nLab
quantale

Context

(0,1)(0,1)-Category theory

Monoidal categories

Contents

Definition

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\yx\backslash y and y/xy/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 a 1-object quantaloid, i.e., a 1-object category enriched in suplattices.

Quantales and Frames

Additional conditions often imposed on a quantale include:

  • Commutativity: xy=yxx\otimes y = y\otimes x
  • Idempotence: xx=xx\otimes x = x
  • Affineness: the unit for \otimes is the top element: 1=1=\top.

If all three of commutativity, idempotence, and affineness are assumed, they force \otimes to be the meet and therefore the quantale to be a frame. 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.)

The following construction gives a simple means for passing from commutative affine quantales to frames:

Lemma

Let (Q,,1)(Q, \cdot, 1) be a commutative affine quantale, and let Idem(Q)Idem(Q) be the subposet of elements xx=xx \cdot x = x. Then Idem(Q)Idem(Q) is a frame, where the meet operation is given by multiplication in QQ. The functor IdemIdem is right adjoint to the forgetful functor from commutative affine quantales to frames.

Proof

Notice that xxx1=xx \cdot x \leq x \cdot 1 = x for any xQx \in Q, so the interest is in the other condition xxxx \leq x x. If x,yx, y are idempotent, we easily have xyx y idempotent using commutativity, and xyx1=xx y \leq x 1 = x and xy1y=yx y \leq 1 y = y by affineness. Thus zxyz \leq x y implies zxz \leq x and zyz \leq y. Conversely, if zz is idempotent and zxz \leq x and zyz \leq y, we have

zzzxyz \leq z z \leq x y

and we now conclude that \cdot is the meet operation on Idem(Q)Idem(Q). Next, we show that Idem(Q)Idem(Q) is closed under taking joins in QQ: if x ix_i is a collection of idempotents, we have

x ix ix i( ix i)( ix i)x_i \leq x_i x_i \leq (\bigvee_i x_i) (\bigvee_i x_i)

for all ii, whence

ix i( ix i)( ix i),\bigvee_i x_i \leq (\bigvee_i x_i) (\bigvee_i x_i),

which is all we need. Since joins in Idem(Q)Idem(Q) are calculated just as they are in QQ, and since multiplication in QQ distributes over arbitrary joins, we have that binary meets distribute over arbitrary joins in Idem(Q)Idem(Q).

Finally, if AA is a frame and QQ is a commutative affine quantale, it is clear that a quantale map f:AQf \colon A \to Q takes elements in AA (which are idempotent under meet) to idempotents in QQ. Hence ff factors uniquely through Idem(Q)QIdem(Q) \hookrightarrow Q, and the map AIdem(Q)A \to Idem(Q) is a frame map. This shows that IdemIdem 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.)

Enrichment over quantales

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,],)([0, \infty], \geq) with ++ taken as tensor product.

Enrichment is often particularly interesting for **-quantales (see below), where one can study **-enriched categories.

Examples

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 Σ\Sigma). The order is the inclusion order on sets, and meet and join are just intersection and union, respectively. Taking ϵ\epsilon to be empty string, and aba \cdot b to the join of two string, the quantalic operations are then:

  • 1={ϵ}1 = \{\epsilon\}
  • LM={lm|lL,mM}L \otimes M = \{ l\cdot m \;|\; l \in L, m \in M \}

This example generalizes as follows: given any monoidal preorder MM (for instance, a monoid equipped with the discrete order, as in the previous example), the collection of down-closed subsets of MM carries a quantale structure given by Day convolution with respect to categories enriched in 2=TV\mathbf{2} = TV, the Heyting algebra of truth values. Explicitly, if ee denotes the unit of MM and \cdot the multiplication, then

  • 1={xM:xe}1 = \{x \in M: x \leq e\}
  • LM={xM: lL mMxlm}L \otimes M = \{x \in M: \exists_{l \in L} \exists_{m \in M} x \leq l \cdot m\}

Another class of examples: internal homs hom sLat(X,X)\hom_{sLat}(X, X) in the closed monoidal category of suplattices. For example, when the suplattice XX is a power set P(S)P(S), one may identify hom sLat(P(S),P(S))\hom_{sLat}(P(S), P(S)) with the poset of binary relations P(S×S)P(S \times S), ordered by inclusion and where the quantalic multiplication is relational composition.

