To let $ A $ Be a dg algebra, say over a field. The homotopy transfer theorem says that $ H (A) $ can not canonically be given the structure of $ A_ infty $algebra, extension of the induced multiplication $ H (A) $, So that $ A $ and $ H (A) $ are almost isomorphic than $ A_ infty $-algebras.
If $ C $ is instead a dg-Coalgebra, then we can also transfer the Coalgebra structure to a quasi-Isomorph $ A_ infty $-Kalgebra structure on $ H (A) $, Unfortunately, the relationship between quasi-isomorphism is less well-behaved for the coalgebras than it is for the algebras, and often one would rather look at the notion of weak equivalence: a coalgebra morphism $ C to C & # 39; $ is called weak equivalence when the map is induced on Cobar constructions $ Omega C to Omega C & # 39; $ is a quasi-isomorphism. Is every dg coalgebra of its cohomology weakly equivalent? $ A_ infty $ Coalgebra?