I am struggling to understand section 2.1., page 20, of these notes by Kontsevich and Soibelman. They read (my comments and questions are inserted in bold):
2.1.Local systems Let $X$ be a topological space (say, a CW complex), $G$ a Lie group. We denote by $G^{\delta}$ the group $G$ equipped with the discrete topology. We will refer to $G^{\delta}$-bundles as "local systems". [Are $G^{\delta}$-bundles the same as principal $G^{\delta}$-bundles?]. One can see three different descriptions of local systems.
A. Sheaf theoretic A local system is given by a covering $U_i$ of $X$ by open sets, transition functions $\gamma_{ij}:U_i\cap U_j\rightarrow G$ [Is $\gamma_{ij}$ required to be continuous and if so with respect to which topology?] which are locally constant and satisfy the $1$-cocycle condition $g_{ij}g_{jk}g_{ki}=id$ [This should probably read $\gamma_{ij}\gamma_{jk}\gamma_{ki}$ and not $g_{ij}g_{jk}g_{ki}$?]. Equivalence of local systems is given by a common refinements of two coverings and a family of maps to $G$ which conjugate one system of transition functions to the other.
B. Group theoretic. Suppose that $X$ is connected. Then equivalence classes of local systems are in one-to-one correspondence with the equivalence classes of homomorphisms of the fundamental group $\pi_1(X)$ to $G$. (If $X$ is not connected, one can use the fundamental groupoid instead of the fundamental group.)
How does this tie in with the definition of local system here? The definition looks similar, but not quite the same.
How are definitions A and B equivalent?
References welcome.
