A quantaloid is a category enriched in the closed symmetric monoidal category of suplattices. Equivalently, it may be defined as a locally posetal bicategory 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 bicategory).
Taking the view that a quantaloid is a closed bicategory, one can study categories enriched in . This can become particularly interesting if is a -quantaloid, i.e., comes equipped with an involution 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 , i.e., -categories
for which . A famous example, due to R.F.C. Walters, is given below.
A toy form of this example: take any frame , and considering the bicategory of spans in : this is evidently a quantaloid, .
Here is a particularly rich source of examples. Let be a quantale. Then there is a quantaloid - of -valued matrices:
Objects are sets ;
Morphisms are functions .
Matrix composition is performed according to the usual rule
where is the multiplication of and the sum is the sup in . This class of examples easily internalizes in a topos, and in that sense it subsumes the first example, , by taking (as an internal frame, hence a quantale).
Slightly more generally, if is a quantal_oid_, there is a quantaloid - of -matrices, as follows:
Objects are sets together with functions ;
Morphisms are matrices that satisfy the typing constraint .
Composition works exactly as before.
The examples in the preceding section carry over straightforwardly:
Next, let be a -quantale. The quantaloid - 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 : we similarly obtain a -quantaloid -, by defining transpose as above.
Let again be a -quantale; for example, could be a frame or it could be a commutative quantale, taking the involution to be the identity.
A -valued set consists of a set and a morphism in - which is
In the case where , 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 - is an allegory, where the modular law holds, one of whose consequences is the inequality . As a result, we have
which together with transitivity guarantees idempotence.
Thus, symmetric idempotents in are what are known as partial equivalence relations (which differ from equivalence relations by dropping reflexivity), or PERs for short.
To be continued…