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Concept# Enriched category

Summary

In , a branch of mathematics, an enriched category generalizes the idea of a by replacing hom-sets with objects from a general . 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 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 , 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 or even , 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 , or the addition operation on morphisms in an )
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 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.

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