# Quantaloid

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{{technical|date=September 2010}}
In [mathematics](/source/mathematics), a '''quantaloid''' is a [category](/source/category_(mathematics)) [enriched](/source/enriched_category) over the category '''Sup''' of [complete lattices with supremum-preserving maps](/source/Complete_lattice).<ref>{{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.&nbsp;15].</ref> In other words, for any [objects](/source/object_(category_theory)) ''a'' and ''b'' the [Hom](/source/Morphism) object between them is not just a [set](/source/set_(mathematics)) but a complete [lattice](/source/Lattice_(order)), in such a way that composition of morphisms preserves all joins:
:<math>(\bigvee_i f_i) \circ (\bigvee_j g_j) = \bigvee_{i,j} (f_i \circ g_j) </math>

The [endomorphism](/source/endomorphism) lattice <math>\mathrm{Hom}(X,X)</math> of any object <math>X</math> in a quantaloid is a [quantale](/source/quantale), whence the name.

==References==
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Category:Category theory

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