Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of suplattices. In other words for any objects a, 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:

( ⋁ i f i ) ∘ ( ⋁ j g j ) = ⋁ i , j ( f i ∘ g j )

The endomorphism lattice H o m ( X , X ) of any object X in a quantaloid is a quantale, whence the name.