nLab codiagonal




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

:XX[Id,Id]X \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

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

In a cartesian bicategory B\mathbf{B}, the pair of terms diagonal, codiagonal refer to the canonical comultiplication Δ:XXX\Delta: X \to X \otimes X and the dual multiplication =Δ *:XXX\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\mathbf{B}), it is the diagonal when seen as belonging to the subcategory of maps (left adjoints), where the restriction of \otimes to Map(B)Map(\mathbf{B}) becomes a 2-product. Similarly, \nabla is not a true codiagonal on B\mathbf{B}, but it becomes a codiagonal in the sense above when seen as belonging to Map(B) opMap(\mathbf{B})^{op}, the opposite obtained by reversing 11-cells but not 22-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.

Last revised on March 6, 2023 at 23:53:48. See the history of this page for a list of all contributions to it.