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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 | |||
: <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> | |||
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. | |||
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. | |||
For example, '''[[Category of sets|Set]]''' is distributive, while '''[[category of groups|Grp]]''' is not, even though it has both products and coproducts. | |||
==References== | |||
{{reflist}} | |||
[[Category:Category theory]] | |||
{{categorytheory-stub}} | |||
Revision as of 21:10, 12 January 2014
In mathematics, a category is distributive if it has finite products and finite coproducts such that for every choice of objects , the canonical map
![{\displaystyle [{\mathit {id}}_{A}\times \iota _{1},{\mathit {id}}_{A}\times \iota _{2}]:A\times B+A\times C\to A\times (B+C)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ab26657683a86bea66546cde94d7e3b8800da698)
is an isomorphism, and for all objects , the canonical map is an isomorphism. Equivalently. if for every object the functor preserves coproducts up to isomorphisms .[1] It follows that and aforementioned canonical maps are equal for each choice of objects.
In particular, if the functor has a right adjoint (i.e., if the category is cartesian closed), it necessarily preserves all colimits, and thus any cartesian closed category with finite coproducts (i.e., any bicartesian closed category) is distributive.
For example, Set is distributive, while Grp is not, even though it has both products and coproducts.
References
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