uconf.constructions.cobar_coalgebra

Cobar construction for a C-coalgebra: Ω_α(V).

Given a twisting morphism α: C → P and a C-coalgebra (V, δ), the cobar construction is the free P-algebra

Ω_α(V) = (T_P(V), d_{T_P} + d_α)

where:

• T_P(V) = ⊕_{n≥1} P(n) ⊗_{S_n} V^{⊗n} is the free P-algebra on V (see uconf.algebraic.free_algebra).

• d_{T_P} = d_P + d_V is the Leibniz differential on the free algebra (Koszul sign rule).

• d_α is the extra twisting differential.

The twisting differential d_α acts on each leaf of a corolla:

d_α(p ⊗ v_1 ⊗ … ⊗ v_n) = Σ_{i=1}^{n} Σ_{k≥2}
    (-1)^{|p| + |v_1| + … + |v_{i-1}|}
    · (p ∘_i α(c_k)) ⊗ v_1 ⊗ … ⊗ v'_1 ⊗ … ⊗ v'_k ⊗ … ⊗ v_n

where δ_k(v_i) = Σ c_k ⊗ v’_1 ⊗ … ⊗ v’_k is the C-coalgebra coaction and p ∘_i α(c_k) ∈ P(n + k - 1) is the operadic composition.

Maurer-Cartan equation ∂α + α ⋆ α = 0 ensures d² = 0.

Ω_α(V) is a dg-P-algebra.

Reference: Loday-Vallette “Algebraic Operads”, Section 11.4.

Classes

CobarCoalgebra(alpha, coalgebra)	Cobar construction Ω_α(V) as a P-algebra.
CobarCoalgebraModule(alpha, coalgebra)	Underlying dg-module of the cobar construction Ω_α(V).

class uconf.constructions.cobar_coalgebra.CobarCoalgebraModule(alpha, coalgebra)

Bases: FreeAlgebraModule

Underlying dg-module of the cobar construction Ω_α(V).

Extends FreeAlgebraModule with the twisting differential d_α. Basis keys are (p_key, m_tuple) pairs inherited from the free algebra, where p_key ∈ P(n)_planar and m_tuple has length n.

The total differential is d = d_{T_P} + d_α where d_{T_P} is the free-algebra Leibniz differential and d_α inserts new vertices via the coalgebra coaction and twisting morphism.

Parameters:

• alpha (TwistingMorphism)

• coalgebra (CooperadCoalgebra)

d_free(elem)

Apply only the free algebra differential d_{T_P}.

d_alpha(elem)

Apply only the twisting differential d_α.

CobarCoalgebraModule.Element

alias of Element

class uconf.constructions.cobar_coalgebra.CobarCoalgebra(alpha, coalgebra)

Bases: OperadAlgebra

Cobar construction Ω_α(V) as a P-algebra.

Wraps CobarCoalgebraModule with the canonical P-algebra structure inherited from the free algebra.

Parameters:

• alpha (TwistingMorphism) – A twisting morphism α: C → P.

• coalgebra (CooperadCoalgebra) – A C-coalgebra (V, δ).

include(v_key)

Return the image of v_key under the inclusion η: V → Ω_α(V).

Parameters:

v_key – A basis key of the coalgebra module V.

Returns:

The element (id_P, (v_key,)) in Ω_α(V).