nLab
codiagonal

Definition

For an object $X$ in a category with coproducts, the codiagonal of $X$ is the canonical morphism

$$\nabla :X\coprod X\stackrel{(\mathrm{Id},\mathrm{Id})}{\to }X.$$\nabla : X \coprod X \stackrel{(Id,Id)}{\to} X \,.

The dual concept is diagonal .

In a cartesian bicategory $B$, the pair of terms diagonal, codiagonal refer to the canonical comultiplication $\Delta :X\to X\otimes X$ and the dual multiplication $\nabla ={\Delta }_{*}:X\otimes X\to X$ on any object. While the comultiplication is not a true diagonal (because $\otimes$ is not a cartesian product in $B$), it is the diagonal when seen as belonging to the subcategory of maps (left adjoints), where the restriction of $\otimes$ to $\mathrm{Map}(B)$ becomes a 2-product. Similarly, $\nabla$ is not a true codiagonal on $B$, but it becomes a codiagonal in the sense above when seen as belonging to $\mathrm{Map}(B{)}^{\mathrm{op}}$, the opposite obtained by reversing $1$-cells but not $2$-cells.

Possible confusion

The term ‘codiagonal’ is also sometimes used in the context of the theory of bisimplicial sets. For this use see total simplicial set, within that entry.

Related concepts

• diagonal

• cylinder object