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In mathematics, a quantaloid is a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) *C ...
enriched over the category Sup of '' suplattices''.. See in particula
p. 15
In other words, for any
objects Object may refer to: General meanings * Object (philosophy), a thing, being, or concept ** Object (abstract), an object which does not exist at any particular time or place ** Physical object, an identifiable collection of matter * Goal, an ai ...
''a'' and ''b'' the
morphism In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
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 lattice \mathrm(X,X) of any object X in a quantaloid is a
quantale In mathematics, quantales are certain partially ordered algebraic structures that generalize locales ( point free topologies) as well as various multiplicative lattices of ideals from ring theory and functional analysis (C*-algebras, von Neumann ...
, whence the name.


References

Category theory {{Cattheory-stub