Let $p:E \to B$ be a fiber bundle over a triangulated base $B$ with fiber $F$, $\sigma$ simplex in $B$, $\sigma \mapsto H_{*}(p^{-1}(\sigma)) \simeq H_{*}(F)$ the obvious map and let $\mathcal{S}$ be the category of simplices in $B$ with inclusions.

Then $\sigma \hookrightarrow \tau$ in $\mathcal{S}$ gives us a map $\mathcal{S} \to H_{*}(p^{-1}(\sigma)) \to H_{*}(p^{-1}(\tau))$. Ie, a morphism in $\mathcal{S}$ gives us an element of $End(H_{*}(F))$

What I'd like to do in this set-up is now construct a map $H_{*}(\Omega B) \to End(H_{*}(F))$ using something like the monodromy representation.

(1) Does this map exist? I'd really love to see a construction.

(2) If the answer to (1) is "yes", is this then a map of $A_{\infty}$-algebras?

Details would be most welcome - this kind of thing is hard to track down in the literature...