# Is there a model structure on (strict?) Monoidal Categories?

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.

I'm not sure about the case of general monoidal categories. (Although I seem to recall a remark that there is no such structure since the category of monoidal categories is not cocomplete and a suitable replacement would be the category of multicategories. Perhaps somebody can confirm this.)

However, the case of strict monoidal categories is known since they are simply monoids internal to the category of categories. It is a theorem of Schwede and Shipley (Theorem 4.1 in Algebras and Modules in Monoidal Model Categories) that if a monoidal model category satisfies so called monoid axiom, then the category of monoids in it inherits a model structure. The monoid axiom is a technical condition which is, in particular, automatically satisfied if all objects are cofibrant as is the case in the canonical model structure on the category of categories.

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