Right A-infinity modules: arity components and component equations #
This file splits the Taylor map of a right A∞ module MM over AA into its arity
components, and spells out the module Stasheff law component by component.
As for the operations m n of an A∞ algebra, the components are indexed by their total arity
n, counting the module input, so b n and m n take n - 1 algebra inputs. There is no
arity-zero operation, and b 0 and m 0 are the junk value 0.
MM.b n : M ⊗ A^⊗(n-1) → Mis the suspended operationb_n^M: the restriction of the Taylor map to the summandsM ⊗ (sA)^⊗(n-1)of the cofree bar comodule.MM.m n : M → A^(n-1) → Mis the unsuspended operationm_n^M, obtained fromMM.b nby the Koszul sign of the degree--1suspension of itsninputs. This is the convention used for the unsuspended operations of anA∞algebra.
The bar differential is the coderivation gradedCoderiv generated by its Taylor map
(barDifferential_eq). Conversely, ofTaylor builds a module from any degree-one Taylor map
whose generated coderivation squares to zero, checked on the Taylor component. Composing with the
Taylor map turns the square-zero law into the suspended module Stasheff equations of every arity
(stasheff_tmul_of_tprod), with the algebra bar differential written out in
stasheff_tmul_of_tprod_splice. On homogeneous inputs, these become
the unsuspended module Stasheff equations in the operations m of the module and of the algebra
(stasheff). Conversely, ofStasheff builds a module from unsuspended operations of the right
degrees satisfying these equations.
The generic module-first unsuspension and the sign calculations shared with module morphisms live
in TauCeti.Algebra.Homology.AInfinity.Module.Right.Suspension.
Main definitions #
TauCeti.AInfinityRightModule.b: the suspended arity components of the Taylor map.TauCeti.AInfinityRightModule.m: the unsuspended module operations.TauCeti.AInfinityRightModule.gradedCoderiv: the coderivation over the algebra bar differential generated by a mapsM ⊗ Tᶜ(sA) → sM.TauCeti.AInfinityRightModule.ofTaylor: construct a module from its Taylor map.TauCeti.AInfinityRightModule.ofStasheff: construct a module from unsuspended operations satisfying the unsuspended module Stasheff equations.
Main results #
TauCeti.AInfinityRightModule.isHomogeneous_b_tmul_tprodandTauCeti.AInfinityRightModule.m_mem_piece: the suspended components have degree one and the unsuspended operation of aritynhas degree2 - n.TauCeti.AInfinityRightModule.ext_bandTauCeti.AInfinityRightModule.ext_m: a module is determined by its grading and either family of operations.TauCeti.AInfinityRightModule.isGradedCoderivationOver_gradedCoderivandTauCeti.AInfinityRightModule.rid_comp_lTensor_counit_comp_gradedCoderiv: the map generated byFis a coderivation over the algebra bar differential with counit componentF.TauCeti.AInfinityRightModule.barDifferential_eqandTauCeti.AInfinityRightModule.ofTaylor_self: the bar differential is the coderivation generated by the Taylor map, and every module is built from its Taylor map.TauCeti.AInfinityRightModule.b_tmul_tprod_of_mem: on homogeneous inputs, the suspended components are the unsuspended operations up to the Koszul sign of the suspension.TauCeti.AInfinityRightModule.stasheff_tmul_of_tprod: the suspended module Stasheff equation of each arity.TauCeti.AInfinityRightModule.stasheff: the unsuspended module Stasheff equation of each arity on homogeneous inputs, withstasheff_arity_oneits arity-one case on arbitrary inputs.
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, Section 4.
The suspended arity-n component b_n^M of a right A∞ module: the Taylor map restricted
to the summand sM ⊗ (sA)^⊗(n-1) of the cofree bar comodule. There is no arity-zero operation,
and b 0 is the junk value 0.
Equations
- MM.b 0 = 0
- MM.b n.succ = MM.taylor ∘ₗ LinearMap.lTensor M (TauCeti.TensorWords.of R A n)
Instances For
The arity-zero component is the junk value 0.
The arity component b (n + 1) is the Taylor map composed with the inclusion of words of
length n.
On a word of length n, the Taylor map is the arity component b (n + 1).
The arity component b (n + 1) has degree one: it sends a homogeneous suspended module
element and n homogeneous suspended letters to the suspended module degree one higher than the
total.
The Taylor maps of two right A∞ modules agree exactly when all their arity components
agree.
Right A∞ modules on a fixed carrier are determined by their grading and their suspended
arity components.
The unsuspended arity-n operation m_n^M of a right A∞ module, with the module input
first: for n = k + 1, the module-first unsuspension of b n. On homogeneous inputs its Koszul
twists multiply to the Koszul sign of the suspension of n inputs. There is no arity-zero
operation, and m 0 is the junk value 0.
