the constant presheaf. Regarded as a local homeomorphism to $U$, it is the space $U\times A$. Its sections over an open set $V\subseteq U$ are the locally constant functions $V\to A$, i.e. functions which are continuous when $A$ has the discrete topology.)

**Definition 2.** Let $Core(Set)$ denote the groupoid of sets and isomorphisms, and $Core(Set_\ast)$ the groupoid of pointed sets and pointed isomorphisms, with forgetful functor $Core(Set_\ast) \xrightarrow{p} Core(Set)$. A **locally constant sheaf** is a pullback of $\Delta(p)$ along a map $\ast\to \Delta Core(Set)$ in the 2-category $Sh_{(2,1)}(X)$ of stacks of groupoids on $X$, where $\Delta$ denotes the constant stack functor $Gpd \to Sh_{(2,1)}(X)$.

This second definition takes a bit of unraveling; we need to think about what a constant stack looks like. Of course, it is a stackification of a constant prestack (= groupoid-valued presheaf, in this case). If $G$ is a groupoid, then a global section of the constant stack $\Delta G$ is determined by

- an open cover $(U_i)$ of $X$,
- for each $i$, an object $a_i\in G$,
- for each $i,j$, a locally constant function $g_{i j}\colon U_i \cap U_j \to Hom_G(a_i,a_j)$,
- such that $g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x)$ for any $i,j,k$ and any $x\in U_i \cap U_j\cap U_k$.

Thus, a global section of $\Delta Core(Set)$ is given by

- an open cover $(U_i)$ of $X$,
- for each $i$, a set $A_i$,
- for each $i,j$, a locally constant function $g_{i j}\colon U_i \cap U_j \to Iso(A_i,A_j)$,
- such that $g_{j k}(x) \circ g_{i j}(x) = g_{i k}(x)$ for any $i,j,k$ and any $x\in U_i \cap U_j\cap U_k$.

Note that we can regard the $g_{i j}$ as sections over $U_i \cap U_j$ of the constant sheaf $\Delta Iso(A_i,A_j)$. The corresponding sheaf on $X$ is obtained by gluing the constant sheaves $\Delta A_i$ on $U_i$ together along the isomorphisms $g_{i j}$. Note that if $X$ is connected, then all the sets $A_i$ must be isomorphic, so we could equivalently give just a single set $A$ together with locally constant functions $g_{i j}\colon U_i \cap U_j \to Aut(A)$. In other words, when $X$ is connected, a locally constant sheaf on $X$ (according to definition 2) is a principal $Aut(A)$-bundle over $X$, for some (discrete) set $A$. Equivalently, we can call this a $\Delta Aut(A)$-torsor in the topos $Sh(X)$.

Now let’s go back to definition 1. We again have an open cover $(U_i)$, and for each $i$ we have $F|_{U_i} \cong \Delta A_i$, for some set $A_i$, so this looks very similar. But now instead of the gluing data $g_{i j}\colon U_i \cap U_j \to Iso(A_i,A_j)$, we are simply given a sheaf $F$ on all of $X$ which restricts to $\Delta A_i$ on each $U_i$. Since the assignment $U\mapsto Sh(U)$ is a stack on $X$, to be given such an $F$ is equivalent to being given isomorphisms $h_{i j}\colon \Delta A_i |_{U_i\cap U_j} \cong \Delta A_j |_{U_i\cap U_j}$ which satisfy the cocycle condition. And again, if $X$ is connected, then we could equivalently give ourselves just one set $A$ and automorphisms $h_{i j} \in Aut_{Sh(X)}(\Delta A|_{U_i\cap U_j})$.

Thus, clearly the difference between definitions 1 and 2 is the difference between an *automorphism of a constant sheaf* and a *global section of a constant sheaf of automorphisms*. What is that difference? Well, since $\Delta$ preserves cartesian products, and $Set$ and $Sh(X)$ are both cartesian closed, for any sets $A$ and $B$ there is a canonical map $\Delta(B^A) \to (\Delta B)^{\Delta A}$. If this map is always an isomorphism, we call $\Delta$ a *cartesian closed functor*. And the “object of isomorphisms” $Iso(A,B)$ in any cartesian closed category can be constructed as an equalizer of a pair of maps $A^B \times B^A \rightrightarrows A^A \times B^B$, and since $\Delta$ also preserves equalizers, if it is a cartesian closed functor, then we have $\Delta Iso(A,B) \cong Iso(\Delta A,\Delta B)$ for any sets $A$ and $B$.

Now it’s a classical fact that $\Delta$ is a cartesian closed functor if and only if $X$ is locally connected. Thus, in this case we have $\Delta Iso(A,B) \cong Iso(\Delta A,\Delta B)$, which implies that the two definitions of “locally constant sheaf” are the same. (One could also give a more abstract argument.) However, if $X$ is not locally connected, then we can have $\Delta Iso(A,B) \ncong Iso(\Delta A,\Delta B)$.

For instance, suppose $X$ is the one-point compactification of $\mathbb{N}$. Then for any set $A$, a global section of $\Delta A$ is a function $\mathbb{N}\to A$ which is eventually constant. In particular, a global section of $Aut(A)$ is an eventually constant sequence of automorphisms of $A$.

On the other hand, since $\Delta A$ is also the coproduct of $A$ copies of the terminal object, a map $\Delta A \to \Delta A$ is just an $A$-indexed family of eventually constant functions $f_x\colon \mathbb{N}\to A$. In particular, an automorphism of $\Delta A$ is such a family such that for every $n$, the function $x \mapsto f_x(n)$ is an automorphism of $A$, as is the limit function $f^\infty$ defined by $f^\infty(x) = f_x(n)$ for sufficiently large $n$. (See below.)

So the difference between a global section of $\Delta Aut(A)$ and an automorphism of $\Delta A$ is that the former is an eventually constant sequence of automorphisms, whereas the latter is a sequence of automorphisms such that for each $x\in A$, the sequence of images of $x$ is eventually constant. Evidently the latter is strictly more general if $A$ is infinite, so we cannot have $\Delta Aut(A) \cong Aut(\Delta A)$.

When I first posted this, I thought I could use this to construct an example of a sheaf on the closed topologist’s sine curve satisfying definition 1 but not definition 2. However, now I don’t think my example quite works any more. But maybe some more similar but more complicated example would work.

So are the two definitions the same? If not, which is the right definition of a locally constant sheaf on a non-locally-connected space, and why?

## Re: Locally Constant Sheaves

Nice post. Haven’t had a chance to think about it yet. None the less, a question:

Isn’t there also a definition in terms of maps from the fundamental groupoid? If so, which of the two definitions you list is it equivalent to?