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

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.

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[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.