We exhibit a subclass of quantales such that the slice categories of members are themselves in the subclass. That is, akin to the fundamental theorem of toposes: “slices of toposes are themselves toposes”, we find X-quantales so that slices of X-quantales are themselves X-quantales.

This is not known to be very useful at the momement. But history shows, and reason expects, it to better to publish something before it is known to be important than it is to publish it after. Either way, I hope this may help someone out or inspire anyone.

Introduction

Quantales are similar to Frames/Locales/Complete Heyting Algebras; in that they are complete partial orders with a distributive (over suprema) monoid structures. The difference is that in the case of Locales (all those others are really equivalent/only differ in the kinds of morphisms considered, not the actual structures) the monoidal operation is actually the infimum.

Let’s spell this out: a quantale is a colection of data \((Q,\leq,\top,\bot,\wedge,\vee,\otimes,I)\) such that \((Q,\leq)\) is a complete partial order, with maximum element \(\top\) , minimum \(\bot\) , infimum \(\wedge\) and supremum \(\vee\) ; and \((Q,\otimes, I)\) is a monoid: that is, \(\otimes\) is an associate operation on \(Q\) with neutral element \(I\) . Furthermore, \[ a\otimes\bigvee_{x\in X} b_x = \bigvee_{x\in X} a\otimes b_x \] \[ \left(\bigvee_{x\in X} a_x\right)\otimes b = \bigvee_{x\in X} a_x\otimes b \]

Quantales arise from ring theory (ideals of unital commutative rings); MV-algebras; Complete Heyting Algebras, as mentioned; multi-parts of sets (that is, for a given \(X\) , $ {f:DN{}   DX} $ ); as well as in \(C^\ast\) -algebras (therefore relating them to weird quantum stuff as well as functional analysis) by taking closed right the ideals of a given algebra.

Quantales come in many different flavors, and while some work has been done on the subject of idempotent and right-sided quantales (as those that arise from functional analysis and quantum mechanics) [see – say – M. Coniglio’s PhD thesis 1997 “A lógica dos feixes sobre quantais right-sided e idempotentes” ¹] there are still “fresh” quantales to talk about.

Any quantale has a right and left implication called \(\multimap\) (we’ll pick a side, say, right implication) given by \[ a\multimap b = \bigvee \{c\ \mid \ a\otimes c \leq b\} \] and this supremum is actually a maximum.

Some Jargon

The category of Quantales as has arrows functions that preserve arbitrary suprema (therefore they send \(\bot\) to \(\bot\) ), preserve the \(\top\) element, and are monoidal morphisms: preserve monoidal product and monoidal unit.

A quantale \(Q\) is said to be integral/semicartesian when \(\top=I\) . That enables us to prove – for instance – that \(a\otimes b\leq a\wedge b\) . The “semicartesian” name comes from category theory, and relates to the fact that there is a “nice” canonical natural transformation \(\otimes\Rightarrow\times\) . This whole definition is somewhat standard and well known.

A quantale \(Q\) is said to be “divisible” or – how I called it when I (re)discovered this (useful but otherwise unremarkable) property – “well residuated” if whenever \(a\leq b\) then there is some \(c\) such that \(a=b\otimes c\) . Well you could say it’s left well-residuated but who cares about it that much?

Finally, it is commutative if \(\otimes\) is a commutative operation.

Statement

In general, Locales have a natural and very canonical local localic structure By which I mean: when regarding \(L\) there is a contravariant functor \(L\to\mathrm{Locale}\) sending objects to their slices, which have a natural locale structure induced from \(L\) .

Namely we take \(a\leq b\) to the functor \(L/b\to L/a\) which takes an object \(c\) and yields \(c\wedge a\) .

This song and dance is not possible on quantales – at least very naturally or as far as I could tell. One big problem being: it’s seems to be hard to preserve the local \(\odot\) s and the units. The question then becomes, can we find a (full) subcategory \[ \mathrm{GoodQuantale} \hookrightarrow \mathrm{Quantale} \] such that the slice functor for every good quantale \(G\) is a \(G\) -indexed category over \(\mathrm{GoodQuantale}\) – alternatively we could ask if [G̷̢͐r̵͍̃o̷̩͝t̸̪̑h̷̲́e̸̞̍n̶̙͊d̶̟̈́ḯ̸̪e̶͕͊c̵̐ͅk̵͖̕ ̶͕͠C̷̖̉ó̸̞n̶̜̄s̶͚͌t̵̤̓r̷̮̈́u̷̥̚c̷̢̏t̸̘͂i̶͙͝o̴̪̾n̶͙̑ noises]. It all comes down to asking if the slices are themselves good quantales and if slicing is contravariantly functorial.

