Note: This started as an nLab article, one that I took some issue with and wanted to rewrite. I’m not sure that a single word of the original remains.
In categorical universal algebra, a variety may refer to one of several things:
An equational class of algebras, carved out as a full subcategory of a category of structures for some algebraic signature $\Sigma$, and consisting of those objects where a given set of equations in the signature is satisfied. (For example, the category of groups.)
Or (similarly but more generally), a full subcategory of a category of models for some algebraic theory, consisting of those objects where a given set of extra equations is satisfied. (For example, the commutative rings is a variety in the category of rings.)
Or, a category that is equivalent to such a full subcategory.
In this article, we are principally concerned with the second and third senses.
A variety of algebras, as a notion from universal algebra, is of course not to be confused with the algebro-geometric notion of algebraic variety?. However, there is a kinship, in that just as an algebraic variety is carved out of an affine space $k^n$ as a solution set of some set of equations, so too is a variety of algebras carved out from a larger category of algebras by specifying some extra algebraic equations that must be met.
The following definitions are adapted from Adamek.
By an abstract (finitary) algebraic theory, we will simply understand a small category $T$ with finite products?. (Compare Lawvere theory?, algebraic theory?.) A model or algebra of $T$ is by definition a product-preserving functor $A: T \to Set$. A homomorphism of models is simply a natural transformation $\theta: A \to B$. The category of $T$-algebras and their homomorphisms will be denoted $Alg_T$.
A category equivalent to one of the form $Alg_T$ may be called an algebraic category. Algebraic categories are locally finitely presentable, hence in particular are complete and cocomplete. (Actually much more is true: algebraic categories are also Barr exact.)
If $T$ is an abstract algebraic theory, an equation in $T$ is a parallel pair of morphisms $u, v: s \stackrel{\to}{\to} t$ in $T$. Following algebraic tradition, such an equation will be written $[u = v]$. If $A: T \to Set$ is a model of $T$, we say that an equation $[u = v]$ is satisfied in $A$ (or $A$ satisfies the equation) if $A(u) = A(v)$.
A full subcategory $V \hookrightarrow Alg_T$ is called a variety if there is a set of equations in $T$ such that the objects of $V$ are precisely those $T$-algebras that satisfy all the equations in the set.
(Frequently it is categorically more convenient or appropriate to loosen this notion, and consider a variety of $T$-algebras to be a fully faithful functor $i: V \to Alg_T$ whose essential image? is the class of $T$-algebras that satisfy the equations.)
A variety, considered as a category in its own right, is an algebraic category.
A variety $V \hookrightarrow Alg_T$ is closed under the following constructions:
Products: If $\{A_i\}_{i \in I}$ is a collection of objects of $V$, then the product $\prod_i A_i$ in $Alg_T$ lies in $V$.
Subobjects: If $i: A \to B$ is a monomorphism and $B$ belongs to $V$, then so does $A$.
It follows that $V$ is closed under limits?.
Images: If $q: B \to A$ is a regular epi and $B$ belongs to $V$, then so does $A$.
Sifted colimits: if $J$ is a small sifted category? and $F: J \to Alg_T$ is a diagram for which $F(j)$ belongs to $V$ for every $j \in Ob(J)$, then the colimit of $J$ also belongs to $V$.
In particular, $V$ is closed under filtered colimits? and reflexive coequalizers?.
(“Birkhoff’s Variety Theorem) A full subcategory $V \hookrightarrow Alg_T$ is a variety if and only if it is closed under products, subobjects, regular quotients, and directed unions.
(To be elaborated more. For example, compared with the classical HSP theorem.)