Diagonal morphism

In category theory, a branch of mathematics, for any object ${\displaystyle a}$ in any category ${\displaystyle {\mathcal {C}}}$ where the product ${\displaystyle a\times a}$ exists, there exists the diagonal morphism

${\displaystyle \delta _{a}:a\rightarrow a\times a}$

satisfying

${\displaystyle \pi _{k}\circ \delta _{a}=id_{a}}$ for ${\displaystyle k\in \{1,2\},}$

where ${\displaystyle \pi _{k}}$ is the canonical projection morphism to the ${\displaystyle k}$-th component. The existence of this morphism is a consequence of the universal property that characterizes the product (up to isomorphism). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The image of a diagonal morphism in the category of sets, as a subset of the Cartesian product, is a relation on the domain, namely equality.

For concrete categories, the diagonal morphism can be simply described by its action on elements ${\displaystyle x}$ of the object ${\displaystyle a}$. Namely, ${\displaystyle \delta _{a}(x)=\langle x,x\rangle }$, the ordered pair formed from ${\displaystyle x}$. The reason for the name is that the image of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism ${\displaystyle \mathbb {R} \rightarrow \mathbb {R} ^{2}}$ on the real line is given by the line that is the graph of the equation ${\displaystyle y=x}$. The diagonal morphism into the infinite product ${\displaystyle X^{\infty }}$ may provide an injection into the space of sequences valued in ${\displaystyle X}$; each element maps to the constant sequence at that element. However, most notions of sequence spaces have convergence restrictions that the image of the diagonal map will fail to satisfy.