  We introduce and study the category of twisted modules over a triangular
differential graded bocs. We show that in this category idempotents split, that
it admits a natural structure of a Frobenius category, that a twisted module is
homotopically trivial if and only if its underlying complex is acyclic, and
that any homotopy equivalence of differential graded bocses determines an
equivalence of the corresponding homotopy categories of twisted modules. The
category of modules over an $A_{\infty}$-algebra is equivalent to the category
of twisted modules over a triangular differential graded bocs, so all preceding
statements lift to the former category.
