category theory

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Definition

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

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

The dual concept is diagonal .

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.

Revised on February 23, 2013 14:03:56 by Tim Porter (95.147.236.135)