Hi Tom, a lot is known about this. I will leave it to more qualified others to give the category-theoretic account. In set-like language, the answer to your question is provided by universal algebra. Denote by Th(G) the theory associated to a particular algebra G (over a given signature). More generally, to a class of algebras S (all over the same, from now on fixed, signature), associate Th(S), the theory of all those equations satisfied by all the algebras in S. Also, to a given theory T, let V(T) be the class of all algebras satisfying the equations in T (also called a variety of algebras). Birkhoff's HSP theorem states that a class C of algebras is of the form V(T), for some T, if and only if C is closed under isomorphism, and under the operations of taking quotient algebras, subalgebras, and cartesian products. (HSP stands for "homomorphic image, subalgebra, product"). As a direct consequence, let C=V(Th(G)), the class of all groups satisfying those equations that a particular group G satisfies. Then C is precisely the class of groups that can be obtained, up to isomorphism, from G by repeatedly taking quotients, subalgebras, and cartesian products. [Proof: certainly, the right-hand side is contained in C. Conversely, by the HSP theorem, the right-hand side class is of the form V(T), for some T. Since G is in the class, T can only contain equations that hold in G, thus T is a subset of Th(G). By contravariance of the "V" operation, it follows that C=V(Th(G)) is a subset of V(T)]. Moreover, since a subalgebra of a quotient is a quotient of a subalgebra, and a cartesian product of quotients [subalgebras] is a quotient [subalgebra] of a cartesian product, the three HSP operations can be taken in this particular order: Thus, a group satisfies all the equations that G satisfies, if and only if it is isomorphic to a quotient of a subalgebra of some (possibly infinite) product G x ... x G. There are generalizations to properties other than equational ones, but I don't remember them as well. A "Horn clause" is an implication between equations, or more precisely a property of the form (forall x1...xn)(P1 and ... and Pn => Q), where P1,...,Pn,Q are equations. Of course, every equation is trivially a Horn clause (for n=0), but not the other way around. A typical example of a Horn clause is cancellability, the property xz=yz => x=y. (This holds in groups, but not in monoids, and cannot be expressed equationally in monoids, because it is not preserved under quotients). If you want to consider the class of algebras (in general smaller than V(Th(G))) that satisfy all the Horn clauses that G satisfies, then you have to drop the homomorphic images. I believe that the algebras in question will be precisely the subalgebras of products of G, but someone might correct me if I remember this wrongly. -- Peter Tom Leinster wrote:
Dear category theorists,
Here's something that I don't understand. People sometimes talk about algebraic structures "satisfying laws". E.g. let's take groups. Being abelian is a law; it says that the equation xy = yx holds. A group G "satisfies no laws" if
whenever X is a set and w, w' are distinct elements of the free group F(X) on X, there exists a homomorphism f: F(X) ---> G such that f(w) and f(w') are distinct.
For example, an abelian group cannot satisfy no laws, since you could take X = {x, y}, w = xy, and w' = yx. There are various interesting examples of groups that satisfy no laws.
To be rather concrete about it, you could define a "law satisfied by G" to be a triple (X, w, w') consisting of a set X and elements w, w' of F(X), such that every homomorphism F(X) ---> G sends w and w' to the same thing. A law is "trivial" if w = w'. Then "satisfies no laws" means "satisfies only trivial laws".
You could then say: given a group G, consider the groups that satisfy all the laws satisfied by G. (E.g. if G is abelian then all such groups will be abelian.) This is going to be a new algebraic theory.
What bothers me is that I feel there must be some categorical story I'm missing here. Everything above is very concrete; for instance, it's heavily set-based. What's known about all this? In particular, what's known about the process described in the previous paragraph, whereby any theory T and T-algebra G give rise to a new theory?
Thanks, Tom