mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, a quantaloid is a

category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...

enriched over the category Sup of complete lattices with supremum-preserving maps.. See in particula
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In other words, for any objects ''a'' and ''b'' the Hom object between them is not just a

set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...

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_ (f_i \circ g_j)

The

endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a g ...

lattice

\mathrm(X,X)

of any object

X

in a quantaloid is a quantale, whence the name.

References

Category theory {{Cattheory-stub