nLab codiagonal

Definition

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

$$\nabla : X \sqcup X \stackrel{[Id,Id] }{\to} X$$

given by copairing the identity morphism with itself. The dual concept is diagonal . More generally, one can consider the iterated codiagonal

$$\nabla : X \sqcup X \sqcup \cdots \sqcup X \stackrel{[Id,Id,\ldots,Id]}{\to} X.$$

In a cartesian bicategory $\mathbf{B}$, 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 $\mathbf{B}$), it is the diagonal when seen as belonging to the subcategory of maps (left adjoints), where the restriction of $\otimes$ to $Map(\mathbf{B})$ becomes a 2-product. Similarly, $\nabla$ is not a true codiagonal on $\mathbf{B}$, but it becomes a codiagonal in the sense above when seen as belonging to $Map(\mathbf{B})^{op}$, 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