Comparing dg category models for path spaces via $A_\infty$-functors

Abstract

We construct a many-object dual version of Chen's iterated integral map. For any topological space X, the construction takes the form of an A-infinity functor between two dg categories whose objects are the points of X: the domain has as morphisms the singular (cubical) chains on the space of (Moore) paths in X and the codomain has morphisms arising by totalizing a cosimplicial chain complex determined by the dg coalgebra of singular (simplicial) chains in X. When X is simply connected, we show this construction defines a homotopy inverse to a classical map of Adams, which sends ordered sequences of singular simplices in X linked by shared vertices to cubes of paths in X. When X is not necessarily simply connected, following an idea of Irie, we incorporate the fundamental groupoid of X into the construction and deduce analogous results. Along the way, we provide an elementary and new proof of the fact that the (direct-sum) cobar construction of the chains in X, suitably interpreted, models the dg category of paths in X, an extension of Adams's cobar theorem established by Rivera-Zeinalian using different methods.