Quantale Theory 101 [was: is 0 prime?]

Quantale Theory 101 [was: is 0 prime?]

[Jeff Egger]

--- 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.



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Re: Quantale Theory 101 [was: is 0 prime?]

[Vaughan Pratt]

> As to the question of "why?", I have a very biased and unscientific
> answer: Sup is the most awesome category.

Oh, *there*'s the problem.  I was getting quite puzzled about all this
stuff.  Presumably by Sup you mean what Peter Johnstone calls CSLat,
complete semilattices, which is a lovely self-dual category.  (If not
ignore the following.)

According it the status of "the most awesome" however is a symptom of
not yet having come to grips with the joy of Chu, a more awesome
self-dual category (fully) embedding CSLat in a duality-preserving and
concrete-preserving way while exhibiting that duality as simply matrix
transposition, yet still not *the* most awesome.   And all that just in
Chu(Set,2).  Chu(Set,8) embeds Grp, and concretely at that, which is
more awesome but still not awesome to the max.  More awesome yet is that
you can concretely embed every category of relational structures of
total arity n in Chu(Set,2^n)---Grp fits that description on account of
the group multiplication being a ternary relation, whence Chu(Set,8)).
And so on.

If going up only reduces the awe, then one should instead go down from
CSLat for greater awe.  God and the devil command a degree of awe that
the middle class is hard pressed to match.

Not only am I not a ring theorist but it's never occurred to me even to
play one on the Internet.  On the matter of the ideals of R, it would be
very nice if they were just the endomorphisms of R but presumably that
doesn't work on the ground that not every quotient of R embeds as a
subring of R---if that's wrong then I'm really confused.

I'm not a category theorist either but I do try.  Isn't the obvious
gadget to extract from R not its lattice of ideals but its category of
quotients suitably defined?  Bill, is that what you were getting at?

Vaughan