Diagonal morphism

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 which 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 which is a 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 which the image of the diagonal map will fail to satisfy.

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References

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