What is the definition of homotopy flat connections? What is a definition of a homotopy flat connection - in the context of differential forms with values in a dg algebra
 A: One place where this notion appears (without a formal definition) is in a remark after Corollary 4 in
Cohen, Ralph L.; Stacey, Andrew, Fourier decompositions of Loop bundles, Goerss, Paul (ed.) et al., Homotopy theory: relations with algebraic geometry, group cohomology, and algebraic (K)-theory. Papers from the international conference on algebraic topology, Northwestern University, Evanston, IL, USA, March 24–28, 2002. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 346, 85-95 (2004). ZBL1068.55016.

Remark. The homotopy type of the map of based loop spaces, $\Omega f_\zeta: \Omega M \to U(n)$ can be obtained by taking the holonomy map of a connection on $\zeta$. Therefore this corollary can be interpreted as saying that in order for $L\zeta$ to have a Fourier decomposition, $\zeta$ must admit
a “homotopy flat” connection.

Deciphering what they are saying, my guess is that a collection $\nabla$ on a  (say) principal $G$-bundle $P\to B$  with simply-connected base $B$ is homotopy flat if the holonomy map
$$
\Omega(M)\to G 
$$
is null-homotopic. (In the case of a flat bundles this map is, of course, constant.) Maybe one of the authors, Andrew Stacey (@user:45) can confirm (or deny) that this is what they meant.
