Nonunital A-infinity algebras #
An uncurved nonunital A∞ algebra on an internally ℤ-graded module consists of operations
m n of degree 2 - n, with m 0 = 0, whose suspended Taylor map extends to a square-zero
degree-one coderivation of the reduced tensor coalgebra. This file packages that definition and
exposes both of its standard presentations: the bar differential and the unsuspended Stasheff
identities.
The comparison between the presentations uses the degree--1 suspension convention. In
particular, the arity-two identity has the sign (-1)^|a| on m₂(a,m₁(b)), while the arity-three
identity becomes ordinary associativity when m₃ vanishes. Arity zero is explicitly forced to
vanish, rather than being unconstrained data hidden from the bar construction.
Main definitions #
TauCeti.AInfinityAlgebra: an uncurved nonunitalA∞algebra.TauCeti.AInfinityAlgebra.barDifferential: its square-zero bar coderivation.TauCeti.AInfinityAlgebra.ofStasheff: construct an algebra from the unsuspended identities.TauCeti.AInfinityAlgebra.ofTaylor: construct an algebra from a degree-one Taylor map whose bar coderivation squares to zero.TauCeti.AInfinityAlgebra.differential: the unary operation as a linear endomorphism.TauCeti.AInfinityAlgebra.mul: the binary operation as a bilinear map.
Main results #
TauCeti.AInfinityAlgebra.stasheff: the unsuspended Stasheff identities, withstasheff_arity_onethroughstasheff_arity_fourevaluated verbatim.TauCeti.AInfinityAlgebra.taylor_of_twoandTauCeti.AInfinityAlgebra.barDifferential_of_two: the Taylor map and the bar differential on a two-letter word.
References #
- E. Getzler and J. D. S. Jones, A-infinity algebras and the cyclic bar complex, Sections 1--2.
- B. Keller, Introduction to A-infinity algebras and modules, Sections 3.1 and 3.6.
An uncurved nonunital A∞ algebra over a commutative ring.
The Taylor map is stored alongside the operations because suspension depends on the degrees of
homogeneous inputs. taylor_isSuspension determines it uniquely from grading and m, as
recorded by AInfinity.IsSuspension.taylor_eq. Its graded Taylor expansion is the primary bar
coderivation, and bar_square_zero is the stored Stasheff law.
- grading : InternalGrading R A
The internal cohomological grading of the carrier.
- m (n : ℕ) : MultilinearMap R (fun (x : Fin n) => A) A
The unsuspended arity-
noperation. The arity-zero operation vanishes: the algebra is uncurved.
- m_degree (n : ℕ) : 0 < n → MultilinearMap.IsHomogeneous (self.m n) (fun (x : Fin n) => self.grading.piece) self.grading.piece (2 - ↑n)
The operation
m nhas cohomological degree2 - n. The Taylor map from nonempty suspended tensor words to suspended letters.
- taylor_isSuspension : AInfinity.IsSuspension self.grading self.taylor self.m
The Taylor map is the suspension of the displayed operations.
- bar_square_zero : ReducedTensorWords.gradedCoderiv (self.grading.shift 1) self.taylor 1 ∘ₗ ReducedTensorWords.gradedCoderiv (self.grading.shift 1) self.taylor 1 = 0
The degree-one coderivation extended from
taylorsquares to zero.
Instances For
The degree-one coderivation of the reduced bar construction.
Equations
Instances For
The bar differential is the coderivation generated by the stored Taylor map.
The bar differential is a graded coderivation for the suspended grading.
The bar differential has cohomological degree one.
The letter component of the bar differential is its stored Taylor map.
The bar differential squares to zero.
The square-zero bar law is equivalent to all unsuspended Stasheff identities on homogeneous inputs.
The unsuspended operations of an A∞ algebra satisfy every Stasheff identity on homogeneous
inputs.
Construct an A∞ algebra from homogeneous operations satisfying the unsuspended Stasheff
identities and a Taylor map realizing their suspension.
Equations
- TauCeti.AInfinityAlgebra.ofStasheff G m hm0 hm F hFm hSI = { grading := G, m := m, m_zero := hm0, m_degree := hm, taylor := F, taylor_isSuspension := hFm, bar_square_zero := ⋯ }
Instances For
A∞ algebras are determined by their grading and unsuspended operations; the Taylor map is
forced by the suspension relation and all remaining fields are propositions.
Construct an A∞ algebra from a Taylor map of degree one for the suspended grading whose bar
coderivation squares to zero. Its operations are the desuspension of the Taylor map.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The bar differential of the algebra built from a Taylor map is the coderivation it generates.
The operations of an A∞ algebra are the desuspension of its Taylor map.
Every A∞ algebra is built from its own Taylor map.
Low-arity identities #
The arity-one identity is m₁ m₁ = 0.
The arity-two identity is the graded Leibniz rule, with sign (-1)^(d 0) on the second
differentiated input.
The arity-three Stasheff identity, with the Koszul signs determined by the degrees of its first two inputs.
The arity-four Stasheff identity, with the Koszul signs determined by the degrees of its first three inputs.
Low-arity identities for arbitrary inputs #
The unary A∞ operation, regarded as a linear differential on the total module.
Equations
- 𝒜.differential = (𝒜.m 1).curryRight ![]
Instances For
Evaluating the differential is evaluating the unary operation.
The unary operation raises the degree by one.
On a single letter the Taylor map is the unary operation: the suspension sign of a word of length one is trivial.
The bar differential sends a single letter to the single letter given by the unary operation.
The binary A∞ operation, regarded as a linear map in each argument.
Instances For
Evaluating the bilinear product is evaluating the binary operation.
The binary operation has degree zero.
On a pure tensor word of length n the Taylor map evaluates the arity-n operation after
twisting the i-th letter by the Koszul twist of parameter n - 1 - i; on homogeneous letters
these twists multiply to the suspension sign (-1) ^ suspExp n d.
On a two-letter word the Taylor map is the binary operation, with the suspension sign carried by the degree-one Koszul twist of the first letter.
On a three-letter word the Taylor map is the ternary operation, with the suspension sign carried by the degree-one Koszul twist of the middle letter: the first letter is twisted by the trivial parameter two.
The bar differential of a two-letter word: the unary operation applied to either letter, and the collapse of both letters to the binary operation. Koszul twists carry the suspension signs.
The unary operation squares to zero.
The unary operation anticommutes with the degree-one Koszul twist.
The binary operation has degree zero, so it commutes with every Koszul twist.
The graded Leibniz rule for arbitrary inputs, with the sign on the second term carried by the degree-one Koszul twist of the left factor.
The arity-three Stasheff identity for arbitrary inputs, with the Koszul signs of the first two inputs carried by the degree-one twist. It expresses the associator of the binary operation as a unary boundary, modulo the three terms in which the ternary operation meets a unary boundary.
If the ternary operation vanishes, the arity-three identity says that m₂ is associative.