Graded coderivations over a right comodule #
An operator D on a right comodule over a coalgebra with operator b is a coderivation
over b when its coaction satisfies the signed co-Leibniz rule. On a homogeneous left
tensor factor x, the term applying b to the right factor has the Koszul coefficient
(-1) ^ (q * |x|), where q is the twist parameter (the degree of b in graded
applications). Homogeneity of D, with its own degree, is imposed separately.
The condition is formulated for any right comodule, so it applies to the cofree bar
comodule sM ⊗ Tᶜ(sA) without constructing a second comodule API. An odd homogeneous
coderivation over a square-zero b has a square which is a comodule morphism. This is
the algebraic step needed to read module Stasheff identities from the components of D².
Likewise, the graded commutator of a homogeneous comodule morphism with two coderivations over
the same operator b is a comodule morphism. Its counit component therefore detects compatibility
with the differentials when the target is cofree.
The sign convention follows Getzler--Jones, A-infinity algebras and the cyclic bar complex, Sections 1--2, and Keller, Introduction to A-infinity algebras and modules, Sections 3--4.
The signed co-Leibniz law for an endomorphism D of a right comodule over an operator
b on the coalgebra, with twist parameter q. Homogeneity (including the degree of D)
and square-zero conditions are separate: this predicate records exactly the compatibility
with the coaction.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The signed co-Leibniz law evaluated on one comodule element.
The signed co-Leibniz law can be checked on individual comodule elements.
If the coalgebra operator squares to zero and (-1) ^ (q * r) = -1, the square of a
degree-r comodule coderivation with twist q commutes with the coaction. In the bar
construction this makes
D² a comodule morphism, so its vanishing can be checked on its cogenerator component.
The square of a degree-one coderivation over a square-zero coalgebra operator commutes with the coaction.
When (-1) ^ (q * r) = -1 in the coefficient ring, the square of a degree-r
coderivation over a square-zero coalgebra operator is a comodule endomorphism.
Equations
- h.squareHomOfNegOnePowEqNegOne hD hqr hb = { toLinearMap := D ∘ₗ D, map_coact := ⋯ }
Instances For
The square of an odd coderivation over a square-zero coalgebra operator is a comodule endomorphism.
Equations
- h.squareHom hD hb = h.squareHomOfNegOnePowEqNegOne hD ⋯ hb
Instances For
The underlying map of the square comodule endomorphism under the sign hypothesis.
The underlying map of the square comodule endomorphism is the square of the coderivation.
On a cofree comodule, the square under the sign hypothesis vanishes exactly when its component obtained by applying the coalgebra counit vanishes. This is the universal property that reduces module Stasheff identities to Taylor components.
For a degree-one coderivation, the square vanishes if and only if its cofree counit component vanishes.
Two coderivations over the same coalgebra operator on a cofree comodule are equal when their counit components agree.
The graded commutator of a degree-r comodule morphism with coderivations over the same
coalgebra operator is a comodule morphism. The two coalgebra terms cancel by Koszul naturality.
Equations
- f.coderivationComm G H hf D E hD hE = { toLinearMap := E ∘ₗ f.toLinearMap - ↑↑(q * r).negOnePow • f.toLinearMap ∘ₗ D, map_coact := ⋯ }
Instances For
The underlying map of the coderivation commutator is the signed operator commutator.