nLab quantale



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

Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory



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 maps of sets that preserve arbitrary joins.

In most recent works, quantales are defined more generally as semi-group objects in the monoidal category of suplattices.

In more detail, a quantale is a poset having all joins and an associative, tensor product \otimes which distributes over joins (the internal-homs then come automatically by the adjoint functor theorem); this tensor product is unital if we understand quantales in a narrower sense as monoids.

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 an 1-object quantaloid, i.e., an 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.

(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 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 object in the cartesian monoidal category of posets is idempotent and affine, then the monoid multiplication is the meet operation.


Let (M,,1)(M, \otimes, 1) be the monoid. If axa \leq x and aya \leq y, then a=aaxya = a \otimes a \leq x \otimes y. On the other hand, we have xyx1=xx \otimes y \leq x \otimes 1 = x and similarly xyyx \otimes y \leq y, so axya \leq x \otimes y implies axa \leq x and aya \leq y by transitivity. Thus axya \leq x \otimes y iff axa \leq x and aya \leq y, i.e., xyx \otimes y satisfies the defining property of the meet xyx \wedge y.

Along similar lines, the following construction provides passage from commutative affine quantales to frames:


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.


If x,yx, y are idempotent, then so is xyx \cdot y using the fact that \cdot is commutative. Thus Idem(Q)Idem(Q) is an idempotent affine submonoid of QQ, which by Proposition forces \cdot to be the meet. 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 (the opposite inequality is automatic since aaa1=aa \cdot a \leq a \cdot 1 = a for all aQa \in Q). 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 by 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.

Quantale congruences and nuclei

A quantale congruence is simply an equivalence relation ()Q×Q({\equiv}) \hookrightarrow Q \times Q on a quantale QQ that respects the quantale multiplication and the taking of sups: if aba \equiv b and xyx \equiv y, then axbya x \equiv b y, and if x iy ix_i \equiv y_i for all iIi \in I, then iIx i iIy i\bigvee_{i \in I} x_i \equiv \bigvee_{i \in I} y_i.

Consequently, Q/Q/{\equiv} defines a quantale Q˜\tilde{Q} with operations inherited along a quantale quotient map q:QQ˜q: Q \to \tilde{Q}. Since qq preserves arbitrary sups, it has a right adjoint r:Q˜Qr: \tilde{Q} \to Q by the adjoint functor theorem applied to posets. Thus we have a monad or closure operator rq:QQr q: Q \to Q, and the algebras/fixpoints of the monad/closure operator are identified with the elements of Q˜\tilde{Q}, i.e., r:Q˜Qr: \tilde{Q} \to Q is isomorphic to the inclusion Fix(rq)QFix(r q) \hookrightarrow Q.

The monad rqr q is a quantale nucleus in the sense of the following definition:


A function j:QQj: Q \to Q on a quantale QQ is a nucleus if it is a monad (xyx \leq y implies j(x)j(y)j(x) \leq j(y), xj(x)x \leq j(x), jj(x)=j(x)j j(x) = j(x) for all x,yQx, y \in Q) and is lax monoidal: 1j(1)1 \leq j(1) and j(x)j(y)j(xy)j(x) \cdot j(y) \leq j(x \cdot y) for all x,yQx, y \in Q.

There is a natural bijective correspondence between congruences on a quantale QQ and nuclei on QQ.

Enrichment over quantales

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,],)([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. See the automata references. A very simple example is the power set of strings (i.e., the power set 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 the empty string, and aba \cdot b to be the join of two strings, the quantalic operations are then:

  • 1={ϵ}1 = \{\epsilon\}
  • IJ={ijiI,jJ}I \otimes J = \{ i\cdot j \mid i \in I, j \in J \}
  • K/J={i jJijK}K/J = \{ i \mid \forall_{j\in J} i\cdot j \in K \}
  • I\K={j iIijK}I\backslash K = \{ j \mid \forall_{i\in I} i\cdot j \in K \}

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={xMxe}1 = \{x\in M \mid x \leq e\}
  • IJ={xM iI jJxij}I \otimes J = \{x\in M \mid \exists_{i \in I} \exists_{j \in J} x \leq i \cdot j\}
  • K/J={iM jJijK}K/J = \{i \in M \mid \forall_{j\in J} i\cdot j \in K \}
  • I\K={jM iIijK}I\backslash K = \{j \in M \mid \forall_{i\in I} i\cdot j \in K \}

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.

An example in algebra is given by the lattice of ideals of a commutative ring, with the tensor product given by ideal multiplication, which makes it into a commutative affine quantale. Residuation in this case is ideal division (𝔞:𝔟)={x|x𝔟𝔞}(\mathfrak{a} : \mathfrak{b}) = \{x | x\mathfrak{b} \subseteq \mathfrak{a}\}.

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

  • 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}.


The initial paper to use the term `quantale' was

Discussion of quantales as a model for linear logic and quantum logic:

A monograph on quantales:

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

Relations with automata and process semantics

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.
  • Pedro Resende, Lectures on étale groupoids, inverse semigroups and quantales, Lecture Notes for the GAMAP IP Meeting, Antwerp, 4–18 September, 2006, 115 pp.; pdf
  • Pedro Resende, Groupoid sheaves as quantale sheaves, J. Pure Appl. Algebra 216 (2012), 41–70; arxiv/0807.4848


  • 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

A connection between topoi, “Grothendieck” quantales and C *C^\ast-algebras

  • Simon Henry, Toposes, quantales and C* algebras in the atomic case, arxiv/1311.3451

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

Last revised on December 12, 2023 at 11:38:43. See the history of this page for a list of all contributions to it.