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 $.

Comments

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