View source for Distributive category

{{cleanup|reason=there's more than one proposed notion under this name, see last ref in further reading|date=July 2014}}
In [[mathematics]], a [[category (category theory)|category]] is '''distributive''' if it has finite [[product (category theory)|product]]s and finite [[coproduct]]s and such that for every choice of [[object (category theory)|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 (where 0 denotes the [[initial object]]). Equivalently, if for every object <math>A</math> the [[endofunctor]] <math>A \times -</math> defined by <math>B\mapsto A\times B</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 [[colimit]]s, and thus any cartesian closed category with finite coproducts (i.e., any [[bicartesian closed category]]) is distributive.

== Example ==
The [[category of sets]] is distributive. Let {{var|A}}, {{var|B}}, and {{var|C}} be [[set (mathematics)|sets]]. Then
:<math>\begin{align}
  A\times (B\amalg C) 
  &= \{(a,d) \mid a \in A\text{ and }d \in B \amalg C\} \\
  &\cong \{(a,d) \mid a \in A\text{ and }d \in B\} \amalg \{(a,d) \mid a \in A\text{ and }d \in C\} \\
  &= (A \times B) \amalg (A \times C)
\end{align}</math>
where <math>\amalg</math> denotes the coproduct in '''Set''', namely the [[disjoint union]], and <math>\cong</math> denotes a [[bijection]].  In the case where {{var|A}}, {{var|B}}, and {{var|C}} are [[finite set]]s, this result reflects the [[distributive property]]: the above sets each have [[cardinality]] <math>|A|\cdot (|B|+|C|)=|A|\cdot|B| + |A|\cdot|C|</math>.

The categories '''[[Category of groups|Grp]]''' and '''[[Category of abelian groups|Ab]]''' are not distributive, even though they have both products and coproducts.

An even simpler category that has both products and coproducts but is not distributive is the category of [[pointed set]]s.<ref name="LawvereSchanuel2009">{{cite book|author1=F. W. Lawvere|author2=Stephen Hoel Schanuel|title=Conceptual Mathematics: A First Introduction to Categories|year=2009|publisher=Cambridge University Press|isbn=978-0-521-89485-2|edition=2nd|pages=[https://archive.org/details/conceptualmathem00lawv/page/296 296–298]|url-access=registration|url=https://archive.org/details/conceptualmathem00lawv/page/296}}</ref>

==References==
{{reflist}}

==Further reading==
{{refbegin}}
* {{cite journal | doi = 10.1017/S0960129500000232 | title=Introduction to distributive categories | journal=Mathematical Structures in Computer Science | date=1993 | volume=3 | issue=3 | pages=277–307 | first=J. R. B. | last=Cockett| doi-access=free }}
* {{cite journal | doi = 10.1016/0022-4049(93)90035-R | title=Introduction to extensive and distributive categories | journal=Journal of Pure and Applied Algebra | date=1993 | volume=84 | issue=2 | pages=145–158 | first=Aurelio | last=Carboni| doi-access=free }}
{{refend}}

[[Category:Category theory]]


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