Let $G$ be a topological group. Here are two definitions of $G$-equivariant sheaves on a $G$-space $X$.
(a) Define an $G$-equivariant sheaf by a sheaf $F$ (étalé space) equipped with a $G$-action compatible with the natural projection $F \to X$.
(b) Following Bernstein-Lunts (Part I §0), we consider (part of) the simplicial space $X_\bullet$ associated to the $G$-space: $X_n := G^n \times X$, $s_0:X_0 \to X_1$ defined by $s_0(x)=(1,x)$ and $d_i:X_n \to X_{n-1}$ defined by
- $d_0(g_1,\dots,g_n,x) = (g_2,\dots,g_n,g_1^{-1}x)$,
- $d_i(g_1,\dots,g_n,x) = (g_1,\dots,g_ig_{i+1},\dots,g_n,x)$ if $0<i<n$,
- $d_n(g_1,\dots,g_n,x) = (g_1,\dots,g_{n-1},x)$.
Define a $G$-equivariant sheaf by a pair $(F,\theta)$, where $F$ is a sheaf on $X$ and $\theta:d_1^*F \to d_0^*F$ is an isomorphism of sheaves satisfying the cocycle conditions: $d_2^*\theta \circ d_0^*\theta = d_1^*\theta$ and $s_0^*\theta = \mathrm{id}_F$.
At the next page, they make a remark on the definition: If $G$ is a discrete group, then Definition (a) is equivalent to Definition (b).
Question: Why is not (a)=(b) true for a non-discrete groups $G$ (e.g. classical Lie groups)?
I thought that (a), which is the most natural definition, was always equivalent to (b), which is a natural definition in the context of simplicial spaces. What is the difference?