uconf.algebraic.coalgebra

Coalgebras over dg-cooperads (C-coalgebras).

A coalgebra over a dg-cooperad C is a dg-module V together with a collection of costructure maps

δ_n : V → C(n) ⊗_{S_n} V^{⊗n}

satisfying: - Counit: (ε ⊗ id) ∘ δ_1 = id where ε : C(1) → k is the cooperadic counit. - Cocomposition: coassociativity dual to the operad composition axioms. - Coequivariance: δ_n is S_n-equivariant. - Chain map: ∂_V ∘ δ_n = δ_n ∘ ∂_V (Leibniz rule with cooperad boundary).

Reference: Loday-Vallette “Algebraic Operads”, Chapter 12.

Classes

CoactionMap(*args, **kwargs)	Type contract for cooperad coaction callables.
CooperadCoalgebra(module, cooperad_cls, ...)	A dg-module equipped with a C-coalgebra structure.

class uconf.algebraic.coalgebra.CoactionMap(*args, **kwargs)

Bases: Protocol[CoalgebraElementInputType, CoactionValueType]

Type contract for cooperad coaction callables.

class uconf.algebraic.coalgebra.CooperadCoalgebra(module, cooperad_cls, coaction_map)

Bases: Generic[CoalgebraElementType, CoactionValueType]

A dg-module equipped with a C-coalgebra structure.

Wraps an underlying CombinatorialFreeModule (the module V) and a cooperad provider cooperad_cls (satisfying uconf.cooperad.CooperadComponent) together with an explicit costructure map.

The coaction map is supplied as a callable via coaction_map:

    coalg = CooperadCoalgebra(module, CoAssociative, my_map)

where ``my_map(v_element, n) → (C(n) ⊗ V^{⊗n}).Element``.

Parameters:

• module – Underlying dg-module (a CombinatorialFreeModule).

• cooperad_cls (CooperadLike) – Cooperad provider (class or factory, CooperadComponent-compatible).

• coaction_map (CoactionMap[CoalgebraElementType, CoactionValueType]) – Callable implementing the C-coalgebra coaction δ_n.

coact(v_element, n)

Apply the C-coaction δ_n(v) ∈ C(n) ⊗_{S_n} V^{⊗n}.

Parameters:

• v_element (CoalgebraElementType) – An element of the coalgebra module V.

• n (int) – The coaction arity (number of factors in the coaction).

Returns:

An element of C(n) ⊗ V^{⊗n}.

Raises:

• ValueError – If n <= 0.

• TypeError – If coaction_map does not return an iterable object.

Return type:

CoactionValueType

boundary(v)

Apply the differential ∂_V to a coalgebra element.

Parameters:

v – An element of the coalgebra module.

Returns:

The boundary ∂_V(v) in the module.