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algebraic theory

Algebraic theories

Idea
Metaphor
In higher category theory
References

Idea

An algebraic theory is a concept in universal algebra that describes a specific type of algebraic gadget, such as groups or rings. An individual group or ring is a model of the appropriate theory. Roughly speaking, an algebraic theory consists of a specification of operations and laws that these operations must satisfy.

Traditionally, algebraic theories were described in terms of logic. But finitary algebraic theories (that is, those involving only finitary operations) can be understood category-theoretically as Lawvere theories. More generally, algebraic theories involving only operations of arity bounded by some cardinal number can be understood category-theoretically with a suitably generalization of Lawvere theories. However, there are also algebraic theories with operations of unbounded arity, such as the theory of algebras in which arbitrary sums are possible (one model of which is $[0, ∞]$ ), or the theory of complete lattices; these can also be modeled by certain ‘large Lawvere theories,’ but the notion is not as well-behaved as in the bounded case; see this thread.

Algebraic theories (finitary or otherwise) can also be understood through monads; in fact, some category theorists define an algebraic category to be one that is monadic over Set. For example, the theory of a compact Hausdorff space $X$ can be seen as algebraic, with one operation for each directed set $D$ that takes an ultranet indexed by $D$ to its limit; this is a finitary operation only when $D$ is finite (which is really the trivial case). There is no Lawvere theory for compact Hausdorff spaces (at least, no small one), yet there is a monad for it, which maps each set $S$ to its set of ultrafilters.

Here is the connection between the logical and categorial descriptions, based on Johnstone, §§3.7&8. Say that a category $C$ is:

algebraic if it is given by a monad on the category of (small) sets;
small algebraic if it is given by a (small) set of operation symbols and equations;
large algebraic if it is given by a (possibly proper) class of operation symbols and equations.

Then any small algebraic category is algebraic, and any algebraic category is large algebraic, but neither implication may be reversed.

Essentially algebraic theories allow for partially-defined operations. Just as finitary algebraic theories can be understood as Lawvere theories, which live in the doctrine of cartesian monoidal categories, so finitary essentially algebraic theories can be understood by a generalisation to finitely complete categories.

Commutative algebraic theories form an important subclass. Their categories of models are closed: the Hom sets have a natural model-structure, and the enriched Hom-functor has a left adjoint, tensor-product. The theory of complete lattices and suprema-preserving functions is an interesting (non-finitary) example.

Metaphor

Ring theory is a branch of mathematics with a well-developed terminology. A ring $A$ determines and is determined by an algebraic theory, whose models are left $A$ -modules and whose n-ary operations have the form

(x_{1}, \dots, x_{n}) \to a_{1} x_{1} + \dots + a_{n} x_{n}

for some n-ple $(a_{1}, \dots, a_{n})$ of elements of $A$ . We may call such an algebraic theory annular. The pun model/module is due to Jon Beck. The notion that an algebraic theory is a generalized ring is often a fertile one, that automatically provides a slew of suggestive terminology and interesting problems. Many fundamental ideas of ring/module-theory are simply the restriction to annular algebraic theories of ideas that apply more widely to algebraic theories and their models.

Let us denote the category of models and homomorphisms (in $Set$ ) of an algebraic theory $A$ by $A Mod$ . Then compare the following to their counterparts in ring theory:

In higher category theory

(∞,1)-algebraic theory

References

Peter Johnstone, Stone Spaces

Someone should improve this article so that it gives a definition of ‘algebraic theory’ before considering special cases such as ‘commutative algebraic theory’.