Let $(C,d_C)$ be a dg-coalgebra with comultiplication $\Delta$ and $(A,d_A)$ a dg-algebra with multiplication $\mu$. A twisting cochain is a morphism $\tau:C\to A[1]$ such that the following Maurer-Cartan equation holds:
Notice that the last, perturbation term describes the square $\tau\star\tau$ in the convolution algebra of homogeneous maps in $\mathrm{Hom}(C,A)$.
Let $\mathrm{Cogc}$ be the category of cocomplete dg-co(al)gebras and $\mathrm{Alg}$ the category of dg-algebras. There is a bar-construction functor $B :\mathrm{Alg}\to\mathrm{Cogc}$ which is a right adjoint to the cobar-construction functor $\Omega:\mathrm{Cogc}\to\mathrm{Alg}$. Starting from a map $f\in\mathrm{Cogc}(C,B A)$, one constructs a twisting cochain $\tau_f$ by postcomposing $f: C\to B A$ by the natural projection $B A\to A[1]$; the Maurer-Cartan equation for $\tau_f$ translates to saying that $f$ is a chain map, $d_{B A}\circ f = f\circ d_C$. One then replaces $\tau_f$ by the composition of the evident canonical map $\tau_0:\Omega C\to C[-1]$ (called the canonical twisting cochain) and $\tau_f[-1]:C[-1]\to A$ to obtain a morphism $f':\Omega C\to A$. The Maurer–Cartan equation for $\tau$ is equivalent also to saying that $f'$ is a chain map, i.e. $d_A\circ f'=f'\circ d_{\Omega C}$.
A twisting cochain is a datum used to define the twisted tensor product $L\otimes_\tau M$ for any right $C$-comodule $L$ and any left $A$-module $M$, as well as the twisted module of homomorphisms $\mathrm{Hom}_\tau(N,P)$ where $N$ is a left $C$-dg-comodule and $P$ a left $A$-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 $\tau$, one always has a pair of adjoint functors $\otimes_\tau A$ and $\otimes_\tau C$ between the derived category of modules over $A$ and the coderived category of comodules over $C$ (where $C$ is in $\mathrm{Cogc}$ 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 $\otimes_{\tau_0}\Omega C$ where $\tau_0:\Omega C\to C[-1]$ is the canonical twisting cochain). This pair of adjoint functors is an adjoint equivalence iff the composition $\Omega C\to C[-1]$ by $\tau[-1]:C[-1]\to A$ (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 $(C,A,\tau)$ the Koszul–Moore triple. Lefevre-Hasegawa’s thesis (pdf) asserts that in that case $A$ determines $C$ up to a weak equivalence (defined above) and $C$ determines $A$ up to a quasi-isomorphism. Moreover,
where $k$ 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.
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 $A_\infty$-categories, Grothendieck duality on complex manifolds (Toledo-Tong) and so on.
An old query archived here.
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