The Yoneda lemma for right A-infinity modules #
Let A be an A∞ algebra with a strict unit e, and let M be a right A∞ module over A on
which e is a right unit for the binary operation, m₂(x, e) = x. The free module of rank one
A represents the underlying complex of M: evaluation at e is a homotopy equivalence from the
morphism complex Hom(A, M) to the underlying complex (M, m₁) of M. This is the A∞
counterpart of the differential graded Yoneda lemma, where the corresponding map is an isomorphism;
for A∞ modules only a homotopy equivalence survives, because a morphism out of A has higher
components which are not determined by its value at e.
All constructions are made on the cofree bar comodules sA ⊗ Tᶜ(sA) and sM ⊗ Tᶜ(sA), through
TauCeti.TensorWords.tmulConcat, which sends y ⊗ w to x ⊗ y w.
- Evaluation at
e,TauCeti.AInfinityRightModule.evalUnit, applies the Taylor map of a cochain to the one-letter worde. It commutes with the differentials as soon asm₁(e) = 0. - The Yoneda cochain
TauCeti.AInfinityRightModule.yonedaCochainofxhas Taylor mapy ⊗ w ↦ b^M(x ⊗ y w), that is(a₁, …, aₙ) ↦ m_{n+1}(x, a₁, …, aₙ)up to the suspension signs. By the module Stasheff identities, the assignmentx ↦ yonedaCochain xcommutes with the differentials, which makes it the map of complexesTauCeti.AInfinityRightModule.yoneda; evaluating the Yoneda cochain ofxatereturnsm₂(x, e) = x. - Prepending
eis a contracting homotopy of the bar complex of the free module of rank one (TauCeti.AInfinityAlgebra.barDifferential_toRightModule_comp_tmulConcat_add), becauseeis a strict unit. Precomposing with it gives the homotopyTauCeti.AInfinityRightModule.yonedaHomotopybetween the identity ofHom(A, M)and the composite of evaluation with the Yoneda cochains.
Main definitions #
TauCeti.AInfinityRightModule.yonedaEval: evaluation at a degree-zero cycle, such as the unit, as a map of cochain complexes.TauCeti.AInfinityRightModule.yoneda: the Yoneda cochains, a map of cochain complexes.TauCeti.AInfinityRightModule.homotopyYonedaEvalCompYoneda: the homotopy from the identity of the morphism complex to evaluation followed by the Yoneda cochains.TauCeti.AInfinityRightModule.yonedaHomotopyEquiv: the Yoneda lemma, a homotopy equivalence.
Main results #
TauCeti.AInfinityRightModule.evalUnit_homDifferential,TauCeti.AInfinityRightModule.homDifferential_yonedaCochains: both maps commute with the differentials.TauCeti.AInfinityRightModule.evalUnit_yonedaCochain: evaluating the Yoneda cochain ofxategivesm₂(x, e), so evaluation is a left inverse of the Yoneda cochains.TauCeti.AInfinityRightModule.homDifferential_yonedaHomotopy_add: the homotopy identityδ K F + K δ F = F - Y (F e).TauCeti.AInfinityRightModule.quasiIso_yonedaEval: evaluation at the unit is a quasi-isomorphism.
Implementation notes #
The cochain-level constructions allow the ground ring, the algebra and the module to live in
different universes. The complex-level statements compare the morphism complex, whose terms live
in the universe of linear maps between the bar comodules, with the underlying complex of the
module; they put all three in one universe u, as the differential graded category of right
A∞ modules does.
References #
- B. Keller, Introduction to A-infinity algebras and modules, Section 4.
- P. Seidel, Fukaya categories and Picard–Lefschetz theory, Chapter I (the Yoneda embedding).
The bar differential of a module after concatenation behind x: the cut before the first
letter applies the differential to x, the other cuts give the cofree lift of the Taylor map, and
the algebra bar differential acts on the concatenated word with the Koszul sign of x.
Prepending a strict unit to a nonempty word and applying the Taylor map of the free module of rank one returns the first letter on one-letter words and vanishes on longer words.
Prepending a strict unit is a contracting homotopy of the cofree bar comodule of the free
module of rank one: b ∘ h + h ∘ b = 1 for h (y ⊗ w) = e ⊗ y w.
The Yoneda cochain of x: the cochain out of the free module of rank one whose Taylor map
sends y ⊗ w to the module Taylor map of x ⊗ y w, after the degree-one Koszul twist of x.
On unsuspended operations it is (a₁, …, aₙ) ↦ m_{n+1}(x, a₁, …, aₙ) up to the suspension
signs.
Instances For
The Yoneda cochain of x is the cofree lift of the module Taylor map after concatenation
behind the Koszul twist of x.
The Yoneda cochain of an element of degree p is a cochain of degree p.
The Yoneda cochains of the elements of degree p.
Instances For
The degreewise Yoneda cochains are the Yoneda cochains.
Evaluation of a cochain at the unit: the Taylor map of the cochain at the one-letter word
e.
Instances For
Evaluation at e applies the cochain to e ⊗ 1 and then the counit.
A cochain out of the free module of rank one sends e ⊗ 1 to its value at the unit, followed
by the empty word.
Evaluation at a unit of degree zero sends cochains of degree p to elements of degree p.
Evaluation at a cycle e commutes with the differentials: the module bar differential of
F (e ⊗ 1) is its unary part, and the bar differential of the free module kills e ⊗ 1.
Evaluating the Yoneda cochain of x at e gives the binary module operation m₂(x, e).
The Yoneda cochains form a chain map: the differential of the Yoneda cochain of x is the
Yoneda cochain of the differential of x.
Evaluation at a unit of degree zero on the cochains of degree p.
Instances For
Degreewise evaluation at the unit is evaluation at the unit.
The contracting homotopy of the Yoneda lemma, from cochains of degree p to cochains of degree
q = p - 1: up to the sign (-1)^p, the cofree lift of the Taylor map of a cochain after
prepending the unit e.
Instances For
The homotopy is (-1)^p times the cofree lift of the Taylor map after prepending e.
The homotopy of the Yoneda lemma. For a strict unit e, the composite of evaluation at
e with the Yoneda cochains is homotopic to the identity of the morphism complex out of the free
module of rank one: δ K F + K δ F = F - Y (F e).
Evaluation at a cycle e of degree zero, as a map of cochain complexes from the morphism
complex out of the free module of rank one to the underlying complex of the module.
Instances For
In degree p, the evaluation map of complexes is evaluation at the unit.
The Yoneda cochains, as a map of cochain complexes from the underlying complex of the module to the morphism complex out of the free module of rank one.
Instances For
In degree p, the Yoneda map of complexes is the degree-p Yoneda cochains.
If a degree-zero cycle e is a right unit for the binary module operation, evaluation at e
is a left inverse of the Yoneda cochains.
The contracting homotopy of the Yoneda lemma, as a homotopy from the identity of the morphism complex out of the free module of rank one to evaluation at the unit followed by the Yoneda cochains.
Instances For
The Yoneda lemma for right A∞ modules. Over an A∞ algebra with a strict unit e,
for a module on which e is a right unit for the binary operation, evaluation at e is a
homotopy equivalence from the morphism complex out of the free module of rank one to the
underlying complex of the module, with homotopy inverse the Yoneda cochains.
Instances For
Evaluation at a strict unit e is a quasi-isomorphism from the morphism complex out of the
free module of rank one to the underlying complex of a module on which e is a right unit for the
binary operation.