cocommutative coalgebra in nLab

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Contents

Idea

The notion of cocommutative coalgebra is the formal dual of commutative algebra.

Cocommutative coalgebras form the category CocommCoalg.

Definition

For (C,⊗,I,σ)(C,\otimes, I, \sigma) a symmetric monoidal category, a comonoid object in CC is an object AA equipped with a morphism

Δ:A→A⊗A \Delta : A \to A \otimes A

(“comultiplication”) and a morphism

ε:A→I \varepsilon: A \to I

(“counit”) that satisfy the coassociativity and counitality equations:

A →Δ A⊗A Δ↓ ↓ Id⊗Δ A⊗A →Δ⊗Id A⊗A⊗A A Id↙ ↓ Δ ↘ Id A ←Id⊗ε A⊗A →ε⊗Id A. \array{ A &\stackrel{\Delta}{\to}& A \otimes A \\ ^\mathllap{\Delta} \downarrow && \downarrow^{\mathrlap{Id \otimes \Delta}} \\ A \otimes A &\underset{\Delta \otimes Id}{\to}& A \otimes A \otimes A } \qquad \qquad \qquad \array{ & & A & & \\ & ^\mathllap{Id} \swarrow & \downarrow^\mathrlap{\Delta} & \searrow^\mathrlap{Id} & \\ A & \underset{Id \otimes \varepsilon}{\leftarrow} & A \otimes A & \underset{\varepsilon \otimes Id}{\to} & A } \,.

This is co-commutative if we have a commuting diagram

A⊗A Δ↗ ↘ σ A,A A →Δ A⊗A, \array{ && A \otimes A \\ & {}^{\mathllap{\Delta}}\nearrow && \searrow^\mathrlap{\sigma_{A,A}} \\ A &&\underset{\Delta}{\to}&& A \otimes A } \,,

where σ A,A:A⊗A→A⊗A\sigma_{A,A} : A \otimes A \to A \otimes A is the braiding isomorphism of CC.

In the case of the symmetric monoidal category of modules over a commutative ring RR, such an object is called a cocommutative coalgebra over RR. In the case of the symmetric monoidal category of chain complexes (or differential graded spaces), such an object is called a DG coalgebra.

Sometimes “cocommutative coalgebra” in a symmetric monoidal category is used as a synonym for cocommutative comonoid object. We will follow that convention below. If MM is symmetric monoidal, then CocommCoalg(M)CocommCoalg(M) denotes the category of cocommutative algebras and coalgebra maps in MM.

Relation to cartesian monoidal categories

Cocommutative coalgebras form a bridge between the doctrines of symmetric monoidal categories and cartesian monoidal categories. Notice that every object in a cartesian monoidal category carries a unique cocommutative comonoid structure (the counit is the unique map to the terminal object, and the counicity equations then force the comultiplication map to be the diagonal), and every morphism in a cartesian monoidal category is thereby a coalgebra map.

If CC and DD are two cocommutative coalgebras, then C⊗DC \otimes D becomes a cocommutative coalgebra with comultiplication

C⊗D→Δ C⊗Δ DC⊗C⊗D⊗D→Id⊗σ⊗IdC⊗D⊗C⊗DC \otimes D \stackrel{\Delta_C \otimes \Delta_D}{\to} C \otimes C \otimes D \otimes D \stackrel{Id \otimes \sigma \otimes Id}{\to} C \otimes D \otimes C \otimes D

and counit

C⊗D→ε C⊗ε DI⊗I≅I.C \otimes D \stackrel{\varepsilon_C \otimes \varepsilon_D}{\to} I \otimes I \cong I.

• comonoid object

• monoid, commutative monoid, category of monoids

• gs-monoidal category