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.
See also: regular epimorphism, regular category, exact category
Last revised on July 12, 2017 at 12:05:07. See the history of this page for a list of all contributions to it.