nLab
category of V-enriched categories

Idea

For $𝒱$ a suitable context of enrichment (notbaly: for $𝒱$ a cosmos) there is a $2$-category $𝒱\mathrm{Cat}$ whose

• objects are $V$-enriched categories;

• morphisms are $V$-enriched functors; and

• $2$-morphisms are $V$-enriched natural transformations.

Sometimes one also considers $𝒱\mathrm{Cat}$ as a mere category by dropping the $2$-morphisms (and using enriched strict categories).

Examples

• For $𝒱=($Set$,\times )$, $𝒱\mathrm{Cat}\simeq$ Cat, the $2$-category of locally small categories.

• For $𝒱=($Cat$,\times )$, $𝒱\mathrm{Cat}\simeq$ Str2Cat, the $2$-category of strict 2-categories.

Related concepts

$(n+1,r+1)$-categories of (n,r)-categories

• Pos

• Set

  • Rel, Prof

• Grpd, ∞Grpd

• Cat

  • VCat

  • Operad

• 2Cat

• (∞,1)Cat

  • (∞,1)Operad

• (∞,n)Cat

category: category