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

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

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 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 $id$ on an object $X$ is the diagonal morphism $(id,id)$ of $X$ and has a coequalizer isomorphic to $X$ itself.

Related concepts

• regular epimorphism

• regular category

• exact category