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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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Revision as of 22:28, 25 February 2014
Hello and welcome. My name is Figures Wunder. His spouse doesn't like it the way he does but what he really likes doing is to do aerobics and he's been performing it for fairly a whilst. Puerto Rico is exactly where he and his wife live. Managing individuals is his profession.
Feel free to surf to my weblog home std test kit