(0,1)-category theory: logic, order theory
proset, partially ordered set (directed set, total order, linear order)
distributive lattice, completely distributive lattice, canonical extension
category with duals (list of them)
dualizable object (what they have)
ribbon category, a.k.a. tortile category
monoidal dagger-category?
A quantale is a closed monoidal suplattice. Equivalently, it is a monoid object in the closed symmetric monoidal category of suplattices where the morphisms are the set maps that preserve arbitrary joins. This means it is a poset having all joins and an associative, unital tensor product $\otimes$ which distributes over joins (the internal-homs then come automatically by the adjoint functor theorem). The internal-homs in a quantale are sometimes called residuations and written $x\backslash y$ and $y/x$. Unitality is skipped by some authors; in that case we can talk about subclass of unital quantales.
As a semigroup (monoid if unital) in suplattices, a quantale is essentially the same thing as an 1-object quantaloid, i.e., an 1-object category enriched in suplattices.
Additional conditions often imposed on a quantale include:
(On affineness: see also semicartesian monoidal category.) If idempotence and affineness are assumed, then $\otimes$ is forced to be the meet (whence commutativity is automatic) and the quantale is thereby a frame; see Proposition 1 below. General quantales are sometimes considered to be a “noncommutative” version of a frame, whose opposite category would be a category of “noncommutative locales.”
(This is the origin of the name “quantale,” a portmanteau of “quantum” and “locale”. Note, though, that quantales seem to be generally treated in the literature more as “quantum frames” than “quantum locales,” and in particular their morphisms usually go in the “frame direction.” Possibly this can be explained by the fact that in the past, it was common to use the word “locale” for what we now call a “frame” and simply distinguish between “locale homomorphisms” (now called “frame homomorphisms”) and “continuous maps.” The name “quantale” was introduced by C.J. Mulvey.)
If a monoid in the cartesian monoidal category of posets is idempotent and affine, then the monoid multiplication is the meet operation.
Let $(M, \otimes, 1)$ be the monoid. If $a \leq x$ and $a \leq y$, then $a = a \otimes a \leq x \otimes y$. On the other hand, we have $x \otimes y \leq x \otimes 1 = x$ and similarly $x \otimes y \leq y$, so $a \leq x \otimes y$ implies $a \leq x$ and $a \leq y$ by transitivity. Thus $a \leq x \otimes y$ iff $a \leq x$ and $a \leq y$, i.e., $x \otimes y$ satisfies the defining property of the meet $x \wedge y$.
Along similar lines, the following construction provides passage from commutative affine quantales to frames:
Let $(Q, \cdot, 1)$ be a commutative affine quantale, and let $Idem(Q)$ be the subposet of elements $x \cdot x = x$. Then $Idem(Q)$ is a frame, where the meet operation is given by multiplication in $Q$. The functor $Idem$ is right adjoint to the forgetful functor from commutative affine quantales to frames.
If $x, y$ are idempotent, then so is $x \cdot y$ using the fact that $\cdot$ is commutative. Thus $Idem(Q)$ is an idempotent affine submonoid of $Q$, which by Proposition 1 forces $\cdot$ to be the meet. Next, we show that $Idem(Q)$ is closed under taking joins in $Q$: if $x_i$ is a collection of idempotents, we have
for all $i$, whence
which is all we need (the opposite inequality is automatic since $a \cdot a \leq a \cdot 1 = a$ for all $a \in Q$). Since joins in $Idem(Q)$ are calculated just as they are in $Q$, and since multiplication in $Q$ distributes over arbitrary joins, we have that binary meets distribute over arbitrary joins in $Idem(Q)$.
Finally, if $A$ is a frame and $Q$ is a commutative affine quantale, it is clear that a quantale map $f \colon A \to Q$ takes elements in $A$ (which are idempotent under meet) to idempotents in $Q$. Hence $f$ factors uniquely through $Idem(Q) \hookrightarrow Q$, and the map $A \to Idem(Q)$ is a frame map. This shows that $Idem$ is the right adjoint as claimed.
In fact, we may also observe that the forgetful functor from commutative affine quantales to commutative quantales also has a right adjoint, just be passing from a commutative quantale to the principal downset given by the quantale unit. (However, the forgetful functor from commutative quantales to quantales does not have a right adjoint.)
On very general grounds, the forgetful functor from the category of frames to the category of quantales has a left adjoint (see here). This forgetful functor is full and faithful, and as a result the unit of the adjunction is a regular epi described by a suitable quantale congruence; see the next section.
A quantale congruence is simply an equivalence relation $({\equiv}) \hookrightarrow Q \times Q$ on a quantale $Q$ that respects the quantale multiplication and the taking of sups: if $a \equiv b$ and $x \equiv y$, then $a x \equiv b y$, and if $x_i \equiv y_i$ for all $i \in I$, then $\bigvee_{i \in I} x_i \equiv \bigvee_{i \in I} y_i$.
Consequently, $Q/{\equiv}$ defines a quantale $\tilde{Q}$ with operations inherited along a quantale quotient map $q: Q \to \tilde{Q}$. Since $q$ preserves arbitrary sups, it has a right adjoint $r: \tilde{Q} \to Q$ by the adjoint functor theorem applied to posets. Thus we have a monad or closure operator $r q: Q \to Q$, and the algebras/fixpoints of the monad/closure operator are identified with the elements of $\tilde{Q}$, i.e., $r: \tilde{Q} \to Q$ is isomorphic to the inclusion $Fix(r q) \hookrightarrow Q$.
