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 $Q$ is a closed 2-category, one can study categories enriched in $Q$. This can become particularly interesting if $Q$ is a $*$-quantaloid, i.e., comes equipped with an involution $*: Q \to Q$ 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 $Q$, i.e., $Q$-categories
for which $d(y, x) = d(x, y)^*$. A famous example, due to R.F.C. Walters, is given below.
Despite the exotic-sounding name (a portmanteau of quantale and groupoid, i.e., a many-object quantale), quantaloids are quite commonplace.
A toy form of this example: take any frame $H$, and consider the bicategory of spans in $H$: this is evidently a quantaloid, $Span(H)$.
Here is a particularly rich source of examples. Let $Q$ be a quantale. Then there is a quantaloid $Q$-$Mat$ of $Q$-valued matrices:
Objects are sets $X$;
Morphisms $X \to Y$ are functions $M: X \times Y \to Q$.
Matrix composition is performed according to the usual rule
where $\cdot$ is the multiplication of $Q$ and the sum is the sup in $Q$. This class of examples easily internalizes in a topos, and in that sense it subsumes the first example, $Rel(E)$, by taking $Q = \Omega$ (as an internal frame, hence a quantale).
Slightly more generally, if $Q$ is a quantaloid, there is a quantaloid $Q$-$Mat$ of $Q$-matrices, as follows:
Objects are sets $X$ together with functions $p: X \to Q_0$;
Morphisms $(X, p) \to (Y, q)$ are matrices $M: X \times Y \to Q_1$ that satisfy the typing constraint $M(x, y): p(x) \to q(y)$.
Composition works exactly as before.
The examples in the preceding section carry over straightforwardly:
Next, let $Q$ be a $*$-quantale. The quantaloid $Q$-$Mat$ is as before; this becomes a $*$-quantaloid by defining the transpose of a matrix as
This easily carries over to the case where we start with a $*$-quantaloid $Q$: we similarly obtain a $*$-quantaloid $Q$-$Mat$, by defining transpose as above.
Let $Q$ again be a $*$-quantale; for example, $Q$ could be a frame or it could be a commutative quantale, taking the involution to be the identity.
A $Q$-valued set consists of a set $X$ and a morphism $M: X \to X$ in $Q$-$Mat$ which is
Symmetric: $M^* = M$,
Idempotent: $M \circ M = M$
In the case where $Q = \Omega$, 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 $Q$-$Mat = Rel(E)$ is an allegory, where the modular law holds, one of whose consequences is the inequality $M \leq M M^* M$. As a result, we have
which together with transitivity guarantees idempotence.
Thus, symmetric idempotents in $Rel(E)$ 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.