TITLE: Completeness of 2-category of Monoidal Categories
QUESTION [8 upvotes]: Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?

REPLY [11 votes]: It depends on what you mean by "the 2-category of monoidal categories" and also what you mean by "complete".

The 2-category of monoidal categories and strict monoidal functors is complete as a Cat-enriched category in the sense of enriched category theory, hence also complete as a bicategory.
The 2-category of monoidal categories and strong monoidal functors is not complete as a Cat-enriched category, but it has all PIE-limits and hence all pseudo-limits, and therefore is complete as a bicategory (has all "bilimits", i.e. bicategorical limits).
The 2-category of monoidal categories and lax (resp. colax) monoidal functors is not even complete as a bicategory, but it does have an interesting class of limits, including all colax (resp. lax) limits.