For
concrete categories, the diagonal morphism can be simply described by its action on elements of the object . Namely, , the
ordered pair formed from . 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 on the
real line is given by the line that is the
graph of the equation . The diagonal morphism into the
infinite product may provide an
injection into the
space of sequences valued in ; 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 in a category where the
coproducts exists, the co-diagonal[3][2][7][5][6] is the canonical morphism
satisfying
for
where is the injection morphism to the -th component.
Let be a morphism in a category with the
pushout is an
epimorphism if and only if the codiagonal is an isomorphism.[8]
Awodey, s. (1996). "Structure in Mathematics and Logic: A Categorical Perspective". Philosophia Mathematica. 4 (3): 209–237.
doi:
10.1093/philmat/4.3.209.