nLab twisting cochain

Twisting cochains

Twisting cochains


Let (C,d C)(C,d_C) be a dg-coalgebra with comultiplication Δ\Delta and (A,d A)(A,d_A) a dg-algebra with multiplication μ\mu. A twisting cochain is a morphism τ:CA[1]\tau:C\to A[1] such that the following Maurer-Cartan equation in the convolution algebra holds:

d Aτ+τd C+μ(ττ)Δ=0.d_A\circ\tau+\tau\circ d_C+\mu\circ(\tau\otimes\tau)\circ\Delta = 0.

Notice that the last, perturbation term describes the square ττ\tau\star\tau in the convolution algebra of homogeneous maps in Hom(C,A)\mathrm{Hom}(C,A).

Relation to the adjunction bar-cobar

Let Cogc\mathrm{Cogc} be the category of cocomplete dg-co(al)gebras and Alg\mathrm{Alg} the category of dg-algebras. There is a bar-construction functor B:AlgCogcB :\mathrm{Alg}\to\mathrm{Cogc} which is a right adjoint to the cobar-construction functor Ω:CogcAlg\Omega:\mathrm{Cogc}\to\mathrm{Alg}. Starting from a map fCogc(C,BA)f\in\mathrm{Cogc}(C,B A), one constructs a twisting cochain τ f\tau_f by postcomposing f:CBAf: C\to B A by the natural projection BAA[1]B A\to A[1]; the Maurer-Cartan equation for τ f\tau_f translates to saying that ff is a chain map, d BAf=fd Cd_{B A}\circ f = f\circ d_C. One then replaces τ f\tau_f by the composition of the evident canonical map τ 0:ΩCC[1]\tau_0:\Omega C\to C[-1] (called the canonical twisting cochain) and τ f[1]:C[1]A\tau_f[-1]:C[-1]\to A to obtain a morphism f:ΩCAf':\Omega C\to A. The Maurer–Cartan equation for τ\tau is equivalent also to saying that ff' is a chain map, i.e. d Af=fd ΩCd_A\circ f'=f'\circ d_{\Omega C}.

Some usages of twisting cochains

A twisting cochain is a datum used to define the twisted tensor product L τML\otimes_\tau M for any right CC-comodule LL and any left AA-module MM, as well as the twisted module of homomorphisms Hom τ(N,P)\mathrm{Hom}_\tau(N,P) where NN is a left CC-dg-comodule and PP a left AA-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 τ\tau, one always has a pair of adjoint functors τA\otimes_\tau A and τC\otimes_\tau C between the derived category of modules over AA and the coderived category of comodules over CC (where CC is in Cogc\mathrm{Cogc} 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 τ 0ΩC\otimes_{\tau_0}\Omega C where τ 0:ΩCC[1]\tau_0:\Omega C\to C[-1] is the canonical twisting cochain). This pair of adjoint functors is an adjoint equivalence iff the composition ΩCC[1]\Omega C\to C[-1] by τ[1]:C[1]A\tau[-1]:C[-1]\to A (compare reasoning above) is a quasi-isomorphism. This can also be expressed by saying that the canonical map

A τC τAA A\otimes_\tau C\otimes_\tau A\to A

is a quasiisomorphism. In that case, Keller calls the triple (C,A,τ)(C,A,\tau) the Koszul–Moore triple. Lefevre-Hasegawa’s thesis (pdf) asserts that in that case AA determines CC up to a weak equivalence (defined above) and CC determines AA up to a quasi-isomorphism. Moreover,

H *C=Tor * A(k,k)andH *A=Ext C(k,k)H_* C = \mathrm{Tor}^A_*(k,k)\quad \text{and}\quad H^* A = \mathrm{Ext}_C(k,k)

where kk 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.

Alternative guises

The work of O’Brian, Toledo, and Tong throughout the 70s and 80s focused on a Čech version of twisting cochains, using them as a tool to circumvent the non-existence of global resolutions of coherent analytic sheaves by vector bundles in the holomorphic setting. The definition of a twisting cochain in this sense is related to the one above, but not obviously the same. For them, a twisting cochain is a Maurer–Cartan element in some Čech bicomplex given by the endomorphism algebra of a graded vector bundle.

The definition by Bondal and Kapranov (BK91) of a twisted complex turns out to be a generalisation of the O’Brian–Toledo–Tong notion. They show that, given an arbitrary (not necessarily pretriangulated) dg-category 𝒜\mathcal{A}, the “smallest” dg-category 𝒜\mathcal{A}' that contains it and in which there exist shifts and functorial cones is exactly the category of twisted complexes in 𝒜\mathcal{A}. Furthermore, if 𝒜\mathcal{A} is actually pretriangulated then this embedding is a quasi-equivalence (thus allowing us to pull back the shifts and cones, which give a triangulated structure on the homotopy category of 𝒜\mathcal{A}).

For a history and exposition of this side of the twisting cochain story, see e.g. (Hosgood, §G).


John C. 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.

A twisting function is an analogue of a twisting cochain in the context of simplicial sets.

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 A A_\infty-categories, Grothendieck duality on complex manifolds (Toledo-Tong) and so on.

An old query archived here.


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  • Henri Gillet, The KK-theory of twisted complexes, in “Applications of algebraic KK-theory to algebraic geometry and number theory”, Part I, II (Boulder, Colo., 1983), 159–191, Contemp. Math., 55, AMS 1986.

  • Richard M. Hain, Twisted cochains and duality between minimal algebras and minimal Lie algebras, Trans. AMS 277, 1 (1983) 397–411.

Last revised on April 18, 2024 at 15:07:46. See the history of this page for a list of all contributions to it.