Variety in a category

A notion generalizing that of a variety of universal algebras. Let $ \mathfrak K $ be a bicategory with products. A full subcategory $ \mathfrak M $ of $ \mathfrak K $ is called a variety if it satisfies the following conditions: a) if $ \mu : A \rightarrow B $ is an admissible monomorphism and $ B \in \mathop{\rm Ob} \mathfrak M $, then $ A \in \mathop{\rm Ob} \mathfrak M $; b) if $ \nu : A \rightarrow B $ is an admissible epimorphism and $ A \in \mathop{\rm Ob} \mathfrak M $, then $ B \in \mathop{\rm Ob} \mathfrak M $; c) if $ A _ {i} \in \mathop{\rm Ob} \mathfrak M $, $ i \in I $, then $ A = \prod _ {i \in I } A _ {i} \in \mathop{\rm Ob} \mathfrak M $.

If $ \mathfrak K $ is a well-powered category, that is, the admissible subobjects of any object form a set, then every variety is a reflective subcategory of $ \mathfrak K $. This means that the inclusion functor $ I : \mathfrak M \rightarrow \mathfrak K $ has a left adjoint $ S : \mathfrak K \rightarrow \mathfrak M $. The unit of this adjunction, the natural transformation $ \eta : I _ {\mathfrak K } \rightarrow T = S I $, has the property that for each $ a \in \mathop{\rm Ob} {\mathfrak K } $ the morphism $ \eta _ {A} : A \rightarrow T ( A) $ is an admissible epimorphism. In many important cases the functor $ T $ turns out to be right-exact, that is, it transforms the cokernel $ \nu $ of a pair of morphisms $ \alpha , \beta : A \rightarrow B $ into the cokernel of the pair of morphisms $ T ( \alpha ) , T ( \beta ) $, if $ ( \alpha , \beta ) $ is a kernel pair of the morphism $ \nu $. Moreover, right exactness and the presence of the natural transformation $ \eta : I \rightarrow T $ are characteristic properties of $ T $.

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