# Kernel pair

of a morphism in a category

A categorical generalization of the equivalence relation induced by a mapping of one set into another. A pair of morphisms $\epsilon _ {1} , \epsilon _ {2} : R \rightarrow A$ in a category $\mathfrak K$ is called a kernel pair of the morphism $\alpha : A \rightarrow B$ if $\epsilon _ {1} \alpha = \epsilon _ {2} \alpha$, and if for any pair of morphisms $\phi , \psi : X \rightarrow A$ for which $\phi \alpha = \psi \alpha$ there is a unique morphism $\gamma : X \rightarrow R$ such that $\phi = \gamma \epsilon _ {1}$ and $\psi = \gamma \epsilon _ {2}$.

Let $\mathfrak Y$ be an arbitrary category of universal algebras of a given type and all homomorphisms between them that is closed with respect to finite products, and let $\epsilon _ {1} , \epsilon _ {2} : R \rightarrow A$ be a kernel pair of a homomorphism $f: A \rightarrow B$ in $\mathfrak Y$. Then the image of the homomorphism

$$\epsilon _ {1} \times \epsilon _ {2} : R \rightarrow A \times A$$

induced by the pair $\epsilon _ {1} , \epsilon _ {2}$ is a congruence on the algebra $A$( cf. also Congruence (in algebra)). Conversely, if $R \subset A \times A$ is an arbitrary congruence on $A$, $i$ is the imbedding of $R$ into $A \times A$, and $p _ {1} , p _ {2}$ are the projections of $A \times A$ onto $A$, then the pair of homomorphisms $ip _ {1} , ip _ {2} : R \rightarrow A$ is a kernel pair of the natural homomorphism of $A$ onto the quotient algebra $A/R$.

In an arbitrary category with finite products and kernels of pairs of morphisms (cf. Kernel of a morphism in a category), the kernel pair of a morphism $\alpha : A \rightarrow B$ is constructed as follows. One chooses a product $A \times A$ with the projections $\pi _ {1}$ and $\pi _ {2}$, and determines the kernel $\mu$ of the pair of morphisms $\pi _ {1} \alpha , \pi _ {2} \alpha : A \times A \rightarrow B$. Then the pair of morphisms $\mu \pi _ {1} , \mu \pi _ {2}$ is a kernel pair of $\alpha$.

A cokernel pair is defined dually.

#### References

 [a1] F.G. Manes, "Algebraic categories" , Springer (1976) pp. Chapt. 2, §1 [a2] H. Schubert, "Kategorien" , 2 , Springer (1970) pp. Sect. 18.4 [a3] S. MacLane, "Categories for the working mathematician" , Springer (1971) pp. Sects. 3.3, 3.4
How to Cite This Entry:
Kernel pair. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_pair&oldid=47492
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article