nLab
kernel pair

Idea

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.

Definition

The kernel pair of a morphism $f:X\to Y$ in a category $C$ is a pair of morphisms $R\rightrightarrows X$ which form a limit of the diagram

$$\begin{array}{ccccc}X& & & & X\\ & {\searrow }^f& & {\swarrow }_f\\ & & Y\\ \end{array}$$\array{ X & & & & X \\ & \searrow^f & & \swarrow_f \\ & & Y \\ }

We can think of this as the fiber product $X{\times }_{Y}X$ of $X$ with itself over $Y$, or as the pullback of $f$ along itself.

Properties

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

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

If a pair of parallel morphisms is a kernel pair and has 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