🥎ミ 🐕三 Good Quantale!

Well, unsurprisingly, the (a) subcategory of quantales which are good in that regard is the one comprised of (commutative) semicartesian and divisible quantales. What a twist! The only concepts I’ve introduced are the ones we needed?? How very efficient and un-textbooklike of us, dear reader.

For any quantale, it’s not hard to show that \[ x\otimes (x\multimap (x\otimes a)) = x\otimes a \]

Proof: this quantity, whatever it may be, call it \(\gamma\) . \[ x\otimes\bigvee\{c\ \mid\ x\otimes c\leq x\otimes a\} = \bigvee\{x\otimes c\ \mid\ x\otimes c\leq x\otimes a\} = x\otimes a \]

If a quantale is divisible, then one may prove the following – neater – result: \[ a \leq b \implies b\otimes (b\multimap a) = a \] as \(a = b\otimes c\) for some \(c\) , and therefore we may apply the previous lemma and obtain what we set out for. A useful property of such quantales is the fact that we have a “canonical” such \(c\) , in the form of \(b\multimap a\) as provided by what we just proved.

Not only canonical, but universal, as \(b\multimap a\) is the biggest quantity that when tensored by \(b\) on the left yields less than \(a\) , and any aspiring \(c\) must do that.

I found out about this property, as one often does, in the shower – contemplating the quantale \[ (\mathbb R\cup\{\infty\}, \leq^{\mathrm{op}}, 0, \infty, \max,\min, +, 0) \]

And this led me to find out about divisible quantales, a very much not novel concept. Given a divisible quantale, we may try to define a monoidal structure on one of its slices then. Take \(b\in Q\) , define, for \(a, a'\leq b\) \[ a\otimes_b a' = b\otimes (b\multimap a)\otimes(b\multimap a') \]

Unit and Integrality

From this definition, it’s clear that the following equations hold: \[ a\otimes_b b = b\otimes (b\multimap a)\otimes (b\multimap b) = a \otimes \top \] \[ b\otimes_b a = b\otimes (b\multimap b)\otimes (b\multimap a) = b\otimes \top \otimes (b\multimap a) \]

If we additionally demand \(Q\) be semicartesian, then \(b\) becomes the unit of \(\otimes\) as well as the top element on \(Q/b\) .

Distributivity over Suprema

\[ \begin{aligned} \left(\bigvee_i z_i\right)\otimes_b a &= b\otimes\left(b\multimap\bigvee_i z_i\right)\otimes(b\multimap a)\\ \text{but this supremum is smaller than $b$ }\\ &= \left(\bigvee_i z_i\right)\otimes(b\multimap a)\\ &= \bigvee_i z_i\otimes(b\multimap a)\\ &= \bigvee_i b\otimes(b\multimap z_i)\otimes(b\multimap a)\\ &= \bigvee_i z_i\otimes_b a \end{aligned} \] So we have proved one side of the distributivity, we need the other side of course. But… yes, it seems that it doesn’t quite hold. Maybe I’m right, maybe I’m wrong, but it is a sure thing that I didn’t manage to coax it from just integrality and divisibility.

So let’s say that the monoid is commutative – as most good things are. So good, in fact, that we have finished proving the distributivity if that is the case, since if \(\otimes\) is commutative, then so is \(\otimes_b\) and we are done.

Associativity

We ought to prove \(x\otimes_b(y\otimes_b z)\) is the same as \((x\otimes_b y)\otimes_b z\) this is easy to accomplish with commutativity but simply wouldn’t be the case otherwise, so it further confirms its actual necessity.

\[ \begin{aligned} x\otimes_b(y\otimes_bz) &= b\otimes (b\multimap x)\otimes ( b\multimap[b\otimes(b\multimap y)\otimes(b\multimap z)] )\\ &= (b\multimap x)\otimes b\otimes ( b\multimap[b\otimes(b\multimap y)\otimes(b\multimap z)] )\\ &= (b\multimap x)\otimes b\otimes(b\multimap y)\otimes(b\multimap z) \\ &= b\otimes(b\multimap x)\otimes (b\multimap y)\otimes(b\multimap z) \\ &= [b\otimes(b\multimap x)\otimes (b\multimap y)]\otimes(b\multimap z) \\ &= b\otimes(b\multimap[b\otimes(b\multimap x)\otimes (b\multimap y)])\otimes(b\multimap z) \\ &= b\otimes(b\multimap[x\otimes_b y])\otimes(b\multimap z) \\ &= (x\otimes_b y)\otimes_b z \end{aligned} \]

