(Non-trivial) local coefficient system which is not a bundle of groups

Let me first give the definition of a local coefficient system (see also [2, p. 257] and [3, p. 35]):

Let $X$ be a topological space. A local coefficient system is a functor from the category $\Pi_1(X)$ (= the fundamental groupoid) to the category AbGrp of abelian groups. Such a functor assigns to each $x \in X$ an abelian group $G(x)$ and to each homotopy class $\xi \in \pi_1(X; x_1, x_2)$ (the set of all endpoint-preserving homotopy classes of paths from $x_1$ to $x_2$) a homomorphism $G(\xi): G(x_2) \to G(x_1)$; these are required to satisfy

(i) if $\xi \in \pi_1(X, x) = \pi_1(X; x, x)$ is the identity, then $G(\xi): G(x) \to G(x)$ is the identity;

(ii) if $\xi \in \pi_1(X; x_1, x_2)$, $\eta \in \pi_1(X; x_2, x_3)$, then $G(\xi \eta) = G(\xi) \circ G(\eta): G(x_3) \to G(x_1)$

So my question is:

It is easy to see that every bundle of groups (defined in [1, p. 330]) is a local coefficient system, but I think the converse is not true (as stated without proof in [3, p. 35]), so I am looking for a (non-trivial) local coefficient system which is not a bundle of groups.

Please note: A good choice for the space $X$ in the above definition of a local coefficient system could for example be the Hawaiian earrings.

References:

[1] Hatcher, Allen: Algebraic Topology, 2002, https://www.math.cornell.edu/~hatcher/AT/AT.pdf, p. 330

[2] Whitehead, George: Elements of Homotopy Theory, New York: Springer, 1978 (Graduate Texts in Mathematics Vol. 61), p. 257

[3] Hutchings, Michael: Introduction to higher homotopy groups and obstruction theory, 2011, https://math.berkeley.edu/~hutching/teach/215b-2011/homotopy.pdf, p. 35

• The converse is true for spaces that have a universal cover, so counterexamples will be quite pathological. – Andy Putman Aug 1 '16 at 20:01
• An alternative definition is that a local system is a locally constant sheaf of abelian groups on $X$. This agrees with your definition for nice $X$ but I think not in general. – Qiaochu Yuan Aug 1 '16 at 23:31

Strip away the group structure and you get the simpler question: Does every functor from the fundamental groupoid of $X$ to Set correspond to a covering space (a bundle of sets)? As a special case this includes the question, does every subgroup of the fundamental group of a path-connected space come from a connected covering space.The answer to all of these is no. A standard example is when $X$ is the "Hawaiian earring" a.k.a. "clamshell space" a.k.a. "shrinking wedge of circles" (union of infinite sequence of circles in the plane with diameters tending to zero and one point in common).
I like to organize these ideas like this: There is a functor, obviously faithful, from the category $Cov(X)$ of covering spaces of $X$ to the category $Fun(\Pi_1(X),Set)$. It is not always fully faithful, but it is if $X$ is locally path-connected. In this case it is not always an equivalence of categories, but it is if $X$ is, as they say, "semi-locally simply connected".
• You should check that a necessary condition for a local coefficient system $G$ to be a bundle of groups is that every point in $X$ has a neighborhood $U$ such that for all loops $\xi \in \pi_1(U,x)$ the map $G(\xi)$ is the identity of $G(x)$. – user95545 Aug 3 '16 at 11:51
• One can write down an example with the Hawaiian earrings $X$ where the fiber at the special point is $G(x)=\prod_{j=1}^\infty \mathbb Z$, and for each loop $\xi\in \pi_1(X,x)$, the map $G(\xi)$ multiplies the $j$-th component of $G(x)$ with $(-1)^{w_j}$, where $w_j$ is the winding number of $\xi$ around the $j$-th circle. – user95545 Aug 3 '16 at 11:56