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A notion generalizing that of a [[Variety of universal algebras|variety of universal algebras]]. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962801.png" /> be a [[Bicategory(2)|bicategory]] with products. A [[Full subcategory|full subcategory]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962802.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962803.png" /> is called a variety if it satisfies the following conditions: a) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962804.png" /> is an admissible monomorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962805.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962806.png" />; b) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962807.png" /> is an admissible epimorphism and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962808.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v0962809.png" />; c) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628010.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628011.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628012.png" />.

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628013.png" /> is well-powered, that is, the admissible subobjects of any object form a set, then every variety is a [[Reflective subcategory|reflective subcategory]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628014.png" />. This means that the inclusion functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628015.png" /> has a left adjoint <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628016.png" />. The unit of this adjunction, the natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628017.png" />, has the property that for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628018.png" /> the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628019.png" /> is an admissible epimorphism. In many important cases the functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628020.png" /> turns out to be right-exact, that is, it transforms the cokernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628021.png" /> of a pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628022.png" /> into the cokernel of the pair of morphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628023.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628024.png" /> is a [[Kernel pair|kernel pair]] of the morphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628025.png" />. Moreover, right exactness and the presence of the natural transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628026.png" /> are characteristic properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096280/v09628027.png" />.

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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.Sh. Tsalenko,   E.G. Shul'geifer,   "Fundamentals of category theory" , Moscow  (1974)  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Fröhlich,   "On groups over a d.g. near ring II. Categories and functors"  ''Quart. J. Math.'' , '''11'''  (1960)  pp. 211–228</TD></TR></table>



====Comments====
In a [[Topos|topos]], one also considers exponential varieties [[#References|[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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Freyd,   "All topoi are localic, or why permutation models prevail"  ''J. Pure Appl. Alg.'' , '''46'''  (1987)  pp. 49–58</TD></TR></table>