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:

(\underset{i}{⋁} f_{i}) \circ (\underset{j}{⋁} g_{j}) = \underset{i, j}{⋁} (f_{i} \circ g_{j})

The endomorphism lattice $H o m (X, X)$ of any object $X$ in a quantaloid is a quantale, whence the name.

References

^ Page Module:Citation/CS1/styles.css has no content.Rosenthal, Kimmo I. (1996), The theory of quantaloids, Pitman Research Notes in Mathematics Series, vol. 348, Longman, Harlow, ISBN 0-582-29440-1, MR 1427263. See in particular p. 15.