Unbounded twisted complexes

We define unbounded twisted complexes and bicomplexes generalising the notion of a (bounded) twisted complex over a DG category [BK90]. These need to be considered relative to another DG category B admitting countable direct sums and shifts. The resulting DG category of unbounded twisted complexes has a fully faithful convolution functor into Mod-B which factors through B if the latter is closed under twisting. As an application, we rewrite definitions of A∞-structures in terms of twisted complexes to make them work in an arbitrary monoidal DG category or a DG bicategory.