Kerodon

Proof. For $n \neq 0$, the desired result follows from immediately from the definitions. Let us therefore assume that $n = 0$, so that $\operatorname{\mathcal{D}}$ is isomorphic to the nerve of a partially ordered set. If $\operatorname{\mathcal{C}}$ is also isomorphic to the nerve of a partially ordered set, then any fiber product $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ has the same property (since the formation of nerves commutes with fiber products). Conversely, suppose that $F$ is a $0$-categorical inner fibration. Then $F$ is also a $1$-categorical inner fibration, so the preceding argument shows that $\operatorname{\mathcal{C}}$ is isomorphic to the nerve of a category $\operatorname{\mathcal{C}}_0$. We will complete the proof by showing that $\operatorname{\mathcal{C}}_0$ is a partially ordered set. For this, we must verify the following:

• Let $u,v: X \rightarrow Y$ be morphisms in $\operatorname{\mathcal{C}}$ having the same source and target; we wish to show that $u = v$. Our assumption that $\operatorname{\mathcal{D}}$ is a $0$-category guarantees that $F(u) = F(v)$. The desired result now follows from the observation that the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$ is a $0$-category.

• Let $X$ and $Y$ be isomorphic objects of $\operatorname{\mathcal{C}}$; we wish to show that $X = Y$. Fix morphisms $u: X \rightarrow Y$ and $v: Y \rightarrow X$. Since $\operatorname{\mathcal{D}}$ is a $0$-category, we have $F(u) = \operatorname{id}_{D} = F(v)$ for some object $D \in \operatorname{\mathcal{D}}$. In this case, we can regard $u$ and $v$ as morphisms of the $\infty $-category $\operatorname{\mathcal{C}}_{D} = \{ D\} \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}$. Our assumption that $F$ is a $0$-categorical inner fibration guarantees that $\operatorname{\mathcal{C}}_{D}$ is a $0$-category, so that $X = Y$.

Proof. For $n < 0$, this follows immediately from the definitions. We may therefore assume that $n \geq 0$. Using Remark 4.8.6.10, we can reduce to the case where $\operatorname{\mathcal{E}}= \Delta ^ m$ is a standard simplex. In this case, our assumption on $G$ guarantees that $\operatorname{\mathcal{D}}$ is an $n$-category. We wish to show that $\operatorname{\mathcal{C}}$ is an $n$-category if and only if the inner fibration $F$ is $n$-categorical, which follows from Proposition 4.8.6.9. $\square$

Proof of Proposition 4.8.6.12. If $n \leq -2$, then $F$ is an isomorphism of simplicial sets and therefore an equivalence of $\infty $-categories (Example 4.5.1.11). If $n = -1$, then $F$ is an isomorphism from $\operatorname{\mathcal{C}}$ onto a full subcategory of $\operatorname{\mathcal{D}}$, and therefore fully faithful (Example 4.8.3.2). We may therefore assume without loss of generality that $n \geq 0$. By virtue of Proposition 4.8.3.36, we may assume without loss of generality that $\operatorname{\mathcal{D}}$ is a standard simplex; in this case, we wish to show that $\operatorname{\mathcal{C}}$ is locally $(n-1)$-truncated. This follows from Warning 4.7.7.21, since $\operatorname{\mathcal{C}}$ is an $n$-category (Proposition 4.8.6.9). $\square$

Proof. For $n < 0$, this is immediate from the construction. We may therefore assume without loss of generality that $n \geq 0$. Using Remarks 4.8.6.10 and 4.8.6.16, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In particular, $\operatorname{\mathcal{D}}$ is an $n$-category. In this case, Example 4.8.6.14 guarantees that the simplicial set $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \simeq \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}})}}$ is an $n$-category. The desired result now follows from Proposition 4.8.6.9. $\square$

