Variety in a category

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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 well-powered, 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.


[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


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.


[a1] P.J. Freyd, "All topoi are localic, or why permutation models prevail" J. Pure Appl. Alg. , 46 (1987) pp. 49–58
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Variety in a category. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by M.Sh. Tsalenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article