Quantaloid Explained

In mathematics, a quantaloid is a category enriched over the category Sup of suplattices.^[1] In other words, for any objects a and b the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins:

( v ee _{i} f _{i)} \circ ( v ee _{j} g _{j)} = v ee _{i,j} (f _{i} \circ g _{j)}

The endomorphism lattice

Hom (X,X) of any object X in a quantaloid is a quantale, whence the name.

Notes and References

1. . See in particular p. 15.