The following seems so obvious that I suspect it's not what Tom is really asking for; but it seems to me to be an answer to his question. A law in Tom's sense is just a parallel pair of arrows F(X) \rightrightarrows F(1) in the algebraic theory T under consideration (thinking of T as the dual of the category of finitely-generated free algebras). To get the theory of algebras satisfying a given set S of laws, you just need to construct the product-respecting congruence on T generated by S (i.e., the usual closure conditions for a congruence, plus the condition that f ~ f' and g ~ g' imply f x g ~ f' x g'), and factor out by it. Now any T-algebra A (in a category C, say) corresponds to a product- preserving functor F: T --> C; and the set of laws satisfied by A is just the (necessarily product-respecting) congruence generated by F, i.e. the set of parallel pairs in T having the same image under F. Is there anything more to it than that? Peter Johnstone ------------ On Mon, 7 Aug 2006, 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