A simple answer to Tom Leinster's question involves the Galois connection well-analyzed by Michel Hebert at Whitepoint (2006): in a fixed category an object A can "satisfy" a morphism q: F->Q iff q*: (Q,A) -> (F,A) is a bijection. Then for any class of objects A there is the class of "laws" q satisfied by all of them, and reciprocally. If the category itself is mildly exact, one could instead of morphisms q consider their kernels as reflexive pairs. For example, if there is a free notion, a reflexive pair F' =>F has a coequalizer which could be taken as a law q.
However, the "categorical story" that Tom was missing is not told well by the "Universal Algebra" of 75 years ago. Unfortunately, Galois connections in the sense of Ore are not "universal" enough to explicate
Thanks to Bill Lawvere for pointing out the close connection between the questions I was asking and the recent work of Jiri Adamek, Michel Hebert and Lurdes Sousa, presented at both CT06 and the Glasgow PSSL. It must have lodged itself in my mind in some subliminal way; apologies for not mentioning it earlier. Perhaps the following is rather basic, but I'm failing to understand one of the points in Bill's message. As I read it, he's saying that the Galois connection of traditional universal algebra (connecting sets of laws and varieties of algebras) is contained within the structure-semantics adjunction of his thesis. I don't see how this works. I *do* understand the following points: 1. Traditional universal algebra - given a signature S, one has the set E of equations between S-terms and the class V of S-algebras, and the relation "satisfaction" gives a Galois connection between the power-sets/classes P(E) and P(V). A Galois connection is, of course, a contravariant adjunction on the right between posets. 2. Categorical algebra - structure and semantics form a contravariant adjunction on the right between the category Th of Lawvere theories and (roughly speaking) the category K of categories over Set. One is a section of the other: if T is a theory then Struc(Sem(T)) = T. 3. If T is a theory, any equation between the operations in T can be construed as a pair of parallel arrows in T, and so induces a map from T to the quotient theory T' obtained by imposing this equation. Such a map T ---> T' is an epimorphism, although not every epi in the category of theories arises in this way. 4. Given a contravariant adjunction on the right, both functors turn colimits into limits, hence epis into monos. In particular, any set of equations between the operations of a theory T induces an epi T ---> T', hence a mono Sem(T') ---> Sem(T) between the categories of models, which may perhaps be the inclusion of a full subcategory. This makes it look as if there's going to be a Galois connection between the poset of quotient objects of T (i.e. epis out of T) and subobjects of Sem(T), for every theory T. But there seem to be two problems: (i) the functors in an adjunction on the right don't in general turn monos into epis, so I don't see why the structure-semantics adjunction is going to turn subcategories of Sem(T) into quotient theories of T; (ii) even if this did work, epis out of T are more general than equations, and monos into Sem(T) are more general than full subcategories, so it wouldn't exactly recover the classical Galois connection of universal algebra. I guess I've made a wrong turn somewhere; can someone put me right? Thanks, Tom the related universal phenomena in algebra, algebraic geometry, and functional analysis. The mere order-reversing maps between posets of classes are usually restrictions of adjoint functors between categories, and noting this explicitly gives further information. For example, Birkhoff's theorem does not apply well to the question:
"Do groups form a variety of monoids?"
Indeed, does the word "variety" mean a kind of category or a kind of
course the category of groups does become a variety if we adjoin an additional operation to the theory of monoids.
In my thesis (1963) (now available on-line as a TAC Reprint, and extensively elaborated on by Linton and others in SLNM 80) I isolated an adjoint
"Structure/Semantics" strictly analogous to the basic "Function algebra/Spectrum) pairs occurring in algebraic geometry and in functional analysis. In that context, note that the epimorphisms in the category of
given by equations, dual semantically to Birkhoff subvarieties) as well as localizations (laws given by adjoining inverses to previously given operations, semantically corresponding to "open" algebraic subcategories). Can these "open" inclusions between algebraic categories be characterized semantically?
The technical notion "Structure of" was motivated by the example of cohomology operations: in general, the totality of natural operations on
than those of the codomain category. The example illustrates that such adjoints are of much broader interest than the mere perfect duality that one might obtain by restricting both sides (one does not expect to recover a space from its cohomology, and the category of spaces studied is not even an algebraic category).
As an important further example of a large adjoint which specializes both to Galois connections in each space as well as to a perfect duality on suitable subcategories, consider Stone's study of the relation between spaces and real commutative algebras; for computational purposes, the spaces of the form C(X) need to receive morphisms from algebras A (like
inclusion functor? In algebraic geometry, an analogous question concerns whether an algebraic space that is a subspace of another one is closed (i.e. definable by equations) or not. Often instead it is defined by inverting some global functions, giving an open subscheme, not a subvariety, but still a good subspace. The analogy goes still further; a typical open subspace of X is actually a closed subspace of X x R, and of pair theories (categories with finite products) include both surjections (laws the values of a given functor involves both more operations and more laws polynomial algebras) that are not of that form; such homomorphisms are by adjointness equivalent to continuous maps X --> Spec(A), where Spec(A) would map further to R^n if n were a chosen parameterizer for generators of A in a presentation.
Best wishes to all.
Bill
************************************************************ F. William Lawvere Mathematics Department, State University of New York 244 Mathematics Building, Buffalo, N.Y. 14260-2900 USA Tel. 716-645-6284 HOMEPAGE: http://www.acsu.buffalo.edu/~wlawvere ************************************************************