 Diagonal Map
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Diagonal Map

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

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

$\pi _{k}\circ \delta _{a}=id_{a}$ for $k\in \{1,2\},$ where $\pi _{k}$ is the canonical projection morphism to the $k$ -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 $x$ of the object $a$ . Namely, $\delta _{a}(x)=\langle x,x\rangle$ , the ordered pair formed from $x$ . 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 $\mathbb {R} \rightarrow \mathbb {R} ^{2}$ on the real line is given by the line that is the graph of the equation $y=x$ . The diagonal morphism into the infinite product $X^{\infty }$ may provide an injection into the space of sequences valued in $X$ ; 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.