Traditionally, $A$-$\infty$ algebras are defined as objects $A$ in the monoidal category of graded modules over the base field or ring with a collection of operations $m_n: A^n \to A$, where the first operation $m_1$ gives a differential on $A$. We introduce more general notions of $A$-$\infty$ algebras, coalgebras, modules, and comodules in an arbitrary DG monoidal category or, more generally, a DG bicategory. The operations now do not include the differential, instead our definition uses the language of unbounded twisted complexes which implicitly makes use of the intrinsic differential of each object as afforded by the Yoneda embedding. For A-infinity algebras and their modules, we introduce the notions of strong homotopy unitality and construct the Free-Forgetful homotopy adjunction, the Kleisli category and the derived category. Analogous constructions, with some subtleties, exist for A-infinity coalgebras and comodules. Finally, we define the notion of homotopy adjoint A-infinity coalgebra and algebra, and prove for these the derived comodule-module equivalence. One of the principal applications is to the DG bicategories of DG categories and of enhanced triangulated categories, giving rise to the notion of A-infinity monad/comonad and enhanced exact monad/comonad. Given an adjoint pair $(F,R)$ of enhanced functors, we write down strictly associative but strong homotopy unital enhancements of adjunction monad $RF$ and comonad $FR$. Given an adjoint triple $(L,F,R)$ we show that enhanced comonad $LF$ and enhanced monad $RF$ are homotopy adjoint and hence derived comodule-module equivalent.

This is joint work with Rina Anno (Kansas) and Sergey Arkhipov (Aarhus).