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-algebra, $A$ a dg-coalgebra, $\tau:C\to A$ 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