Lift Chen’s and Hain’s iterated-integral map via oo-categorical functors

Establish that, for k = R and a pointed smooth manifold (M, x), the composite canonical map induced by the oo-functors S → cCAlg(k) → CAlg(k)^op, namely Hom_S(pt, {x} ×_M M) → Hom_{CAlg(R)^op}(A(pt), A({x} ×_M M)) → Hom_{CAlg(R)^op}(A(pt), A({x}) × A(M)), is a lift of the classical iterated-integral map appearing in Chen’s theorem and Hain’s theorem that connects the (rational) homotopy of M with the (co)homology of the smooth loop space Loop(M, x).

Background

In Section 4 the paper revisits Chen’s and Hain’s results that relate rational homotopy groups of a smooth manifold to (co)homology of its loop space via iterated integrals, and connects these to the simplicial iterated integral framework developed earlier in the paper.

The author then proposes an oo-categorical perspective: using the sequence of oo-functors from spaces to cocommutative coalgebras and then to commutative algebras, one obtains canonical maps between Hom-sets. The expectation is that, when specialized to k = R and to a pointed smooth manifold (M, x), this abstract construction should recover (lift) the classical iterated-integral map underlying Chen’s and Hain’s theorems.

The text explicitly notes that this expectation lacks a proper proof and marks it as future work, thereby formulating an open problem to verify that the oo-categorical map indeed coincides with the classical iterated-integral map in the real-coefficient case.