The diagonal of an object in a category with products is the canonical morphism
The dual concept is codiagonal .
Recall that the diagonal of a set is a subset of its cartesian square . If is now an object in some cartesian monoidal category , then the diagonal of is now a subobject of its categorial square . (Actually, need not be cartesian monoidal, as long as the product exists.)
Specifically, the diagonal morphism of is a morphism given (using the universal property of the product) by the identity morphism from to itself, taken twice. That is, is the universal solution to
If is (the category of sets), then this diagonal morphism is precisely the diagonal function of .
The diagonal morphism is always a regular monomorphism, since it is the equaliser of the two projection maps . (In fact, it is a split monomorphism, since it is also a section of either projection map.) Thus, it makes into a regular subobject of , the diagonal subobject. When is the , this recovers the original notion of the diagonal subset of .
In the category Set the diagonal is the function for all . See diagonal subset.
In the category Top of topological spaces, an object is a Hausdorff space if and only if its diagonal subobject is a closed subspace of ; this fact can be generalised to other notions of space.
In Cat the diagonal morphisms are diagonal functors.
Revised on December 13, 2011 10:12:29
by Urs Schreiber