nLab quantaloid




The notion of quantaloid is the horizontal categorification of that of quantale: a quantaloid is a quantale with many objects.


A quantaloid is a category enriched in the closed symmetric monoidal category of suplattices. Equivalently, it may be defined as a locally posetal 2-category in which each hom-poset is cocomplete, and which admits right adjoints to composing with an arrow on either side (making it a (bi)closed 2-category).

Taking the view that a quantaloid QQ is a closed 2-category, one can study categories enriched in QQ. This can become particularly interesting if QQ is a **-quantaloid, i.e., comes equipped with an involution *:QQ*: Q \to Q which is the identity on 0-cells, reverses the direction of 1-cells, and preserves the direction of 2-cells. In that case one can study **-enriched categories over QQ, i.e., QQ-categories

(X,p:XQ 0,d:X×XQ 1)(X, p: X \to Q_0, d: X \times X \to Q_1)

for which d(y,x)=d(x,y) *d(y, x) = d(x, y)^*. A famous example, due to R.F.C. Walters, is given below.

Examples of quantaloids

Despite the exotic-sounding name (a portmanteau of quantale and groupoid, i.e., a many-object quantale), quantaloids are quite commonplace.

  • Let EE be a Grothendieck topos. Then the bicategory of relations in EE, Rel(E)Rel(E), is a quantaloid.

A toy form of this example: take any frame HH, and consider the bicategory of spans in HH: this is evidently a quantaloid, Span(H)Span(H).

Here is a particularly rich source of examples. Let QQ be a quantale. Then there is a quantaloid QQ-MatMat of QQ-valued matrices:

  • Objects are sets XX;

  • Morphisms XYX \to Y are functions M:X×YQM: X \times Y \to Q.

Matrix composition is performed according to the usual rule

(MN)(x,z)= y:YM(x,y)N(y,z)(M N)(x, z) = \sum_{y: Y} M(x, y) \cdot N(y, z)

where \cdot is the multiplication of QQ and the sum is the sup in QQ. This class of examples easily internalizes in a topos, and in that sense it subsumes the first example, Rel(E)Rel(E), by taking Q=ΩQ = \Omega (as an internal frame, hence a quantale).

Slightly more generally, if QQ is a quantaloid, there is a quantaloid QQ-MatMat of QQ-matrices, as follows:

  • Objects are sets XX together with functions p:XQ 0p: X \to Q_0;

  • Morphisms (X,p)(Y,q)(X, p) \to (Y, q) are matrices M:X×YQ 1M: X \times Y \to Q_1 that satisfy the typing constraint M(x,y):p(x)q(y)M(x, y): p(x) \to q(y).

Composition works exactly as before.

Examples of **-quantaloids

The examples in the preceding section carry over straightforwardly:

  • For EE a Grothendieck topos, Rel(E)Rel(E) is a **-quantaloid where the **-operator takes a relation from XX to YY, i.e., a subobject i:RX×Yi: R \hookrightarrow X \times Y, to its opposite obtained by composing ii with the symmetry isomorphism X×YY×XX \times Y \cong Y \times X.

Next, let QQ be a **-quantale. The quantaloid QQ-MatMat is as before; this becomes a **-quantaloid by defining the transpose of a matrix as

  • M *(x,y)=(M(y,x)) *M^*(x, y) = (M(y, x))^*

This easily carries over to the case where we start with a **-quantaloid QQ: we similarly obtain a **-quantaloid QQ-MatMat, by defining transpose as above.

Connection with Q-valued sets

Let QQ again be a **-quantale; for example, QQ could be a frame or it could be a commutative quantale, taking the involution to be the identity.

A QQ-valued set consists of a set XX and a morphism M:XXM: X \to X in QQ-MatMat which is

  • Symmetric: M *=MM^* = M,

  • Idempotent: MM=MM \circ M = M

In the case where Q=ΩQ = \Omega, the subobject classifier in a topos with its locale structure, symmetry and idempotence of a relation is equivalent to the a priori weaker condition of symmetry and transitivity. The reason is that QQ-Mat=Rel(E)Mat = Rel(E) is an allegory, where the modular law holds, one of whose consequences is the inequality MMM *MM \leq M M^* M. As a result, we have

MMM *M=symMMMtransMMM \leq M M^* M \stackrel{sym}{=} M M M \stackrel{trans}{\leq} M M

which together with transitivity guarantees idempotence.

Thus, symmetric idempotents in Rel(E)Rel(E) are what are known as partial equivalence relations (which differ from equivalence relations by dropping reflexivity), or PERs for short.

To be continued…

Last revised on February 6, 2024 at 12:05:48. See the history of this page for a list of all contributions to it.