TITLE: Is there a model structure on (strict?) Monoidal Categories?
QUESTION [8 upvotes]: Basically, what the title says. 
Presumably, one could use the fact that monoidal categories (resp. strict monoidal categories) are one-object bicategories (resp. 2-categories) and use the Lack model structure on those, but I am unsure if this would work or not.

REPLY [4 votes]: There is a model structure on the category of monoidal categories and strict monoidal functors.  A strict monoidal functor is a fibration / weak equivalence just when its underlying functor is a fibration / weak equivalence in Cat.
More generally for T a 2-monad with rank on Cat you can lift the model structure on Cat to the category of strict T-algebras and strict T-algebra morphisms: this generalises the above example.  
For these results, see the discussion in Section 1.7 and Theorem 4.5 of Steve Lack's paper "Homotopy theoretic aspects of 2-monads": https://arxiv.org/abs/math/0607646