My question is about the relationship between the free loop space LX of a space X and the (appropriately defined) Borel construction $PX \times_{\Omega X} \Omega X$ which is a homotopy equivalent fibration. The shortest formulation of my question is:

Is this homotopy equivalence fibre-wise deloopable? (equivalently) Is it a fibre-wise $A_\infty$ equivalence?

So,

The claim is that there is a model for the free loop space LX given by the Borel construction. For this we take based Moore loops in X and Moore paths in X starting at the base point. The Borel construction is then the space of equivalence classes $[p,\alpha]$ where $p$ is a Moore path in X, $\alpha$ is a Moore loop at the base point. The equivalence relation is given by $[p \beta, \alpha] = [p, \beta \alpha \beta^{-1}]$ (where $\beta$ is also a loop at base point).

What is the homotopy equivalence between this and LX? Is this homotopy equivalence fibre wise an $A_\infty$ equivalence? What is the algebraic structure on the fibres of the Borel space?

There is a natural map we can write down from the Borel space at least to the space of free **Moore** loops in which we map $[p,\alpha]$ to $p\alpha p^{-1}$. However the the image of this map is not even closed under composition in each fibre! (just regard the composition of $p\alpha p^{-1}$ and $p'\alpha p^{'-1}$ where $p$ and $p'$ are different.

a.