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

${\displaystyle (\bigvee _{i}f_{i})\circ (\bigvee _{j}g_{j})=\bigvee _{i,j}(f_{i}\circ g_{j})}$

The endomorphism lattice ${\displaystyle \mathrm {Hom} (X,X)}$ of any object ${\displaystyle X}$ 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, vol. 348, Longman, Harlow, ISBN 0-582-29440-1, MR 1427263. See in particular p. 15.

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