Divisibility

We know that divisibility gives a canonical candidate: if \(a\leq b\) then \(b\otimes(b\multimap a) = a\) ; if \(x\leq y\leq b\) then we ought to find some \(z\leq b\) such that \(y\otimes_b z=x\) . We have a good guess: \[ b \otimes (y\multimap x)\leq b \] which is less than \(b\) due to integrality!

\[ \begin{aligned} y\otimes_b (b\otimes(y\multimap x)) &= b\otimes(b\multimap y)\otimes(b\multimap[b\otimes(y\multimap x)])\\ &=(b\multimap y)\otimes b\otimes(b\multimap[b\otimes(y\multimap x)])\\ &=(b\multimap y)\otimes [b\otimes(y\multimap x)]\\ &=b\otimes(b\multimap y)\otimes(y\multimap x)\\ &=y\otimes(y\multimap x)\\ &=x\\ \end{aligned} \] as desired.

Putting it together

Trivially, \(Q/b\) is a complete partial order by inheriting from \(Q\) . We’ve defined an associative operation over it which distributes over suprema and has a unit – which coincides with the maximum of the partial order. Furthermore, that operation is commutative and enjoys the divisibility property.

Therefore, if \(Q\) is a commutative integral and divisible quantale then \(Q/b\) is one such quantale as well. Moreover, it is immediate to see that \(Q/\top\) is simply \(Q\) – I swear it’s easy ok, I’m not dodging the proof.

A Matter of Coherence

We already know how to take some \(b\) in a good quantale to a good quantale \(Q/b\) but there is more work to be done. What does the fact \(a\leq b\) imply – in terms of connections – between \(Q/a\) and \(Q/b\) . Can we write functors (or – really – quantale morphisms) between them? Can we do it functorially?

Suppose \(a\leq b\) as above, then consider the map \[ \begin{aligned} Q/(a\leq b): Q/b&\to Q/a\\ c&\mapsto a\otimes_b c \end{aligned} \]

We know \(\otimes_b\) is an increasing map – because it distributes over suprema etc. – and exactly because it distributes over suprema, it also sends suprema to suprema, since \(Q/a\) ’s order is inherited from \(Q/b\) ’s.

Even better, \(a\otimes_b b\) is \(a\) , so \(\otimes_b\) sends the top element of \(Q/b\) to the top element of \(Q/a\) . Really, all we need to show is that \(a\otimes_b\_\) is a monoid morphism and we will have produced a quantale morphism between them; which is going to contravariantly vary on \(Q\) .

Morphism is Monoidal

Take \(x,y\in Q/b\) ,²

\[ \begin{aligned} (a\otimes_b x)\otimes_a(a\otimes_b y) &= a\otimes (a\multimap [a\otimes_b x])\otimes (a\multimap [a\otimes_b y])\\ &= a\otimes (a\multimap [b\otimes (b\multimap a)\otimes (b\multimap x)]) \otimes (a\multimap [a\otimes_b y])\\ &= a\otimes (a\multimap [a \otimes (b\multimap x)])\otimes (a\multimap [a\otimes_b y])\\ &= a \otimes (b\multimap x)\otimes (a\multimap [a\otimes_b y])\\ &= (b\multimap x)\otimes a \otimes (a\multimap [a\otimes_b y])\\ &= (b\multimap x)\otimes (a\otimes_b y)\\ &= (b\multimap x)\otimes b\otimes(b\multimap a) \otimes(b\multimap y)\\ &= b\otimes(b\multimap a)\otimes (b\multimap x) \otimes(b\multimap y) \end{aligned} \] and the above quantity, we know – from the proof of associativity – to be \[ a\otimes_b(x\otimes_b y) \] and so, just as desired, the map \(Q/(a\leq b)\) is a quantale morphism. In fact what we have shown is just that the canonical self-indexing (the slice pseudo functor) factors through \(\mathrm{GoodQuantale}\) .

Further Directions

One could investigate how arrows in \(\mathrm{GoodQuantale}\) themselves act on slices; this ought to be straightforward: the local monoidal structure is defined in terms of the total structure, so it stands to reason that arrows act on slices in the natural way. Additional properties of those interactions could be investigated.