Quantaloid

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

[math]\displaystyle{ (\bigvee_i f_i) \circ (\bigvee_j g_j) = \bigvee_{i,j} (f_i \circ g_j) }[/math]

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