A-infinity categories #
An uncurved nonunital A∞ category on a graded linear quiver C over a commutative ring R
has, for every composable string X₀, …, Xₙ of objects with n ≥ 1, an operation
mₙ : Hom(Xₙ₋₁, Xₙ) ⊗ ⋯ ⊗ Hom(X₀, X₁) ⟶ Hom(X₀, Xₙ)
of degree 2 - n, subject to the Stasheff identities on every composable string.
The structure is stored as an A∞ algebra on the total module of morphisms
⨁ (X, Y), Hom(X, Y) with its degreewise grading, whose operations are path-compatible
(TauCeti.GradedLinearQuiver.IsPathCompatible): they send a composable string to a morphism
between its endpoints, and every other string to zero. A path-compatible operation is determined
by its values on composable strings, so this is exactly the data of the operations mₙ above.
The stored law is therefore the square-zero law b ∘ b = 0 of the suspended bar coderivation of
the total module, as for TauCeti.AInfinityAlgebra. On a word of morphisms which is not
composable every term of the Stasheff identities vanishes, so these identities on the total module
are exactly the Stasheff identities on composable strings. Constructions on A∞ algebras, such as
the bar differential and the unsuspended Stasheff identities, apply to the total algebra
directly.
The operation mₙ on a single composable string is
TauCeti.AInfinityCategory.pathOperation, a TauCeti.GradedLinearQuiver.PathOperation of
degree 2 - n, with inputs in Keller's order (aₙ, …, a₁). The arity-one and arity-two
operations are the differential TauCeti.AInfinityCategory.homDifferential of each hom module
and the composition TauCeti.AInfinityCategory.comp, with m₂(g, f) = g ∘ f. The
differential squares to zero and satisfies the Leibniz rule
d (g ∘ f) = d g ∘ f + (-1)^{|g|} g ∘ d f.
Main definitions #
TauCeti.AInfinityCategory: an uncurved nonunitalA∞category on a graded linear quiver.TauCeti.AInfinityCategory.pathOperation: the operationmₙon a composable string.TauCeti.AInfinityCategory.homDifferential: the differentialm₁of a hom module.TauCeti.AInfinityCategory.comp: the compositionm₂.
Main results #
TauCeti.AInfinityCategory.homInclusion_pathOperation: the operation on a composable string is the total operation on the included morphisms.TauCeti.AInfinityCategory.ext_pathOperation: anA∞category is determined by its operations on composable strings.TauCeti.AInfinityCategory.homDifferential_homDifferential: the differential squares to zero.TauCeti.AInfinityCategory.homDifferential_comp: the Leibniz rule.
References #
- B. Keller, Introduction to A-infinity algebras and modules, Sections 3.1 and 7.1.
An uncurved nonunital A∞ category on a graded linear quiver C: an A∞ algebra on
the total module of morphisms ⨁ (X, Y), Hom(X, Y), graded degreewise, whose operations are
path-compatible. Its operation mₙ on a composable string X₀, …, Xₙ is
TauCeti.AInfinityCategory.pathOperation.
- m (n : ℕ) : MultilinearMap R (fun (x : Fin n) => GradedLinearQuiver.TotalHom R C) (GradedLinearQuiver.TotalHom R C)
- taylor_isSuspension : AInfinity.IsSuspension self.grading self.taylor self.m
- bar_square_zero : ReducedTensorWords.gradedCoderiv (self.grading.shift 1) self.taylor 1 ∘ₗ ReducedTensorWords.gradedCoderiv (self.grading.shift 1) self.taylor 1 = 0
The grading of the total algebra is the degreewise grading of the morphisms.
Every operation of positive arity sends composable strings to morphisms between their endpoints, and other strings to zero. The nullary operation vanishes, so it is path-compatible automatically (
TauCeti.AInfinityCategory.isPathCompatible_m).
Instances For
A∞ categories on a graded linear quiver are determined by their operations.
Every operation of an A∞ category sends composable strings to morphisms between their
endpoints, and other strings to zero.
The operation mₙ sends inputs of degrees dᵢ to an element of degree ∑ dᵢ + 2 - n.
The operation mₙ on the composable string X₀, …, Xₙ: a multilinear map of degree 2 - n
from the homogeneous morphisms Xₙ₋₁ ⟶ Xₙ, …, X₀ ⟶ X₁ to the morphisms X₀ ⟶ Xₙ.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The operation on a composable string is the component, between the endpoints of the string, of the total operation on the included morphisms.
The operation on a composable string is the total operation. Including the value of the operation on a composable string into the total module gives the total operation on the included morphisms.
A∞ categories on a graded linear quiver are determined by their operations on composable
strings.
The differential and the composition #
The differential m₁ of the morphisms X ⟶ Y of an A∞ category.
Equations
Instances For
The differential of a morphism is the component of the unary operation of the included morphism.
The differential of a morphism, included into the total module, is the unary operation of the included morphism.
The differential of an A∞ category raises the degree by one.
The differential of an A∞ category squares to zero.
The composition m₂ of an A∞ category: 𝒞.comp X Y Z g f is the composite
g ∘ f : X ⟶ Z of g : Y ⟶ Z and f : X ⟶ Y.
Equations
- 𝒞.comp X Y Z = (𝒞.mul.compl₁₂ (TauCeti.GradedLinearQuiver.homInclusion Y Z) (TauCeti.GradedLinearQuiver.homInclusion X Y)).compr₂ (TauCeti.GradedLinearQuiver.homProjection X Z)
Instances For
The composite of two morphisms is the component of the binary operation of the included morphisms.
The composite of two morphisms, included into the total module, is the binary operation of the included morphisms.
The composite of morphisms of degrees p and q has degree p + q.
The Leibniz rule. For g : Y ⟶ Z of degree p and f : X ⟶ Y, the differential of
g ∘ f is d g ∘ f + (-1)^p g ∘ d f.