I am trying to understand two related notions:
- $(\infty,2)$-category as in Definition 5.5.1.3, Kerodon
- weak $\infty$-bicategory as in Definition 4.1.1 in "$(\infty,2)$-Categories and the Goodwillie Calculus I (Lurie, J.)".
I would like to show (as claimed in Remark 1.21, "On the Equivalence of all Models of $(\infty,2)$-categories (Gagna, et al.)" that an $(\infty,2)$-category (which I identify with the scaled simplicial set given by the underlying simplicial set along with its thin $2$-cells) is necessarily a weak $\infty$-bicategory.
Question: I am having trouble in understanding why an $(\infty,2)$-category should lift against the following collection of generating scaled anodyne maps as given in (C), Definition 3.1.3,"$(\infty,2)$-Categories and the Goodwillie Calculus I (Lurie, J.)".
The inclusion $$(\Lambda_0^n\bigsqcup _{\Delta^{\{0,1\}}} \Delta^0,T) \subseteq (\Delta^n\bigsqcup _{\Delta^{\{0,1\}}} \Delta^0,T)$$ where $n>2$ and $T$ is the collection of all degenerate $2$-simplices of $\Delta^n\bigsqcup _{\Delta^{\{0,1\}}} \Delta^0$ together with the image of the simplex $\Delta^{\{0,1,n\}}$.
It is mentioned in "On the Equivalence of all Models of $(\infty,2)$-categories (Gagna, et al.)" that this follows by definition and I see that the following condition ((3), Definition 5.5.1.3, Kerodon) satisfied by an $(\infty,2)$-category is very similar.
Let $n\geqslant 3$ and let $\sigma_0:\Lambda_0^n \rightarrow \mathcal{C}$ be a morphism of simplicial sets with the property that the $2$-simplex $\sigma_0|_{N_{\bullet}(0<1<n)}$ is left-degenerate. Then $\sigma_0$ can be extended to an $n$-simplex of $\mathcal{C}$
However, since left-degeneracy of the respective $2$-simplex is a stronger requirement than the edge $\Delta^{\{0,1\}}$ being degenerate, I do not see how this would result in an immediate implication.
I also see that this is very similar to Joyal's special outer horn lifting theorem in $\infty$-categories and I could show this for when $n=3$. However, I am not able to put all the details together in the general case.
On a related note, I would also like to know what the implications are if the definition of scaled anodyne maps is modified by replacing the above mentioned collection of generating scaled anodyne maps by those where the image of the simplex $\Delta^{\{0,1,n\}}$ is left-degenerate.
Any help is much appreciated.
