The kernel pair of a morphism in a category is the fiber product of the morphism with itself.
The dual notion is that of cokernel pair.
The kernel pair of a morphism in a category is a pair of morphisms which form a limit of the diagram
We can think of this as the fiber product of with itself over , or as the pullback of along itself.
The kernel pair is always a congruence on ; informally, is the subobject of consisting of pairs of elements which have the same value under (sometimes called the ‘kernel’ of a function in ).
The coequalizer of the kernel pair, if it exists, is supposed to be the “object of equivalence classes” of the internal equivalence relation . In other words, it is the quotient object of in which generalized elements are identified if they are mapped by to equal values in . In a regular category (at least), this can be identified with a subobject of called the image of .
If a morphism has a kernel pair and is a coequalizer, then it is the coequalizer of its kernel pair. This is a special case of the correspondence of generalized kernels in enriched categories.
In any category with binary pullbacks, the kernel pair of the identity morphism on an object is the diagonal morphism of and has a coequalizer isomorphic to itself.
Last revised on June 12, 2021 at 20:26:01. See the history of this page for a list of all contributions to it.