co-associativity in nLab

Contents

Contents

Idea

The formal dual of associativity.

Definition

Given a monoidal category (𝒞,⊗)(\mathcal{C}, \otimes) and an object AA in 𝒞\mathcal{C} equipped with morphism (“co-multiplication”) Δ:A⟶A⊗A\Delta \colon A \longrightarrow A \otimes A, then this is co-associative if the following diagram commutes

A ⟶Δ A⊗A ↓ ↓ Δ⊗id A⊗A ⟶id⊗Δ A⊗A⊗A. \array{ A &\overset{\Delta}{\longrightarrow}& A \otimes A \\ \downarrow && \downarrow^{\mathrlap{\Delta \otimes id}} \\ A \otimes A &\underset{id \otimes \Delta}{\longrightarrow}& A \otimes A \otimes A } \,.

If in addition there is a counit on AA for which the coproduct satisfies co-unitality, then AA is called a co-monoid in (𝒞,⊗)(\mathcal{C}, \otimes).