# Todd Trimble Varieties of algebras

Varieties of algebras

# Varieties of algebras

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.

## Idea

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.

## Definitions

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.)

## Basic results

###### Proposition

A variety, considered as a category in its own right, is an algebraic category.

###### Proposition

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?.

###### Theorem

(“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.)

## Examples

• Jiri Adamek, Jiri Rosicky, Enrico M. Vitale, Algebraic theories: a categorical introduction to general algebra, Nov. 20, 2009. (pdf)
Revised on July 29, 2013 at 06:52:18 by Todd Trimble