Equations
Instances For
The arity-zero operation is the junk value 0.
The unsuspended operation evaluates the suspended component on Koszul-twisted inputs.
The suspended component evaluates the unsuspended operation on Koszul-twisted inputs: the
twists defining m are involutions.
On a pure word, the Taylor map evaluates the unsuspended operation on Koszul-twisted inputs.
On homogeneous inputs, the suspended component is the unsuspended operation multiplied by the
Koszul sign of suspending the module input of degree e and the k algebra inputs of degrees
d 0, …, d (k - 1).
The suspended arity components are determined by the grading and the unsuspended operations.
Right A∞ modules on a fixed carrier are determined by their grading and their unsuspended
operations.
The unsuspended operation m_{n+1}^M of arity n + 1 has cohomological degree 1 - n.
The coderivation over the bar differential of AA generated by a map F : sM ⊗ Tᶜ(sA) → sM:
the cofree comodule lift of F, which on x ⊗ w applies F to each prefix of the
deconcatenation of w, plus the algebra bar differential applied to w, with the Koszul sign of
moving it past the suspended module input.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The coderivation generated by F applies F after deconcatenation, and adds the
Koszul-twisted algebra bar differential.
The coderivation generated by F on the word x ⊗ a₀ ⊗ ⋯ ⊗ aₙ₋₁: the sum over the cuts of
the word of F applied to the prefix, tensored with the suffix, plus the Koszul-twisted module
input tensored with the algebra bar differential of the word.
The coderivation generated by a degree-one map has degree one.
The coderivation generated by any map is a coderivation over the algebra bar differential:
its first summand is the comodule morphism lifting F along the cofree coaction, and its second
summand satisfies the co-Leibniz law because the algebra bar differential does.
Applying the coalgebra counit after the coderivation generated by F recovers F: the
algebra bar differential has no counit component.
Construct a right A∞ module from a degree-one Taylor map whose generated coderivation squares
to zero. By cofreeness, the square-zero law is checked on the Taylor component.
Equations
Instances For
The grading of the module built from a Taylor map is the given grading.
The bar differential of the module built from a Taylor map is the coderivation it generates.
The Taylor map of the module built from a Taylor map is that map.
The bar differential is the coderivation generated by the Taylor map: on x ⊗ w it applies
the Taylor map to each prefix of the deconcatenation of w, and adds the algebra bar differential
applied to w, with the Koszul sign of moving it past the suspended module input.
The Taylor map vanishes after the coderivation it generates.
Every right A∞ module is built from its own Taylor map.
The module Stasheff law in operator form: the Taylor map applied after the Taylor map on each prefix, plus the Taylor map applied after the algebra bar differential, vanishes.
The suspended module Stasheff equation of arity n + 1, on the word x ⊗ a₀ ⊗ ⋯ ⊗ aₙ₋₁:
the sum over the cuts of the word of the Taylor map applied after the Taylor map on the prefix,
plus the Taylor map applied after the algebra bar differential of the word, vanishes.
The suspended module Stasheff equation of arity n + 1, with the algebra bar differential
expanded: its terms collapse each nonempty block of letters by the algebra Taylor map, twisting the
letters before the block.
The unsuspended module Stasheff equation of arity n + 1, on a homogeneous module input
x of degree e and homogeneous algebra inputs a 0, …, a (n - 1) of degrees d. Indexing
the inputs x, a 0, …, a (n - 1) by 0, …, n, the term in which an operation of arity s
collapses the block starting at position r carries the sign (-1) ^ (r + s * t), with t
inputs after the block, times the Koszul sign of the degree-2 - s inner operation crossing the
r inputs before it. The first sum collects the terms with r = 0, whose inner operation is
m (k + 1) of MM, and the second those with r = p + 1, whose inner operation is m s of
AA.
The unsuspended module Stasheff equation of arity one: the unary module operation squares to zero.
On the empty algebra word, the Taylor map is the unsuspended unary operation.
On the empty algebra word, the bar differential is the unary module operation tensored with the empty word.
Construct a right A∞ module from unsuspended operations m of degree 1 - n in arity
n + 1 satisfying the unsuspended module Stasheff equations on homogeneous inputs, together with a
Taylor map F evaluating m on Koszul-twisted inputs.
Equations
- TauCeti.AInfinityRightModule.ofStasheff G m hm F hFm hSI = TauCeti.AInfinityRightModule.ofTaylor G F ⋯ ⋯
Instances For
The grading of the module built from unsuspended operations is the given grading.
The Taylor map of the module built from unsuspended operations is the given Taylor map.
The unsuspended operations of the module built from unsuspended operations m are m, as
soon as the junk value m 0 vanishes.