uconf.morphisms.e_comodule_morphism

E-comodule morphism Δ: Ω(C) → E ⊗ Ω(C).

For a quasi-planar cooperad C, the Le Grignou–Roca i Lucio E-comodule structure gives a chain map \(\nu: C \to E \otimes C\) at the cooperad level, where E = BarrattEccles is the Barratt–Eccles operad. This is computed by e_comodule_on_generator().

The extension to an operad morphism \(\Omega(C) \to E \otimes \Omega(C)\) uses the universal property of the free operad \(\Omega(C) = T(s^{-1}\bar C)\): on each generator (single-vertex cobar tree), the cooperad element is mapped via \(\nu\), embedded into \(E \otimes \Omega(C)\) via the canonical inclusion \(\iota: C \hookrightarrow \Omega(C)\), and then composed operadically with child images.

The cooperad \(\mathcal{C}\) must be quasi-planar: its arity-n components must inherit from QuasiPlanarMixin and expose planarize, boundary, and d_sigma. The canonical case is BarConstruction applied to a Hadamard product \(P \otimes \mathcal{E}\) or \(P \otimes \Surj\).

Functions

make_e_comodule_morphism(cooperad_cls)

Build the operad morphism Δ: Ω(C) → E ⊗ Ω(C).

uconf.morphisms.e_comodule_morphism.make_e_comodule_morphism(cooperad_cls)

Build the operad morphism Δ: Ω(C) → E ⊗ Ω(C).

The morphism is defined on generators of the free operad Ω(C) = T(s⁻¹C̄) by composing the cooperad-level E-comodule map e_comodule_on_generator() (which gives ν: C → E ⊗ C) with the canonical inclusion ι: C ↪ Ω(C), then extending via operadic composition.

Parameters

cooperad_clsCooperadLike

A quasi-planar cooperad (class or factory) for which the cobar construction Ω(C) is defined.

Returns

OperadMorphism

The morphism Δ whose source is CobarConstruction(cooperad_cls) and whose target is HadamardProduct(BarrattEccles, CobarConstruction(cooperad_cls)).

Parameters:

cooperad_cls (type[CooperadComponent] | CooperadFactory)

Return type:

OperadMorphism