Variety in a category
A notion generalizing that of a variety of universal algebras. Let be a bicategory with products. A full subcategory
of
is called a variety if it satisfies the following conditions: a) if
is an admissible monomorphism and
, then
; b) if
is an admissible epimorphism and
, then
; c) if
,
, then
.
If is a well-powered category, that is, the admissible subobjects of any object form a set, then every variety is a reflective subcategory of
. This means that the inclusion functor
has a left adjoint
. The unit of this adjunction, the natural transformation
, has the property that for each
the morphism
is an admissible epimorphism. In many important cases the functor
turns out to be right-exact, that is, it transforms the cokernel
of a pair of morphisms
into the cokernel of the pair of morphisms
, if
is a kernel pair of the morphism
. Moreover, right exactness and the presence of the natural transformation
are characteristic properties of
.
A variety inherits many properties of the ambient category. It has the structure of a bicategory, and is complete if the initial category is complete.
In categories with normal co-images, as in the case of varieties of groups, it is possible to define a product of varieties. The structure of the resultant groupoid of varieties has been studied only in a number of special cases.
References
[1] | M.Sh. Tsalenko, E.G. Shul'geifer, "Fundamentals of category theory" , Moscow (1974) (In Russian) |
[2] | A. Fröhlich, "On groups over a d.g. near ring II. Categories and functors" Quart. J. Math. , 11 (1960) pp. 211–228 |
Comments
In a topos, one also considers exponential varieties [a1], which are full subcategories closed under arbitrary subobjects, products and power-objects. Such a subcategory is necessarily closed under quotients as well; it is a topos, and its inclusion functor has adjoints on both sides.
References
[a1] | P.J. Freyd, "All topoi are localic, or why permutation models prevail" J. Pure Appl. Alg. , 46 (1987) pp. 49–58 |
Variety in a category. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Variety_in_a_category&oldid=42559