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!)

  • $\begingroup$ 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. $\endgroup$ – Theo Johnson-Freyd Jul 20 '16 at 23:31
  • 2
    $\begingroup$ 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 ... $\endgroup$ – Simon Henry Jul 21 '16 at 3:17
  • 2
    $\begingroup$ ... 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. $\endgroup$ – Simon Henry Jul 21 '16 at 3:19
  • 4
    $\begingroup$ 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’. $\endgroup$ – Tom Hirschowitz Jul 21 '16 at 7:20
  • 1
    $\begingroup$ 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. $\endgroup$ – Tim Campion Jul 21 '16 at 14:37

Your Answer

By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Browse other questions tagged or ask your own question.