Quantaloid
In mathematics, a quantaloid is a category enriched over the category Sup of complete lattices with supremum-preserving maps. In other words, for any objects a and b the Hom 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 (\bigvee _{i}f_{i})\circ (\bigvee _{j}g_{j})=\bigvee _{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.