On the coalgebraic homotopy transfer theorem Let $A$ be a dg algebra, say over a field. The Homotopy Transfer Theorem says that $H(A)$ can noncanonically be given the structure of $A_\infty$-algebra, extending the induced multiplication on $H(A)$, in such a way that $A$ and $H(A)$ are quasi-isomorphic as $A_\infty$-algebras. 
If $C$ is instead a dg coalgebra then we can also transfer the coalgebra structure to a quasi-isomorphic $A_\infty$-coalgebra structure on $H(A)$. Unfortunately the relation of quasi-isomorphism is less well behaved for coalgebras than for algebras and often one wants to consider instead the notion of weak equivalence: a coalgebra morphism $C \to C'$ is called a weak equivalence if the induced map on cobar constructions $\Omega C\to\Omega C'$ is a quasi-isomorphism. Is any dg coalgebra weakly equivalent to its cohomology as an $A_\infty$ coalgebra?
 A: We worked out some answers to this question in our paper: arXiv:1904.03585 (Edit: the following answer refers to v1 of the paper on the arXiv!)
Here's the short version. There are two possible natural definitions of an $A_\infty$-coalgebra: 
(A) It is a graded vector space $V$ with maps $V \to V^{\otimes n}$, $n \geq 1$, satisfying identities exactly dual to those defining an $A_\infty$-algebra.
(B) It is a graded vector space $V$ and a square zero derivation of the tensor algebra on $V$. 
They are not equivalent; (B) is stronger than (A). In the paper we call them naive and genuine $A_\infty$-coalgebras. There is a homotopy transfer theorem for naive $A_\infty$-coalgebras: for any naive $A_\infty$-coalgebra $C$ there is a transfered naive $A_\infty$-structure on $H(C)$ which is quasi-isomorphic to the one on $C$. This is false for genuine $A_\infty$-algebra structures in general.
We say that an $A_\infty$-coalgebra $C$ is conilpotent if it admits an exhaustive filtration of the form
$$ 0 = F_0 C \subseteq F_1C \subseteq F_2C \subseteq \ldots $$
which is compatible with the coalgebra structure. We call such a filtration a positive filtration. In the conilpotent case, the notions of genuine and naive $A_\infty$-coalgebra coincide. Any filtered quasi-isomorphism of positively filtered $A_\infty$-coalgebras is a weak equivalence. If $C$ is a conilpotent $A_\infty$-coalgebra equipped with a positive filtration, and $C \to V$ is a filtered quasi-isomorphism of chain complexes, then it's possible to transfer the $A_\infty$-coalgebra structure on $C$ to an $A_\infty$-structure on $V$, so that the $A_\infty$-morphism $C \rightsquigarrow V$ is a filtered quasi-isomorphism and hence a weak equivalence. It follows that although $C$ is in general not weakly equivalent to $H(C)$ for any transferred structure, there is always a weak equivalence from $C$ to $H(\operatorname{Gr} C)$ with a transferred $A_\infty$-coalgebra structure on  $H(\operatorname{Gr} C)$. If we fix a positive filtration on $C$ then the $A_\infty$-coalgebra $H(\operatorname{Gr} C)$ with its transferred structure is uniquely determined up to a noncanonical filtered $A_\infty$-isomorphism, and it deserves to be called the filtered minimal model of $C$.
