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


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.
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  • $\begingroup$ Thanks, what about cocompleteness (for the first bullet)? Does MonCat (with strict monoidal functors) have small colimits? $\endgroup$ – user84563 Jul 5 '16 at 16:20
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    $\begingroup$ Yes, because it is the category of algebras for an accessible 2-monad on a locally presentable 2-category (namely Cat). The second bullet is also cocomplete as a bicategory; Blackwell-Kelly-Power proved this using the pseudo-morphism-classifiers and flexibility. I don't know whether the third bullet has any colimits. $\endgroup$ – Mike Shulman Jul 6 '16 at 6:19
  • $\begingroup$ Isn't the category of algebras usually taken as the strict algebras? I guess the possibility of replacing a pseudo algebra with a strict algebra helps here, or am I mistaken in the first place? $\endgroup$ – Kevin Arlin Jul 30 '16 at 15:25
  • $\begingroup$ @KevinCarlson Yes, the category of algebras is the strict algebras. There is in fact a 2-monad whose strict algebras are non-strict monoidal categories; this is how one generally operates in 2-monad theory. $\endgroup$ – Mike Shulman Jul 31 '16 at 7:00

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