Let p:E→B be a fiber bundle over a triangulated base B with fiber F, σ simplex in B, σ↦H∗(p−1(σ))≃H∗(F) the obvious map and let S be the category of simplices in B with inclusions.
Then σ↪τ in S gives us a map S→H∗(p−1(σ))→H∗(p−1(τ)). Ie, a morphism in 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∗(ΩB)→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∞-algebras?
Details would be most welcome – this kind of thing is hard to track down in the literature…
I think one can do something like the following. Let M=map([0,1],B) and e:M→B be evaluation at 0: this is a Hurewicz fibration and a homotopy equivalence. Now form the pullback fibration e∗E→M, and consider the composite ˉE:=e∗E→M→B. This is a fibration fibrewise homotopy equivalent to your original one, and a point in the fibre ˉFb over b∈B is a path γ from b to a b1 and point in p−1(b1). There is an evident A∞ action of the A∞ space ΩbB on this fibre by composing γ with loops at b.
Thus there is an A∞ map ΩbB→End(ˉFb), 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 ΛbB.