nLab twisting cochain

Definition

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 in the convolution algebra holds:

$$d_A\circ\tau+\tau\circ d_C+\mu\circ(\tau\otimes\tau)\circ\Delta = 0.$$

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

Relation to the adjunction bar-cobar

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. $d_A\circ f'=f'\circ d_{\Omega C}$.

Some usages of twisting cochains

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

$$A\otimes_\tau C\otimes_\tau A\to A$$

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,

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

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.

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 $\mathcal{A}$, the “smallest” dg-category $\mathcal{A}'$ that contains it and in which there exist shifts and functorial cones is exactly the category of twisted complexes in $\mathcal{A}$. Furthermore, if $\mathcal{A}$ 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 $\mathcal{A}$).

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

Remarks

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

An old query archived here.

References

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

• Kenji Lefèvre-Hasegawa, Sur les A-infini catégories, (pdf)

• V. A. Smirnov, Simplicial and operad methods in algebraic topology, Transl. Math. Monogr. 198, AMS 2001 (transl. by G. L. Rybnikov)

• D. Barnes and L. A. Lambe, A Fixed Point Approach to Homological Perturbation Theory, with Don Barnes, Proc. AMS, 112 (1991), 881–892.

• K. Hess, The cobar construction: a modern perspective (pdf) (lecture notes from a minicourse at Louvain)

• Alexander I. Efimov, Valery A. Lunts, Dmitri O. Orlov, Deformation theory of objects in homotopy and derived categories, part 1, part 2, part 3)

• N. R. O’Brian, Geometry of twisting cochains. Compositio Math. 63 (1987), no. 1, 41–62.

• D. Toledo, Yue Lin L. Tong, Duality and intersection theory in complex manifolds. I. Math. Ann. 237 (1978), no. 1, 41–77; II. The holomorphic Lefschetz formula. Ann. of Math. (2) 108 (1978), no. 3, 519–538.

• A. I. Bondal, M. M. Kapranov, “Enhanced Triangulated Categories”, Math. USSR Sbornik 70 (1991), pdf

• T. Hosgood, “Chern classes of coherent analytic sheaves: a simplicial approach”, PhD thesis (2020), HAL:tel-02882140

• Henri Gillet, The $K$-theory of twisted complexes, in “Applications of algebraic $K$-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.