Proof. Since $\operatorname{\mathcal{D}}$ is an $\infty $-category and the comparison map $G: \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \rightarrow \operatorname{\mathcal{D}}$ is an inner fibration (Proposition 4.8.6.17), the simplicial set $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ is also an $\infty $-category (Remark 4.1.1.9). Fix an integer $m \leq n+1$; we wish to show that the functor $F'$ is $m$-full. For $n = -2$, there is nothing to prove. If $n = -1$, then $F'$ is surjective on objects (by construction) and therefore essentially surjective. We may therefore assume without loss of generality that $n \geq 0$. Since $F'$ is an inner fibration (Proposition 4.8.6.22), it will suffice to show that for every morphism $\Delta ^1 \rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$, the projection map $\Delta ^{1} \times _{ \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} } \operatorname{\mathcal{C}}\rightarrow \Delta ^1$ is $m$-full (Proposition 4.8.2.25). Using Remark 4.8.6.16, we can replace $F$ by the projection map $\Delta ^1 \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^1$, and thereby reduce to the situation where $\operatorname{\mathcal{D}}= \Delta ^1$ is an $n$-category. In this case, the functor $F'$ exhibits $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ as a homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Example 4.8.6.14), so the desired result follows from Example 4.8.5.19. $\square$

Proof. We may assume without loss of generality that $n \geq 0$ (otherwise, the result follows immediately from the construction). For every morphism of simplicial sets $K \rightarrow \operatorname{\mathcal{D}}$, Remark 4.8.6.18 determines a comparison map

We will prove that each $\theta _{K}$ is an isomorphism of simplicial sets; Proposition 4.8.6.20 then follows by taking $K = \operatorname{\mathcal{D}}$. Note that the construction $K \mapsto \theta _{K}$ carries colimits (in the category of simplicial sets with a morphism to $\operatorname{\mathcal{D}}$) to limits (in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$). By virtue of Remark 1.1.3.13, we can assume without loss of generality that $K$ is a standard simplex. Replacing $F$ by the projection map $K \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow K$ and $\operatorname{\mathcal{D}}'$ by the fiber product $K \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{D}}'$, we are reduced to proving Proposition 4.8.6.20 in the special case where $\operatorname{\mathcal{D}}$ is a standard simplex: in particular, it is an $n$-category. In this case, $\operatorname{\mathcal{D}}'$ is also an $n$-category (Proposition 4.8.6.9), and we can identify $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ with the homotopy $n$-category of $\operatorname{\mathcal{C}}$ (Example 4.8.6.14). Applying Proposition 4.7.9.7, we see that the horizontal maps in the commutative diagram

are isomorphisms. The desired result now follows by passing to fibers of the vertical maps. $\square$

Proof. Without loss of generality, we may assume that $n \geq 0$. Using Remarks 4.1.1.13 and 4.8.6.16, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, $F'$ identifies with the tautological map $\operatorname{\mathcal{C}}\rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}})}}$ (Example 4.8.6.14), so the desired result follows from Corollary 4.7.9.16. $\square$

Proof. We first prove $(1)$. Assume that $F$ is a left fibration, and suppose we are given integers $0 \leq i < n$; we wish to show that every lifting problem

admits a solution. If $m \leq n+2$, then $\sigma _0$ can be lifted to a morphism $\Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}$ (Remark 4.8.6.21), so the desired result follows from our assumption that $F$ is a left fibration. We may therefore assume that $m \geq n+3$. If $n = -2$, then $G$ is an isomorphism and there is nothing to prove. If $n = -1$, then $G$ identifies $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ with a full simplicial subset of $\operatorname{\mathcal{D}}$, and the desired result follows from the observation that $\Lambda ^{m}_{i}$ contains every vertex of $\Delta ^ m$. We may therefore assume that $n \geq 0$. Replacing $F$ by the projection map $\Delta ^ m \times _{\operatorname{\mathcal{D}}} \operatorname{\mathcal{C}}\rightarrow \Delta ^ m$, we can reduce to the case where $\operatorname{\mathcal{D}}= \Delta ^ m$ is a standard simplex. In this case, $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ is an $n$-category (Example 4.8.6.14). In particular, it is an $(n+1)$-coskeletal simplicial set, so the lifting problem (4.90) has a unique solution (since $\Lambda ^{m}_{i}$ contains the $(n+1)$-skeleton of $\Delta ^ m$).

