Quantaloid

In mathematics, a quantaloid is a category enriched over the category Sup of suplattices.^[1] 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:

$(\bigvee_i f_i) \circ (\bigvee_j g_j) = \bigvee_{i,j} (f_i \circ g_j)$

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

References

1. Rosenthal, Kimmo I. (1996), The theory of quantaloids, Pitman Research Notes in Mathematics Series 348, Longman, Harlow, ISBN 0-582-29440-1, MR 1427263. See in particular p. 15.