Homological Algebra – The Carbonaceous Homotopy Transfer Theorem

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?