Map (higher-order function): Difference between revisions
wrong link |
remove redundant redundancy |
||
| Line 1: | Line 1: | ||
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 20:10, 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
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
43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.
- ↑ 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534