# Is every weak $\infty$-bicategory (à la Lurie) an $\infty$-bicategory?

In Definition 4.1.1 of $(\infty,2)$-Categories and the Goodwillie Calculus I, Lurie defines a weak $\infty$-bicategory to be a scaled simplicial set that has the extension property with respect to every scaled anodyne morphism. In Theorem 4.2.7, he defines a model structure on $\operatorname{Set}_{\Delta}^{\operatorname{sc}}$, the category of scaled simplical sets, and in Definition 4.2.8 he defines an $\infty$-bicategory to be a scaled simplicial set that is a fibrant object in the model category $\operatorname{Set}_{\Delta}^{\operatorname{sc}}$.

Every $\infty$-bicategory is a weak $\infty$-bicategory, because every scaled anodyne morphism is a bicategorical equivalence (Proposition 3.1.13). What about the converse? Is every weak $\infty$-bicategory an $\infty$-bicategory?

EDIT:

I think I now have a proof that, indeed, every weak $\infty$-bicategory is an $\infty$-bicategory. I will post it tomorrow.

• A side note: I bigly disliked the morphism of Definition 3.1.3.(B) ( - which, by the way, contains a typo: the first instance of $\Delta^{\{1,3,4\}}$ should be replaced by $\Delta^{\{0,1,4\}}$ - ), but was very happy to find out that we may replace it (without changing the class of scaled anodyne morphisms) by the morphisms $f_1$ and $f_2$ of Remark 3.1.4. Sep 11 '17 at 23:57
• I'm sorry, my remark above that we may replace the morphism in Definition 3.1.3.(B) by the morphisms $f_1$ and $f_2$ of Remark 3.1.4 without changing the class of scaled anodyne morphisms is most probably false. I found a mistake in my calculations. So sad. Sep 12 '17 at 2:51
• The thing to remember about Verity's model is that the complicial horn fillers are all obtained from those in dimension 1 and 2 (which encode transport along equivalences and composition of 1-cells respectively) by taking pushout-joins with cofibrations (specifically, inclusions $\partial \Delta^n \to \Delta^n$. Likewise, the complicial thinness extensions are all obtained from the one in dimension 2 (which encodes 2-out-of-3) by pushout-joins with cofibrations. So the basic principle is just that everything is stable under join. Sep 17 '17 at 15:23
• This is discussed in Observation 38 of Weak Complicial Sets I in the course of showing the related fact that Verity's anodyne extensions are closed under what I referred to as "pushout-joins", which he calls "corner-joins". Sep 17 '17 at 18:38
• You just have to pushout-join on both sides! Sep 18 '17 at 11:21

A few years later, it has been shown by Gagna, Harpaz, and Lanari that the answer is yes. Every weak $$\infty$$-bicategory is an $$\infty$$-bicategory.