nLab kernel pair



Category theory

Limits and colimits



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 f:XYf:X\to Y in a category CC is a pair of morphisms RXR\,\rightrightarrows \, X which form a limit of the diagram

X X f f Y \array{ X & & & & X \\ & \searrow^f & & \swarrow_f \\ & & Y \\ }

We can think of this as the fiber product X× YXX \times_Y X of XX with itself over YY, or as the pullback of ff along itself.


The kernel pair is always a congruence on XX; informally, RR is the subobject of X×XX \times X consisting of pairs of elements which have the same value under ff (sometimes called the ‘kernel’ of a function in Set\Set).

The coequalizer of the kernel pair, if it exists, is supposed to be the “object of equivalence classes” of the internal equivalence relation RR. In other words, it is the quotient object of XX in which generalized elements are identified if they are mapped by ff to equal values in YY. In a regular category (at least), this can be identified with a subobject of YY called the image of ff.

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 idid on an object XX is the diagonal morphism (id,id)(id,id) of XX and has a coequalizer isomorphic to XX 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.