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:

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

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