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

satisfying

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.

单词	Diagonal morphism
释义	Diagonal morphism 英语百科 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 satisfying 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.
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