Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of complete lattices with supremum-preserving maps.¹ 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:

${\displaystyle (\bigvee _{i}f_{i})\circ (\bigvee _{j}g_{j})=\bigvee _{i,j}(f_{i}\circ g_{j})}$

The endomorphism lattice ${\displaystyle \mathrm {Hom} (X,X)}$ of any object ${\displaystyle X}$ in a quantaloid is a quantale, whence the name.