I am trying to motivate using infinity categories by saying that there is no action $\pi_1(B,b_0) \curvearrowright C^*(F)$ where $F \hookrightarrow E \to B$ is a fibration. Define the strict infinity groupoid $\Pi_\infty(B,b_0)$ to have one zero simplex. The $n$ simplexes are singular $n-$ simplexes that map every edge to the basepoint $b_0$ of $B$.
Does $\Pi_\infty(B,b_0)$ act on $C^*(F)$? Does it act on $F$?