Diagonal morphism

In category theory, a branch of mathematics, for every object ${\displaystyle a}$ in every category ${\displaystyle {\mathcal {C}}}$ where the product ${\displaystyle a\times a}$ exists, there exists the diagonal morphism^{[1][2][3][4][5][6]}

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

satisfying

${\displaystyle \pi _{k}\circ \delta _{a}=\operatorname {id} _{a}}$ 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.

The dual notion of a diagonal morphism is a co-diagonal morphism. For every object ${\displaystyle b}$ in a category ${\displaystyle {\mathcal {C}}}$ where the coproducts ${\displaystyle b\sqcup b}$ exists, the co-diagonal^{[3][2][7][5][6]} is the canonical morphism

${\displaystyle \delta _{b}\colon b\sqcup b{\stackrel {[Id,Id]}{\to }}b}$

satisfying

${\displaystyle \delta _{b}\circ \tau _{l}=\operatorname {id} _{b}}$ for ${\displaystyle l\in \{1,2\}.}$

where ${\displaystyle \tau _{l}}$ is the injection morphism to the ${\displaystyle l}$-th component.

Let ${\displaystyle f:X\to Y}$ be a morphism in a category ${\displaystyle {\mathcal {C}}}$ with the pushout is an epimorphism if and only if the codiagonal is an isomorphism.^[8]