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: 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 ) {displaystyle (igvee _{i}f_{i})circ (igvee _{j}g_{j})=igvee _{i,j}(f_{i}circ g_{j})} The endomorphism lattice H o m ( X , X ) {displaystyle mathrm {Hom} (X,X)} of any object X {displaystyle X} in a quantaloid is a quantale, whence the name.