Koszul duality for operads allows for straightforward generalizations of $A$-infinity algebras and $A$-infinity morphisms for the so called Koszul operads $\mathcal{O}$, among which we find the associative operad. Good accounts of this can be found in standard references. However, I've been unable to find a generalization of $A$-infinity homotopies between $A$-infinity morphisms. Does anyone know of any reference(s) where this is treated in the greatest possible generality? Ideally, for Koszul operads over an arbitrary commutative ground ring, but anything is welcome.



Infinity-homotopies
I don't know if you found an answer since you posted the question, but I will write this just in case: there is a "cute" (easy) definition in case of nonsymmetric operads which generalises the A-infinity story rather trivially (derivation homotopy), while for symmetric operads the definition is more involved. There is a comparison of various possibilities in my paper with Poncin <https://link.springer.com/article/10.1007/s10485-015-9407-x> , and an explanation that these possibilities actually come from a suitable model category structure in the definitive treatment by Bruno Vallette <https://aif.centre-mersenne.org/item/AIF_2020__70_2_683_0/>


