In category theory, a branch of mathematics, the diagonal functor
, which maps objects as well as morphisms.
This functor can be employed to give a succinct alternate description of the product of objects within the category
is a universal arrow from
The arrow comprises the projection maps.
More generally, given a small index category
, one may construct the functor category
, the objects of which are called diagrams.
that maps every object in
The diagonal functor
assigns to each object
the natural transformation
is a discrete category with two objects, the diagonal functor
Diagonal functors provide a way to define limits and colimits of diagrams.
, a natural transformation
) is called a cone for
These cones and their factorizations correspond precisely to the objects and morphisms of the comma category
is a terminal object in
, i.e., a universal arrow
Dually, a colimit of
is an initial object in the comma category
, i.e., a universal arrow
has a limit (which will be the case if
is complete), then the operation of taking limits is itself a functor from
The limit functor is the right-adjoint of the diagonal functor.
Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor.
For example, the diagonal functor
described above is the left-adjoint of the binary product functor and the right-adjoint of the binary coproduct functor.
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