The monad $r q$ is a quantale nucleus in the sense of the following definition:
A function $j: Q \to Q$ on a quantale $Q$ is a nucleus if it is a monad ($x \leq y$ implies $j(x) \leq j(y)$, $x \leq j(x)$, $j j(x) = j(x)$ for all $x, y \in Q$) and is lax monoidal: $1 \leq j(1)$ and $j(x) \cdot j(y) \leq j(x \cdot y)$ for all $x, y \in Q$.
There is a natural bijective correspondence between congruences on a quantale $Q$ and nuclei on $Q$.
Aside from being “noncommutative frames”, a different way of thinking about quantales views them as a (0,1)-categorical analogue of a cosmos (in the sense of Benabou). In particular, one can then study enriched categories over a quantale. A classic example is Lawvere metric spaces, seen as categories enriched in the quantale $([0, \infty], \geq)$ with $+$ taken as tensor product.
Enrichment is often particularly interesting for $*$-quantales (see below), where one can study $*$-enriched categories.
Quantales are a surprisingly commonplace structure in computer science. A very simple example is the powerset of strings (i.e., the powerset of the free monoid over some set of characters $\Sigma$). The order is the inclusion order on sets, and meet and join are just intersection and union, respectively. Taking $\epsilon$ to be empty string, and $a \cdot b$ to the join of two string, the quantalic operations are then:
This example generalizes as follows: given any monoidal preorder $M$ (for instance, a monoid equipped with the discrete order, as in the previous example), the collection of down-closed subsets of $M$ carries a quantale structure given by Day convolution with respect to categories enriched in $\mathbf{2} = TV$, the Heyting algebra of truth values. Explicitly, if $e$ denotes the unit of $M$ and $\cdot$ the multiplication, then
Another class of examples: internal homs $\hom_{sLat}(X, X)$ in the closed monoidal category of suplattices. For example, when the suplattice $X$ is a power set $P(S)$, one may identify $\hom_{sLat}(P(S), P(S))$ with the poset of binary relations $P(S \times S)$, ordered by inclusion and where the quantalic multiplication is relational composition.
Quantales, as monoids in the symmetric monoidal category $sLat$, can be tensored to produce new quantales.
A $*$-quantale is a quantale $Q$ equipped with an additional structure of an involution
for which $(x \otimes y)^* = y^* \otimes x^*$ and $1^* = 1$, where $1$ denotes the monoidal unit. (The operator is assumed to be covariant with respect to the poset structure.)
An example of a $*$-quantale is the quantale of binary relations on a set $S$, where the $*$-operation is relational opposite:
Another example is obtained by taking the quantale of down-closed subsets of a $*$-monoidal poset $M$ (which is the same thing as a $*$-monoid in the cartesian monoidal category of posets), with the quantale structure given by Day convolution as described above, and the $*$-operator obtained by cocontinuously extending the $*$-operator on $M$. Explicitly,
A $*$-enriched category over a $*$-quantale $Q$ is a category $(X, d: X \times X \to Q)$ enriched in the underlying quantale, such that
This notion can also be expressed in terms of lax morphisms of $*$-quantales; see below.
A commutative quantale is in particular a symmetric monoidal category (a symmetric monoidal (0,1)-category). As such it may be thought of as a model for linear logic in the general sense. Precisely if it has a dualizing object then it is a star-autonomous category and hence a model for linear logic in the original sense. (see e.g. Yetter 90, page 43). Indeed, quantales have been argued to provide models for quantum logic, see there for more.
There is a variety of notions of morphism of quantale, just as there is a variety of notions of morphism between closed monoidal categories. All the notions considered here are morphisms between the underlying sup-lattices, in other words preserve arbitrary joins, hence are left adjoints as functors between the underlying categories.
At the weak end of the scale, one may consider lax morphisms of quantales, i.e., (lax) monoidal functors of quantales seen as monoidal categories.
A stronger notion is of strong morphisms of quantales seen as monoidal categories. As noted above, all quantale morphisms considered here are already left adjoints in $Cat$, and if the adjunction lifts to $MonCat$ (the 2-category of monoidal categories, lax monoidal functors, and monoidal transformations), then the left adjoint is strong monoidal. This often occurs in practice.
An even stronger notion is where the morphisms also strongly preserve the closed structure, i.e., the internal homs or residuations. (An example is to be developed for buildings.)
There are corresponding notions of morphisms of $*$-quantales, where in each case morphisms strongly respect the $*$ operations. For instance, the notion of $*$-enriched category over a $*$-monoidal poset $M$ can be equivalently recast as a lax morphism between $*$-quantales, $2^d: 2^M \to 2^{X \times X}.$
The initial paper to use the term `quantale' was
Christopher J. Mulvey, &, Rend. Circ. Mat. Palermo, II. Ser., Suppl. 12, 99-104 (1986) Zbl 0633.46065
Discussion of how quantales serve as a model for linear logic and quantum logic is in
A monograph on quantales:
Connections to operator algebras and etale groupoids is discussed in
A connection between topoi, “Grothendieck” quantales and $C^\ast$-algebras
Sheaves on a quantale