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| In [[mathematics]], the [[category (mathematics)|category]] '''Rel''' has the class of [[Set (mathematics)|sets]] as [[object (category theory)|objects]] and [[binary relation]]s as [[morphism (category theory)|morphisms]].
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| A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so {{nowrap| ''R'' ⊆ ''A'' × ''B''}}.
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| The [[Composition of relations|composition]] of two relations ''R'': ''A'' → ''B'' and ''S'': ''B'' → ''C'' is given by: | |
| :(''a'', ''c'') ∈ ''S'' <small>o</small> ''R'' if (and only if) for some ''b'' ∈ ''B'', (''a'', ''b'') ∈ ''R'' and (''b'', ''c'') ∈ ''S''.
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| ==Properties==
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| The category '''Rel''' has the [[category of sets]] '''Set''' as a (wide) [[subcategory]], where the arrow (function) {{nowrap| ''f'' : ''X'' → ''Y''}} in '''Set''' corresponds to the functional relation {{nowrap| ''F'' ⊆ ''X'' × ''Y''}} defined by: {{nowrap|1= (''x'', ''y'') ∈ ''F'' ⇔ ''f''(''x'') = ''y''}}.
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| The category '''Rel''' can be obtained from the category '''Set''' as the [[Kleisli category]] for the [[monad (category theory)|monad]] whose [[functor]] corresponds to [[power set]], interpreted as a covariant functor.
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| Perhaps a bit surprising at first sight is the fact that [[Product (category theory)|product]] in '''Rel''' is given by the [[disjoint union]] (rather than the [[cartesian product]] as it is in '''Set'''), and so is the [[coproduct]].
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| '''Rel''' is [[Closed monoidal category|monoidal closed]], with both the monoidal product <math>A \otimes B</math> and the [[internal hom]] <math>A \Rightarrow B</math> given by [[cartesian product]] of sets.
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| The [[Involution (mathematics)|involutary]] operation of taking the [[Inverse relation|inverse]] (or converse) of a relation, where {{nowrap| (''b'', ''a'') ∈ ''R''<sup>−1</sup> : ''B'' → ''A''}} if and only if {{nowrap| (''a'', ''b'') ∈ ''R'' : ''A'' → ''B''}}, induces a contravariant functor {{nowrap| '''Rel'''<sup>op</sup> → '''Rel'''}} that leaves the objects invariant but reverses the arrows and composition. This makes '''Rel''' into a [[dagger category]]. In fact, '''Rel''' is a [[dagger compact category]].
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| ==See also==
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| *[[Allegory (category theory)]]. The category of relations is the paradigmatic example of an allegory.
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| [[Category:Monoidal categories|Relations]]
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