--- Bill Lawvere <wlawvere@buffalo.edu> wrote:

Indeed, as Jeff points out, we learned from Kummer and Dedekind to replace elements by ideals, but we categorists have been late in providing a clear account of this transition and, in particular, of the reason why the result is not primarily a lattice, but a monoidal closed category with colimits.

In fact, I think that the process of moving from rings to lattices of ideals should be seen in two stages. The first stage is to observe that the functor Ab ---> Sup which maps an abelian group to its lattice of subobjects comes equipped with a natural monoidal structure. [Sup denotes the category of complete lattices and sup-homomorphisms.] Thus monoids in Ab (rings) get mapped to monoids in Sup (quantales). [Of course, one can replace Ab, not just by CMon, but by other interesting categories, such as Ban.] The second stage is to pare the quantale of all (additive) subgroups of a ring down to that of ideals; but (left-, right-, two-sided) ideals are, by definition, precisely the (left-, right-, two-sided) elements of the quantale of subgroups, so all that remains to do is properly describe this process of paring an arbitrary quantale to its "subquantale" of two-sided elements (subquantale in the sense of sub-semigroup, not sub-monoid). Restricting to the category of commutative quantales---which I shall adopt as the case of interest, for the purposes of the present discussion (since it started out with the ring of integers)---we see that this functor is left adjoint to the forgetful functor from the category of {commutative quantales whose unit is top}. [The problem with the general case is that the "obvious" unit map: x |-> T&x&T (where T denotes top and & is quantale multiplication) need not be a quantale homomorphism; there appear to be several ways of fixing this, and I do not yet know which is the best.] The nice thing about this approach is that one then recognises the second stage as leading naturally to a third: namely, collapsing down to the frame of radical ideals (which is the topology of the space of primes I referred to in my previous post). In particular, if one regards this third stage as erroneous

The distributive lattice of radical ideals is refined to the monoidal poset of all ideals.

then one should probably regard the second stage as equally erroneous ---which is the position that the quantale theory community has largely agreed upon. As to the question of "why?", I have a very biased and unscientific answer: Sup is the most awesome category. Cheers, Jeff Egger. Ask a question on any topic and get answers from real people. Go to Yahoo! Answers and share what you know at http://ca.answers.yahoo.com