nLab co-associativity




The formal dual of associativity.


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

A Δ AA Δid AA idΔ AAA. \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).

Last revised on November 13, 2022 at 12:06:11. See the history of this page for a list of all contributions to it.