co-associativity

The formal dual? of associativity.

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 $(\mathcal{C}, \otimes)$.

Created on November 15, 2016 at 10:54:42. See the history of this page for a list of all contributions to it.