definition of the notion of a subcategory of an $\infty$-category

Question

Is the following formulation of the notion of a subcategory of an $\infty$-category ( - in the sense of section 1.2.11 of Lurie's Higher Topos Theory - ) correct?:

Remark. Let $\mathcal{C}$ be an $\infty$-category. A simplicial subset $X \subseteq \mathcal{C}$ is a subcategory (of $\mathcal{C}$) if and only if the inclusion $X \subseteq \mathcal{C}$ is an inner fibration.

Answer 1

Suppose we have a pullback diagram in $\mathbf{sSet}$ of the form below, $$\require{AMScd} \begin{CD} X @>>> Y \\ @VVV @VVV \\ N \tau_1 X @>>> N \tau_1 Y \end{CD}$$ where $\tau_1 : \mathbf{sSet} \to \mathbf{Cat}$ is the left adjoint of the nerve functor $N : \mathbf{Cat} \to \mathbf{sSet}$. Since $\tau_1$ sends every inner horn inclusion in $\mathbf{sSet}$ to an isomorphism in $\mathbf{Cat}$, every morphism in the image of $N : \mathbf{Cat} \to \mathbf{sSet}$ is an inner fibration. Moreover, the class of inner fibrations is closed under pullback, so $X \to Y$ is an inner fibration.

But more is true: $\tau_1$ also inverts boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ for $n \ge 3$, so $X \to Y$ has additional lifting properties beyond those of inner fibrations. Furthermore, in the case where $\tau_1 X \to \tau_1 Y$ is a faithful functor, $N \tau_1 X \to N \tau_1 Y$ has the right lifting property with respect to the boundary inclusion $\partial \Delta^2 \hookrightarrow \Delta^2$ as well. Heuristically, this is saying that a sub-quasicategory is "full" in dimensions $\ge 2$.

Consequently, the right characterisation of sub-quasicategory inclusions by lifting properties should be the following:

• $X \to Y$ is a sub-quasicategory inclusion if and only if it is a monomorphism, an inner fibration, and has the right lifting property with respect to boundary inclusions $\partial \Delta^n \hookrightarrow \Delta^n$ for $n \ge 2$.

We have already verified the "only if" direction, so suppose $X \to Y$ satisfies the above conditions. Then $\tau_1 X \to \tau_1 Y$ is a subcategory inclusion – this is an easy check. The interesting part is to verify that we get the required pullback diagram – for this, work inductively and use the explicit description of $\tau_1 X$ and $\tau_1 Y$.

However, as it turns out, monomorphisms that are inner fibrations automatically have the extra right lifting properties: if we have some commutative square of the form below, $$\begin{CD} \partial \Delta^n @>>> X \\ @VVV @VVV \\ \Delta^n @>>> Y \end{CD}$$ then we can restrict along $\Lambda^n_k \hookrightarrow \partial \Delta^n$ (for some $0 < k < n$) and note that any filler for the inner horn is also a filler for the boundary. Thus your conjecture is correct.