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 in a category
is called a kernel pair of the morphism
if
, and if for any pair of morphisms
for which
there is a unique morphism
such that
and
.
Let 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
be a kernel pair of a homomorphism
in
. Then the image of the homomorphism
![]() |
induced by the pair is a congruence on the algebra
(cf. also Congruence (in algebra)). Conversely, if
is an arbitrary congruence on
,
is the imbedding of
into
, and
are the projections of
onto
, then the pair of homomorphisms
is a kernel pair of the natural homomorphism of
onto the quotient algebra
.
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 is constructed as follows. One chooses a product
with the projections
and
, and determines the kernel
of the pair of morphisms
. Then the pair of morphisms
is a kernel pair of
.
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 |
Kernel pair. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Kernel_pair&oldid=17146