nLab twisted tensor product

This article is about the twisted tensor product of a dg-algebra and a dg-coalgebra (and generalizations). For another notion of twisted tensor product of algebras see there.

Idea

Brown 1959 introduced twisted tensor products to give an algebraic description of fibrations: The chain complex of the total space of a principal fibration is obtained as a small perturbation (at the level of the differential) of the chain complex of the trivial fibration (being the tensor product of chain complexes of the base and of the fiber).

This 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

$$d_\tau = d_L\otimes 1 + 1\otimes d_M + (1\otimes m_M)\circ(1\otimes\tau\otimes 1)\circ(\delta_L\otimes 1).$$

Literature

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

• 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.

• Manuel Rivera, Mahmoud Zeinalian, The colimit of an ∞-local system as a twisted tensor product, Higher Structures 4(1), (2020) higher-structures4.1 arXiv:1805.01264 mpg:pdf