nLab co-associativity

Idea

The formal dual of associativity.

Definition

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

$$\array{ A &\overset{\Delta}{\longrightarrow}& A \otimes A \\ \downarrow && \downarrow^{\mathrlap{\Delta \otimes id}} \\ A \otimes A &\underset{id \otimes \Delta}{\longrightarrow}& A \otimes A \otimes A } \,.$$

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