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 iff 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 the preceding statements lift to the former category.

我们介绍并研究了三角形微分分级 boc 上的扭曲模块的类别。我们证明了在这个范畴中幂等分裂，它承认一个 Frobenius 范畴的自然结构，一个扭曲模块是同调平凡的，如果它的基础复合体是无环的，并且微分分级 bocses 的任何同伦等价决定了相应同伦的等价扭曲模块的类别。模块的一个以上的类别\（A _ {\ infty} \） -代数相当于扭曲模块的上方的三角形差分级BOCS类别，因此，所有前述语句提升到前一类。