CHAPTER 3 Hopf Algebras, Algebraic, Formal, and Quantum Groups

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5. The coalgebra coend

Proposition 3.5.1. Let C be a monoidal category and ω : D ⟶ C be a diagram in C. Assume that there is a universal object coend(ω) and natural transformation δ : ω ⟶ ω ⊗ coend(ω).

Then there is exactly one coalgebra structure on coend(ω) such that the diagrams

[figure: ω ⊗ coend(ω); ω ⊗ coend(ω) ⊗ coend(ω); δ⊗1; ω; ω ⊗ coend(ω); δ; δ; 1⊗Δ]

and

[figure: ω; ω ⊗ coend(ω); δ; id_{ω}; ω ⊗ I; 1⊗ϵ]

commute.

Proof. Because of the universal property of coend(ω) there are structure morphisms Δ : coend(ω) ⟶ coend(ω) ⊗ coend(ω) and ϵ : coend(ω) ⟶ I. This implies the coalgebra property similar to the proof of Corollary 3.3.8.

Observe that by this construction all objects and all morphisms of the diagram ω : D ⟶ C_{0} ⊆ C are comodules or morphisms of comodules over the coalgebra coend(ω). In fact C := coend(ω) is the universal coalgebra over which the given diagram becomes a diagram of comodules.

Corollary 3.5.2. Let (D, ω) be a diagram C with objects in C_{0}. Then all objects of the diagram are comodules over the coalgebra C := coend(ω) and all morphisms are morphisms of comodules. If D is another coalgebra and all objects of the diagram are D-comodules by φ(X) : ω(X) ⟶ ω(X) ⊗ D and all morphisms of the diagram are morphisms of D-comodules then there exists a unique morphism of coalgebras φ̃: coend(ω) ⟶ D such that the diagram

[figure: ω; ω ⊗ coend(ω); δ; φ; ω ⊗ D; 1⊗φ̃]

commutes.

Proof. The morphisms φ(X) : ω(X) ⟶ ω(X) ⊗ D define a natural transformation since all morphisms of the diagram are morphisms of comodules. So the existence and the uniqueness of a morphism φ̃: coend(ω) ⟶ D is clear. The only thing to show is that this is a morphism of coalgebras. This follows from the universal property of C = coend(ω) and the diagram

[figure: ω; ω ⊗ C; δ; δ; 1⊗Δ; 1; 1⊗φ̃; ω ⊗ C; ω ⊗ C ⊗ C; δ⊗1; 1⊗φ̃; 1⊗φ̃⊗φ̃; ω; ω ⊗ D; φ; φ; 1⊗Δ; ω ⊗ D; ω ⊗ D ⊗ D; φ⊗1]

where the right side of the cube commutes by the universal property. Similarly we get that φ̃ preserves the counit since the following diagram commutes

[figure: ω; ω ⊗ C; δ; ω; ω ⊗ D; φ; 1; 1⊗φ̃; 1; 1⊗ε; 1⊗ε; ω ⊗ 𝕂; 1; ω ⊗ 𝕂; 1]