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- Enriched_category abstract "In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms. In an enriched category, the set of morphisms (the hom-set) associated with every pair of objects is replaced by an opaque object in some fixed monoidal category of "hom-objects". In order to emulate the (associative) composition of morphisms in an ordinary category, the hom-category must have a means of composing hom-objects in an associative manner: that is, there must be a binary operation on objects giving us at least the structure of a monoidal category, though in some contexts the operation may also need to be commutative and perhaps also to have a right adjoint (i.e., making the category symmetric monoidal or even cartesian closed, respectively). Enriched category theory thus encompasses within the same framework a wide variety of structures including ordinary categories where the hom-set carries additional structure beyond being a set. That is, there are operations on, or properties of morphisms that need to be respected by composition (e.g., the existence of 2-cells between morphisms and horizontal composition thereof in a 2-category, or the addition operation on morphisms in an abelian category) category-like entities that don't themselves have any notion of individual morphism but whose hom-objects have similar compositional aspects (e.g., preorders where the composition rule ensures transitivity, or Lawvere's metric spaces, where the hom-objects are numerical distances and the composition rule provides the triangle inequality).In the case where the hom-object category happens to be the category of sets with the usual cartesian product, the definitions of enriched category, enriched functor, etc... reduce to the original definitions from ordinary category theory.An enriched category with hom-objects from monoidal category M is said to be an enriched category over M or an enriched category in M, or simply an M-category. Due to Mac Lane's preference for the letter V in referring to the monoidal category, enriched categories are also sometimes referred to generally as V-categories.".
- Enriched_category thumbnail Math-enriched_category_associativity.svg?width=300.
- Enriched_category wikiPageExternalLink tr1.pdf.
- Enriched_category wikiPageExternalLink tr10.pdf.
- Enriched_category wikiPageID "142622".
- Enriched_category wikiPageRevisionID "598443638".
- Enriched_category hasPhotoCollection Enriched_category.
- Enriched_category id "enriched+category".
- Enriched_category title "Enriched category".
- Enriched_category subject Category:Category_theory.
- Enriched_category subject Category:Monoidal_categories.
- Enriched_category type Abstraction100002137.
- Enriched_category type Class107997703.
- Enriched_category type Collection107951464.
- Enriched_category type Group100031264.
- Enriched_category type MonoidalCategories.
- Enriched_category comment "In category theory, a branch of mathematics, an enriched category generalizes the idea of a category by replacing hom-sets with objects from a general monoidal category. It is motivated by the observation that, in many practical applications, the hom-set often has additional structure that should be respected, e.g., that of being a vector space of morphisms, or a topological space of morphisms.".
- Enriched_category label "Catégorie enrichie".
- Enriched_category label "Enriched category".
- Enriched_category label "Verrijkte categorie".
- Enriched_category label "Обогащённая категория".
- Enriched_category sameAs Angereicherte_Kategorie.
- Enriched_category sameAs Catégorie_enrichie.
- Enriched_category sameAs Verrijkte_categorie.
- Enriched_category sameAs m.011zts.
- Enriched_category sameAs Q5379515.
- Enriched_category sameAs Q5379515.
- Enriched_category sameAs Enriched_category.
- Enriched_category wasDerivedFrom Enriched_category?oldid=598443638.
- Enriched_category depiction Math-enriched_category_associativity.svg.
- Enriched_category isPrimaryTopicOf Enriched_category.
