Subject: categories: Laws

From: "Tom Leinster" <[email-masked]>
Date: Mon, 7 Aug 2006 14:36:32 +0100 (BST)

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