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| In [[mathematics]], a [[category (category theory)|category]] is '''distributive''' if it has finite [[product (category theory)|product]]s and finite [[coproduct (category theory)|coproduct]]s such that for every choice of objects <math>A,B,C</math>, the canonical map
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| : <math>[\mathit{id}_A \times\iota_1, \mathit{id}_A \times\iota_2] : A\times B + A\times C\to A\times(B+C)</math> | |
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| is an [[isomorphism]], and for all objects <math>A</math>, the canonical map <math>0 \to A\times 0</math> is an isomorphism. Equivalently. if for every object <math>A</math> the functor <math>A\times -</math> preserves coproducts up to isomorphisms <math>f</math>.<ref>{{cite book|last=Taylor|first=Paul|title=Practical Foundations of Mathematics|publisher=Cambridge University Press|year=1999|page=275}}</ref> It follows that <math>f</math> and aforementioned canonical maps are equal for each choice of objects.
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| In particular, if the functor <math>A\times -</math> has a right [[adjoint functors|adjoint]] (i.e., if the category is [[cartesian closed category|cartesian closed]]), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any [[bicartesian closed category]]) is distributive.
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| For example, '''[[Category of sets|Set]]''' is distributive, while '''[[category of groups|Grp]]''' is not, even though it has both products and coproducts.
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| ==References==
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| {{reflist}}
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| [[Category:Category theory]]
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| {{categorytheory-stub}}
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Latest revision as of 01:59, 3 January 2015
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