Quantales, as monoids in the symmetric monoidal category sLatsLat, can be tensored to produce new quantales.

**-quantales

A **-quantale is a quantale QQ equipped with an additional structure of an involution

*:QQ * : Q \to Q

for which (xy) *=y *x *(x \otimes y)^* = y^* \otimes x^* and 1 *=11^* = 1, where 11 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 SS, where the **-operation is relational opposite:

  • R *={(y,x):(x,y)R}R^* = \{(y, x): (x, y) \in R\}

Another example is obtained by taking the quantale of down-closed subsets of a **-monoidal poset MM (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 MM. Explicitly,

  • L *={x *:xL}L^* = \{x^*: x \in L\}

A **-enriched category over a **-quantale QQ is a category (X,d:X×XQ)(X, d: X \times X \to Q) enriched in the underlying quantale, such that

d(y,x)=d(x,y) *d(y, x) = d(x, y)^*

This notion can also be expressed in terms of lax morphisms of **-quantales; see below.

Relation to linear logic

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.

Morphisms of quantales

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.

    • An important example of this is that categories enriched in a monoidal poset MM, such as Lawvere metric spaces, amount to the same thing as lax quantale morphisms of the form 2 d:2 M2 X×X2^d: 2^{M} \to 2^{X \times X} where the domain is the quantale of upward-closed subsets of MM with the Day convolution structure, and the codomain is the quantale of binary relations on XX, with multiplication being relational composition.
  • 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 CatCat, and if the adjunction lifts to MonCatMonCat (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 MM can be equivalently recast as a lax morphism between **-quantales, 2 d:2 M2 X×X.2^d: 2^M \to 2^{X \times X}.

References

The initial paper to use the term `quantale' was

Discussion of how quantales serve as a model for linear logic and quantum logic is in

  • David Yetter, Quantales and (noncommutative) linear logic, Journal of Symbolic Logic 55 (1990), 41-64.

A monograph on quantales:

  • Kimmo I. Rosenthal, Quantales and their applications, Pitman Res. Notes in Math. Series 234, Longman 1990

Connections to operator algebras and etale groupoids is discussed in

  • Pedro Resende, Étale groupoids and their quantales, Adv. Math. 208 (2007) 147-209; also published electronically: doi; math/0412478
  • M.C. Protin, P. Resende, Quantales of open groupoids, J. Noncommut. Geom. 6 (2012) 199–247.
  • P. Resende, Lectures on ́tale groupoids, inverse semigroups and quantales, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4–18 September, 2006, 115 pp.; pdf
  • P. Resende, Groupoid sheaves as quantale sheaves, J. Pure Appl. Algebra 216 (2012), 41–70; arxiv/0807.4848 doi
  • D. Kruml, J.W. Pelletier, P. Resende, J. Rosický, On quantales and spectra of C *C^\ast-algebras, Appl. Categ. Structures 11 (2003) 543–560.
  • D. Kruml, P. Resende, On quantales that classify C *C^\ast-algebras, Cah. Topol. Geom. Differ. Categ. 45 (2004) 287–296.
  • F. Borceux, J. Rosický, G. Van den Bossche, Quantales and C *C^\ast-algebras, J. London Math. Soc. 40 (1989) 398–404 doi

Sheaves on a quantale

  • Francis Borceux, Rosanna Cruciani, Sheaves on a quantale, Cahiers de Topologie et Géométrie Différentielle Catégoriques (1993) 34:3, page 209-228 pdf

Revised on April 9, 2014 05:56:27 by Tim Porter (2.26.27.237)