nLab twisted tensor product

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


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 of chain complexes of the base and of the fiber). It is the analogue for differential algebra of the twisted cartesian product construction in the theory of simplicial fibre bundles.


Let CC be a dg-coalgebra, AA a dg-algebra, τ:CA\tau:C\to A the twisting cochain, LL a right CC-dg-comodule with coaction δ L:LLC\delta_L:L \to L\otimes C and MM a left AA-dg-module with action m M:MAAm_M:M\otimes A\to A. The twisted tensor product L τML\otimes_\tau M is the chain complex that coincides with the ordinary tensor product LML\otimes M as a graded module over the ground ring, and whose differential d τd_\tau is given by

d τ=d L1+1d M+(1m M)(1τ1)(δ L1). 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).


  • Edgar H. Brown 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 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

Last revised on June 11, 2023 at 12:07:07. See the history of this page for a list of all contributions to it.