Notice that the last, perturbation term describes the square in the convolution algebra of homogeneous maps in .
Let be the category of cocomplete dg-co(al)gebras and the category of dg-algebras. There is a bar-construction functor which is a right adjoint to the cobar-construction functor . Starting from a map , one constructs a twisting cochain by postcomposing by the natural projection ; the Maurer-Cartan equation for translates to saying that is a chain map, . One then replaces by the composition of the evident canonical map (called the canonical twisting cochain) and to obtain a morphism . The Maurer–Cartan equation for is equivalent also to saying that is a chain map, i.e. .
A twisting cochain is a datum used to define the twisted tensor product for any right -comodule and any left -module , as well as the twisted module of homomorphisms where is a left -dg-comodule and a left -dg-module.
B. Keller and his student Kenji Lefèvre-Hasegawa have shown that Koszul duality is closely related to twisting cochains. Given a twisting cochain , one always has a pair of adjoint functors and between the derived category of modules over and the coderived category of comodules over (where is in and the coderived category is just the localization of the category of complexes of comodules at the class of weak equivalences, which are by definition those morphisms which became quasi-isomorphisms after applying where is the canonical twisting cochain). This pair of adjoint functors is an adjoint equivalence iff the composition by (compare reasoning above) is a quasi-isomorphism. This can also be expressed by saying that the canonical map
is a quasiisomorphism. In that case, Keller calls the triple the Koszul–Moore triple. Lefevre-Hasegawa’s thesis (pdf) asserts that in that case determines up to a weak equivalence (defined above) and determines up to a quasi-isomorphism. Moreover,
where is the ground field. Notice that such a formulation of Koszul duality using coalgebras and coderived categories avoids various finiteness conditions present when Koszul duality is phrased as relating algebras to algebras.
Moore was one of the people who studied the subject of ‘differential coalgebra’, including twisting cochains, in the 1960s and 1970s and gave a survey of the area during his ICM address.
There are variants of the notion of twisting cochain in a variety of other contexts.
Apart from original usage for the algebraic models for fibrations, twisting cochains and variants are used in homological perturbation theory (sometimes abbreviated HPT), rational homotopy theory, deformation theory, study of -categories, Grothendieck duality on complex manifolds (Toledo-Tong) and so on.
An old query archived here.
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