nLab local cocompletion

Definition

The local cocompletion of a bicategory $\mathcal{K}$ is the hom-category-wise free cocompletion of $\mathcal{K}$. Abstractly, this can be seen as a change of base for bicategories enriched in a monoidal bicategory, along the left pseudoadjoint from the 2-category of categories to the 2-category of cocomplete categories.

Examples

• Viewing Set as a one-object bicategory, for any category $C$, the

Related pages

• free cocompletion

• Day convolution

• local colimits

References

• Brian Day, An embedding theorem for closed categories, Category Seminar: Proceedings Sydney Category Theory Seminar 1972/1973.

• Max Kelly, A. Labella, V. Schmitt, Ross Street, §5 of: Categories enriched on two sides, Journal of Pure and Applied Algebra 168 1 (2002) 53-98 [doi:10.1016/S0022-4049(01)00048-2]

• Richard Garner, Mike Shulman: Enriched categories as a free cocompletion [arXiv:1301.3191]

See Example 4.6 of:

• Soichiro Fujii and Stephen Lack, The oplax limit of an enriched category, Theory and Applications of Categories 40.14 (2024): 390-412.