Assertion $(2)$ follows by applying $(1)$ to the opposite inner fibration $F^{\operatorname{op}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{D}}^{\operatorname{op}}$. Assertion $(3)$ follows by combining $(1)$ and $(2)$ with Example 4.2.1.5. It remains to prove $(4)$. Fix an object $Y \in \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ and an isomorphism $\overline{e}: \overline{X} \rightarrow G(Y)$ in the $\infty $-category $\operatorname{\mathcal{D}}$; we wish to show that $\overline{e}$ can be lifted to an isomorphism $e: X \rightarrow Y$ of $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$. If $n \leq -2$, then $G$ is an isomorphism and the result is obvious. Otherwise, the comparison map $F': \operatorname{\mathcal{C}}\rightarrow \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ is surjective on vertices, so we can choose an object $\widetilde{Y} \in \operatorname{\mathcal{C}}$ satisfying $F'( \widetilde{Y} ) = Y$. If $F$ is an isofibration, then there exists an isomorphism $\widetilde{e}: \widetilde{X} \rightarrow \widetilde{Y}$ of $\operatorname{\mathcal{C}}$ satisfying $F( \widetilde{e} ) = \overline{e}$. It follows that $e = F'( \widetilde{e} )$ is an isomorphism in $\operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}}$ satisfying $G(e) = \overline{e}$. $\square$

Proof. It follows from Proposition 4.8.6.17 (and Proposition 4.8.6.12) that the comparison map $G: \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \rightarrow \operatorname{\mathcal{D}}$ is $(n+1)$-faithful. By virtue of Remark 4.8.3.29, we can replace $(1)$ by the following condition:

$(1')$

The functor $F'$ is $(n+1)$-faithful: that is, it is $m$-full for $m \geq n+2$.

Since $F'$ is also $m$-full for $m \leq n+1$ (Corollary 4.8.6.19), the equivalence $(1') \Leftrightarrow (2)$ follows from Theorem 4.8.4.1. $\square$

Proof. The implication $(2) \Rightarrow (1)$ follows from Proposition 4.8.6.12 (together with Remark 4.8.2.18). To prove the converse, we may assume without loss of generality that $F$ is an isofibration (Corollary 4.5.3.24). In this case, the factorization $\operatorname{\mathcal{C}}\xrightarrow {F'} \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}/\operatorname{\mathcal{D}})}} \xrightarrow {G} \operatorname{\mathcal{D}}$ of Remark 4.8.6.18 has the desired properties: Proposition 4.8.6.24 guarantees that $F'$ is an equivalence of $\infty $-categories, Proposition 4.8.6.23 guarantees that $G$ is an isofibration, and Proposition 4.8.6.17 guarantees that $G$ is $n$-categorical. $\square$

Proof. We have a commutative diagram of $\infty $-categories

Here $F$ is categorically $(n+1)$-connective by assumption, and the vertical maps are categorically $(n+1)$-connective by virtue of Corollary 4.8.6.19. Applying Proposition 4.8.5.15, we see that the functor $F'$ is also categorically $(n+1)$-connective. We also have a commutative diagram

where the vertical maps are $(n+1)$-faithful (Proposition 4.8.6.17). Using Remark 4.8.3.29, we see that $F'$ is $(n+1)$-faithful. Applying Theorem 4.8.4.1, we see that $F'$ is an equivalence of $\infty $-categories. $\square$

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 4.8.6.17 and Remark 4.8.2.15. The reverse implication follows by applying Proposition 4.8.6.26 in the special case $\operatorname{\mathcal{E}}= \operatorname{\mathcal{D}}$. $\square$