Categorical Enrichment
Enrichment is the categorical method of expressing that morphisms of a certain previewcategory come with extra structure. For instance, monotone maps in can themselves be equipped with a (pointwise) ordering, meaning that for every poset and , is more than a set, it is a poset.
Strict enrichment
For this section, let be a previewmonoidal previewcategory.
Definition Enriched category An -enriched category is
- Objects
- a set of objects ;
- Morphisms
- for every objects , a object ;
- Identity
- for every object , there is a morphism:
- Composition
- for every objects , there is a morphism
- Associativity