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...



Homology of bundles over a triangulated base and $A_\infty$-algebras
I think one can do something like the following. Let $M = map([0,1], B)$ and $e:M \to B$ be evaluation at 0: this is a Hurewicz fibration and a homotopy equivalence. Now form the pullback fibration $e^\*E \to M$, and consider the composite $\bar{E} := e^\*E \to M \to B$. This is a fibration fibrewise homotopy equivalent to your original one, and a point in the fibre $\bar{F}\_b$ over $b \in B$ is a path $\gamma$ from $b$ to a $b\_1$ and point in $p^{-1}(b\_1)$. There is an evident $A\_\infty$ action of the $A\_\infty$ space $\Omega\_b B$ on this fibre by composing $\gamma$ with loops at $b$.


Thus there is an $A\_\infty$ map $\Omega\_b B \to End(\bar{F}\_b)$, which should give you what you want.


Doing the usual Moore loop tricks, one can find an equivalent fibration with an actual action of the grouplike monoid of Moore loops $\Lambda\_b B$.


