uconf.algebraic.cofree_coalgebra

Cofree conilpotent C-coalgebra on a dg-module M.

The cofree conilpotent C-coalgebra on a dg-module M is

\[T^c_C(M) = \bigoplus_{n\geq 1} C(n) \otimes_{S_n} M^{\otimes n}\]

with:

• Degree: deg(c_key, m_tuple) = deg_C(c_key) + Σ_i deg_M(m_i).

• Differential d = d_C + d_M from the Koszul sign rule (Leibniz rule).

• C-coalgebra costructure: the infinitesimal cocomposition Δ^{i;m,n} applies the cooperad’s cocomposition to the C-decoration and splits the M-labels.

The basis keys are pairs (c_key, m_tuple) where:

• c_key is a planar basis key of C(n) for n = len(m_tuple) >= 1.

• m_tuple is a tuple of n values.

Quasi-planar requirement: C must be a quasi-planar cooperad, i.e. each component C(n) satisfies C(n) ≅ C_pl(n) ⊗ k[S_n] and exposes a planarize linear map. The basis uses only planar C-keys, so two corollas (c·σ, m_tuple) and (c, σ·m_tuple) are identified as the same element (coinvariant quotient).

The coprojection pi: T^c_C(M) -> M kills all elements with n >= 2 and maps (id_key, (m,)) |-> m.

Reference: Loday-Vallette “Algebraic Operads”, Section 5.8.

Classes

CofreeCoalgebraModule(cooperad_cls, ...[, name])	Underlying dg-module of the cofree conilpotent C-coalgebra T^c_C(M).
CofreeConilpotentCoalgebra(cooperad_cls, ...)	Cofree conilpotent C-coalgebra on a dg-module M.

class uconf.algebraic.cofree_coalgebra.CofreeCoalgebraModule(cooperad_cls, inner_module, *, name=None)

Bases: CombinatorialFreeModule

Underlying dg-module of the cofree conilpotent C-coalgebra T^c_C(M).

Basis keys are (c_key, m_tuple) pairs where c_key is a planar basis key of C(n) and m_tuple has length n. The differential is the Leibniz rule d = d_C + d_M with Koszul signs. Non-planar keys are automatically normalised via planarize (permuting the m_tuple).

The cooperad C must be quasi-planar: each component C(n) must expose a planarize linear map decomposing elements into planar representative ⊗ symmetric group element.

This class is normally not instantiated directly; use CofreeConilpotentCoalgebra instead.

Parameters:

• cooperad_cls (QuasiPlanarLike)

• inner_module (CombinatorialFreeModule)

• name (ClassVar[str])

__init__(cooperad_cls, inner_module, *, name=None)

Initialize the cofree conilpotent C-coalgebra module T^c_C(M).

Parameters:

• cooperad_cls (type[OperadComponent] | OperadFactory | type[CooperadComponent] | CooperadFactory) – Arity-indexed quasi-planar cooperad provider.

• inner_module (CombinatorialFreeModule) – Cogenerating dg-module M.

• name (str | None) – Display name override. Defaults to T^c_C(M).

Raises:

TypeError – If the cooperad is not quasi-planar (no planarize).

property connectivity: int

Minimum degree of any basis element.

The arity-1 term (id, (m,)) has degree deg_M(m), so the connectivity equals that of the inner module.

degree_on_basis(key)

Degree = deg_C(c_key) + sum_i deg_M(m_i).

Return type:

int

basis_iter(d)

Iterate over planar basis elements of total degree d.

Each yielded element is a single planar term self.term((c_pl, m)) indexed by a planar cooperad key c_pl and m-tuple m.

Uses the isomorphism C(n) ⊗_{S_n} M^{⊗n} ≅ C_pl(n) ⊗ M^{⊗n} and enumerates only planar C(n)-decorations.

Raises:

ValueError – when arity is unbounded.

Parameters:

d (int)

Return type:

Iterator[Any]

basis_weight_iter(d, w)

Iterate over planar basis elements of degree d and weight w.

Each yielded element is a single planar term self.term((c_pl, m)). Weight is additive over leaf-module elements.

Unlike basis_iter(), this method is always finite (w bounds the arity).

Parameters:

• d (int)

• w (int)

Return type:

Iterator[Any]

graded_basis

/home/runner/work/uconf/uconf/src/uconf/algebraic/cofree_coalgebra.py (starting at line 524)

Type:

File

graded_basis_by_weight

/home/runner/work/uconf/uconf/src/uconf/algebraic/cofree_coalgebra.py (starting at line 528) Return a Family of basis elements at degree d and weight w.

Type:

File

CofreeCoalgebraModule.Element

alias of Element

class uconf.algebraic.cofree_coalgebra.CofreeConilpotentCoalgebra(cooperad_cls, inner_module)

Bases: CooperadCoalgebra

Cofree conilpotent C-coalgebra on a dg-module M.

Constructs T^c_C(M) = sum_{n>=1} C(n) x M^{xn} as a CofreeCoalgebraModule and equips it with the canonical C-coalgebra coaction.

Parameters:

• cooperad_cls (QuasiPlanarLike) – Quasi-planar cooperad provider C.

• inner_module – The cogenerating dg-module M.

The coprojection pi: T^c_C(M) -> M is given by project().

Examples:

cofree_coass = CofreeConilpotentCoalgebra(CoAssociative, module_M)
elem = cofree_coass.modul(((1, 2), (m1, m2)))
cofree_coass.coact(elem, 2)

infinitesimal_cocompose(x, i, m, n)

Delta^{i;m,n}: T^c_C(M)(m+n-1) -> T^c_C(M)(m) x T^c_C(M)(n).

For each basis element (c_key, (mu_1, ..., mu_{m+n-1})), applies the cooperad’s infinitesimal cocomposition and distributes M-labels:

• Left factor (arity m): (c_L_key, (mu_1,...,mu_{i-1}, mu_{i+n-1},...,mu_{m+n-1})) with mu_i replaced by a placeholder at position i, corresponding to the right subtree occupying positions i..i+n-1.

• Right factor (arity n): (c_R_key, (mu_i,...,mu_{i+n-1}))

Both factors are constructed directly (no coinvariant planarization).

Parameters:

• x – An element of the cofree module.

• i (int) – Starting position of the right factor (1-indexed).

• m (int) – Arity of the left factor.

• n (int) – Arity of the right factor.

Returns:

An element of T^c_C(M)(m) x T^c_C(M)(n).

project(x)

Coprojection pi: T^c_C(M) -> M, projecting onto arity-1 generators.

Returns the image in the inner module M. Non-zero only for elements of the form (id_key, (m_key,)) where id_key in C(1).

Parameters:

x – An element of the cofree module.

Returns:

An element of the inner module M.