# Is the theory of weak $n$-categories a cofibrant replacement of the theory of strict ones?

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!)

• This doesn't quite answer your question, and isn't fully formed even if it did, so I'll leave it as a comment. Something I learned from Barwick--Schommer-Pries is that there's a very close connection between $(\infty,n)$-categories and 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. – Theo Johnson-Freyd Jul 20 '16 at 23:31
• In some sense, the Batamin-Leinster aproaches to weak $\infty$-category can be describe as such: you start with the "free strict infinity category" monad $T$ on the category of globular set. Then you study $T$-operad, you can say that a morphism of $T$-operade is a "trivial fibration" (has a contraction in Leinster terminology) if the underlying map of globular set have the right lifting property with respect to all the map $S_{n-1} \rightarrow D_n$, or equivalently with respect to all monomorphism. A cofibration is a morphism of operade which has the LLP with respect to all ... – Simon Henry Jul 21 '16 at 3:17
• ... trivial fibration. By the small object argument you get a factorization system on the category of $T$ operad (cofibration/trivial fibration) and Leinster construction of an operad for weak $\infty$-catgory can be describe as a "cofibrant replacement" of $T$ (seen as a $T$ operade) in this sense. But it is not clear if and how this factorization system can be extended to a model structure. See Leinster book (arxiv.org/abs/math/0305049) for more details. – Simon Henry Jul 21 '16 at 3:19
• Is this really in Leinster's book? I learnt it from Garner's later paper ‘A homotopy-theoretic universal property of Leinster’s operad for weak ω-categories’. Quoting from the abstract: ‘[T]he universal and canonical cofibrant replacement of the operad for strict ω-categories is precisely Leinster’s operad for weak ω-categories’. – Tom Hirschowitz Jul 21 '16 at 7:20
• Garner simply induces the cobfibration / trivial fibration half of the Reedy model structure on globular sets along an adjunction to get a weak factorization system on globular operads. He finds that Leinster's model for $\omega$-categories is precisely the cofibrant replacement for the strict $\omega$-categories operad supplied by Garner's small object argument. Inducing the full Reedy model structure will give a full model structure on globular operads. This is exactly what I was looking for, thanks! @TomHirschowitz, SimonHenry if either of you write this as an answer I will accept it. – Tim Campion Jul 21 '16 at 14:37