Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?



Completeness of 2-category of Monoidal Categories
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](http://ncatlab.org/nlab/show/PIE-limit) 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](http://arxiv.org/abs/1104.2111), including all colax (resp. lax) limits.


