One aspect of the categorisation of "equation" is the approach of Banaschewski and Herrlich. This replaces an equation as a pair (s,t) of terms that belong to some free algebra F by the least quotient (coequaliser) e: F -->> F/E identifying s and t. An algebra A satisfies equation (s,t) precisely when every homomorphism F --> A lifts across e to a homomorphism F/E --> A. Thus the notion of an equation becomes that of a regular epi e with free domain, and an object satisfies such an e when it is injective for e. To obtain a notion intrinsic to a given category, the free domains were replaced by domains that are regular-projective, i.e. projective for all regular epis, this being a property enjoyed by free algebras in categories of universal algebras. The approach has been dualised in the coalgebra literature, with a "coequation" being defined as a regular mono with regular-injective codomain, and a "covariety" as the class of coalgebras that are projective for some given class of coequations. Some (not all) relevant references are below. cheers, Rob @Article{ bana:subc76, author = "B. Banaschewski and H. Herrlich", title = "Subcategories Defined by Implications", journal = "Houston Journal of Mathematics", year = "1976", volume = "2", number = "2", pages = "149--171" } @Article{ adam:vari03, author = {Ji{\v{r}}{\'\i} Ad{\'a}mek and Hans-E. Porst}, title = "On Varieties and Covarieties in a Category", journal = "Mathematical Structures in Computer Science", year = "2003", volume = {13}, pages = "201-232" } @Techreport{ awod:coal00, author = "Steve Awodey and Jesse Hughes", title = "The Coalgebraic Dual of {B}irkhoff's Variety Theorem", institution = "Department of Philosophy, Carnegie Mellon University", year = "2000", number = "CMU-PHIL-109", note = "\url{http://phiwumbda.org/~jesse/papers/ index.html}" } On 8/08/2006, at 1:36 AM, 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