I have algebraic models of $n$-categories in mind. By "theory of (weak) $n$-categories", I mean "[monad / operad / whatever] whose algebras are (weak) $n$-categories".

To be more precise: fix an algebraic definition of $n$-category where an $n$-category is an algebra for some monad / operad $T$ on some category $\mathcal{C}$ (like globular sets or something). Then there should be a monad / operad $T_0$ on $\mathcal{C}$ whose algebras are strict $n$-categories, and there should be a canonical map $T \to T_0$, which on algebras induces (contravariantly) the inclusion of strict $n$-categories into weak ones. **My question** is whether there is a natural model structure on monads / operads on $\mathcal{C}$ such that $T \to T_0$ is a cofibrant replacement of $T_0$ by $T$.

Such a model structure would be rather strange since the notion of weak equivalence apparently wouldn't preserve the category of algebras of the monad / operad. But I see a few reasons to expect such a model structure:

Generally, the main intuition I have for weak higher categories is that they have some extra "flab" added to the strict version in order to correct excessive strictness. This sounds a lot like the "flab" you add when cofibrantly replacing a diagram to take its homotopy colimit.

One notion of a weak $n$-functor (due to Garner) is a strict $n$-functor out of a cofibrant replacement of the domain in a suitable model category.

There is a model structure on finitary monads on $\mathsf{Cat}$ (due to Lack) such that the 2-monad for monoidal categories is the cofibrant replacement for the 2-monad for strict monoidal categories.

**ADDED**It seems to be well known that the $E_\infty$ operad is a cofibrant replacement for the commutative operad, while the $A_\infty$ operad is a cofibrant replacement for the associative operad.

I'd be interested in the answer for *any* algebraic notion of $n$-category, and any $n$ with $2 \leq n \leq \infty$. If there's a way to formulate / say something about this for non-algebraic notions of higher category I'd be interested too.

(Of course I'm not suggesting that every weak $n$-category is equivalent to a strict one -- the cofibrant replacement in question lives a category level higher!)

gaunt$n$-categories, which are (necessarily strict) $n$-categories in which all isomorphisms are identities. So if you're hoping for "yes" but receive "no", perhaps you'll win with gaunt-like variations. $\endgroup$ – Theo Johnson-Freyd Jul 20 '16 at 23:31