Quantaloid

{{technical|date=September 2010}}

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

| last = Rosenthal | first = Kimmo I.

| isbn = 0-582-29440-1

| mr = 1427263

| publisher = Longman, Harlow

| series = Pitman Research Notes in Mathematics Series

| title = The theory of quantaloids

| volume = 348

| year = 1996}}. See in particular [https://books.google.com/books?id=O3bno8HpcFAC&pg=PA15 p. 15]. 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:

:$(\backslash bigvee\_i\; f\_i)\; \backslash circ\; (\backslash bigvee\_j\; g\_j)\; =\; \backslash bigvee\_\{i,j\}\; (f\_i\; \backslash circ\; g\_j)$

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