In 1959, Edgar Brown introduced a twisted tensor product to give an algebraic description of a fibration. The chain complex of a total space of a principal fibration is obtained as a small perturbation (at the level of a differential) of the chain complex of the trivial fibration (hence a tensor product). It is the analogue for differential algebra of the twisted cartesian product construction in the theory of simplicial fibre bundles.

Definition

Let $C$ be a dg-coalgebra, $A$ a dg-algebra, $\tau:C\to A$ the twisting cochain, $L$ a right $C$-dg-comodule with coaction$\delta_L:L \to L\otimes C$ and $M$ a left $A$-dg-module with action $m_M:M\otimes A\to A$. The twisted tensor product$L\otimes_\tau M$ is the chain complex that coincides with the ordinary tensor product $L\otimes M$ as a graded module over the ground ring, and whose differential $d_\tau$ is given by

Brown, Edgar H., Jr. Twisted tensor products. I. Annals of Math. (2) 69 1959 223–246.

V. A. Smirnov, Simplicial and operadic methods in algebraic topology, Translations of mathematical monographs 198, AMS, Providence, Rhode Island 2001.

Kenji Lefèvre-Hasegawa, Sur les $A_\infty$-catégories, thesis, (Université Denis Diderot – Paris 7, Paris, November 2003). Corrections, by B. Keller, available here.

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