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:

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 nk^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

The following definitions are adapted from Adamek.

By an abstract (finitary) algebraic theory, we will simply understand a small category TT with finite products?. (Compare Lawvere theory?, algebraic theory?.) A model or algebra of TT is by definition a product-preserving functor A:TSetA: T \to Set. A homomorphism of models is simply a natural transformation θ:AB\theta: A \to B. The category of TT-algebras and their homomorphisms will be denoted Alg TAlg_T.

A category equivalent to one of the form Alg TAlg_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 TT is an abstract algebraic theory, an equation in TT is a parallel pair of morphisms u,v:stu, v: s \stackrel{\to}{\to} t in TT. Following algebraic tradition, such an equation will be written [u=v][u = v]. If A:TSetA: T \to Set is a model of TT, we say that an equation [u=v][u = v] is satisfied in AA (or AA satisfies the equation) if A(u)=A(v)A(u) = A(v).

A full subcategory VAlg TV \hookrightarrow Alg_T is called a variety if there is a set of equations in TT such that the objects of VV are precisely those TT-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 TT-algebras to be a fully faithful functor i:VAlg Ti: V \to Alg_T whose essential image? is the class of TT-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 VAlg TV \hookrightarrow Alg_T is closed under the following constructions:

  • Products: If {A i} iI\{A_i\}_{i \in I} is a collection of objects of VV, then the product iA i\prod_i A_i in Alg TAlg_T lies in VV.

  • Subobjects: If i:ABi: A \to B is a monomorphism and BB belongs to VV, then so does AA.

It follows that VV is closed under limits?.

  • Images: If q:BAq: B \to A is a regular epi and BB belongs to VV, then so does AA.

  • Sifted colimits: if JJ is a small sifted category? and F:JAlg TF: J \to Alg_T is a diagram for which F(j)F(j) belongs to VV for every jOb(J)j \in Ob(J), then the colimit of JJ also belongs to VV.

In particular, VV is closed under filtered colimits? and reflexive coequalizers?.

Theorem

(“Birkhoff’s Variety Theorem) A full subcategory VAlg TV \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

References

Revised on July 29, 2013 at 06:52:18 by Todd Trimble