Distributive category (original) (raw)
In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects , the canonical map is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms . It follows that and aforementioned canonical maps are equal for each choice of objects.
| Property | Value |
|---|---|
| dbo:abstract | In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects , the canonical map is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms . 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. (en) |
| dbo:wikiPageID | 6053993 (xsd:integer) |
| dbo:wikiPageLength | 3566 (xsd:nonNegativeInteger) |
| dbo:wikiPageRevisionID | 1100634274 (xsd:integer) |
| dbo:wikiPageWikiLink | dbr:Mathematics dbr:Coproduct dbr:Functor dbr:Adjoint_functors dbr:Category_(category_theory) dbr:Category_of_sets dbr:Distributive_property dbr:Finite_set dbr:Cardinality dbr:Isomorphism dbr:Product_(category_theory) dbc:Category_theory dbr:Bijection dbr:Disjoint_union dbr:Endofunctor dbr:Initial_object dbr:Cartesian_closed_category dbr:Category_of_abelian_groups dbr:Category_of_groups dbr:Set_(mathematics) dbr:Pointed_set dbr:Bicartesian_closed_category dbr:Colimit dbr:Object_(category_theory) |
| dbp:date | July 2014 (en) |
| dbp:reason | there's more than one proposed notion under this name, see last ref in further reading (en) |
| dbp:wikiPageUsesTemplate | dbt:Cite_journal dbt:Cleanup dbt:Reflist dbt:Var dbt:Categorytheory-stub |
| dct:subject | dbc:Category_theory |
| rdfs:comment | In mathematics, a category is distributive if it has finite products and finite coproducts and such that for every choice of objects , the canonical map is an isomorphism, and for all objects , the canonical map is an isomorphism (where 0 denotes the initial object). Equivalently, if for every object the endofunctor defined by preserves coproducts up to isomorphisms . It follows that and aforementioned canonical maps are equal for each choice of objects. (en) |
| rdfs:label | Distributive category (en) |
| owl:sameAs | freebase:Distributive category wikidata:Distributive category https://global.dbpedia.org/id/4iuM3 |
| prov:wasDerivedFrom | wikipedia-en:Distributive_category?oldid=1100634274&ns=0 |
| foaf:isPrimaryTopicOf | wikipedia-en:Distributive_category |
| is dbo:wikiPageWikiLink of | dbr:Posetal_category dbr:Strict_initial_object dbr:Product_(category_theory) dbr:Semiring dbr:Pointed_set dbr:Rig_category |
| is foaf:primaryTopic of | wikipedia-en:Distributive_category |