I'm currently reading Differential forms in Algebraic Topology by Bott&Tu, and having some trouble understanding section II-13 Monodromy. In particular, pages 144~146.
Let $\mathcal{F}$ be locally constant presheaf with group $G$ on an open cover $\mathcal{U}=\{U_{\alpha}\}$. Then for every loop of the nerve $N(\mathcal{U})$ with a fixed basepoint, say $U_{0}$, we can associate an automorphism of $G$.
The author then claims that if a loop bounds a 2-chain, then the associated automorphism of $G$ is the identity.
I've looked up the definitions of boundaries and chains, but I still can't quite wrap my head around what the author means. Does he mean that when the restriction maps (and their inverses) form a loop, and this loop is the boundary of any 2-chain (i.e. any triangle in the pictures), then the associated automorphism is the identity?
He then quickly generalizes his claim, saying that there is a homomorphism from $\pi_{1}(N(\mathcal{U}))=\{\text{loops}\}/\{\text{bounding loops}\}$ to $Aut G$. From the looks of it, I think bounding loops should be some kind of homotopy?
This must be something basic, but would somebody be kind enough to explain? Thanks!