In category theory, a branch of mathematics, for any object
a
in any category
C
where the product
a
×
a
exists, there exists the diagonal morphism
δ
a
:
a
→
a
×
a
satisfying
π
k
∘
δ
a
=
i
d
a
for
k
∈
{
1
,
2
}
,
where
π
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,
δ
a
(
x
)
=
⟨
x
,
x
⟩
, 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
R
→
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
∞
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.
