Kerodon

Proof. The implication $(1) \Rightarrow (2)$ is a restatement of Corollary 4.8.5.23, and the implication $(2) \Rightarrow (3)$ follows from Corollary 4.5.3.31. To show that $(3)$ implies $(1)$, we may assume without loss of generality that $F$ is an isofibration. Then the comparison map $G: \operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}})}} \rightarrow \operatorname{\mathcal{B}}$ is an $(n-1)$-categorical isofibration (Propositions 4.8.6.17 and 4.8.6.23). If $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{B}}$ is another $(n-1)$-categorical isofibration, then we can use Proposition 4.8.6.20 to identify $\theta _{\operatorname{\mathcal{C}}}$ with the functor $\operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname{\mathcal{B}}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}_{ / \operatorname{\mathcal{B}}}( \operatorname {h}_{\mathit{\leq {}n-1}}{\mathit{(\operatorname{\mathcal{A}}/\operatorname{\mathcal{B}})}}, \operatorname{\mathcal{C}})$ given by precomposition with $G$. If condition $(3)$ is satisfied, then $G$ is an equivalence of $\infty $-categories, so that $F$ is categorically $n$-connective by virtue of Corollary 4.8.6.27. $\square$

Proof. By virtue of Corollary 4.1.3.3, there exists a functor $Q: \operatorname{Set_{\Delta }}\rightarrow \operatorname{Set_{\Delta }}$ which commutes with filtered colimits and a natural transformation of functors $u: \operatorname{id}_{\operatorname{Set_{\Delta }}} \rightarrow Q$ with the property that, for every simplicial set $A$, the simplicial set $Q(A)$ is an $\infty $-category and the morphism $u_ A: A \rightarrow Q(A)$ is inner anodyne. For every morphism of simplicial sets $f: A \rightarrow B$, we have a commutative diagram

where the horizontal maps are categorical equivalences. It follows that $f$ is categorically $n$-connective if and only if $Q(f)$ is categorically $n$-connective. The desired result now follows from the analogous statement for filtered colimits of $\infty $-categories (Remark 4.8.5.11). $\square$

Proof. Combine Proposition 4.8.8.10 with Remark 4.8.8.7. $\square$

Proof. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-faithful functor of $\infty $-categories, and consider the cubical diagram

Our assumption that $f$ is categorically $n$-connective guarantees that the right face is a categorical pullback square, and our assumption on (4.91) guarantees that the front and back faces are categorical pullback squares. Applying Proposition 4.5.3.19, we conclude that the left face is also a categorical pullback square. $\square$

Proof. We will prove the following stronger assertion: for every morphism of simplicial sets $S' \rightarrow S$, the induced map

is categorically $n$-connective. By virtue of Corollary 4.8.8.11 (and Remark 1.1.4.4), we may assume without loss of generality that $S'$ has dimension $\leq k$ for some integer $k \geq -1$. We proceed by induction on $k$. In the case $k=-1$, the simplicial set $S'$ is empty and there is nothing to prove. Assume therefore that $k \geq 0$. Let $S''$ denote the $(k-1)$-skeleton of $S'$ and let $I$ be the set of nondegenerate $k$-simplices of $S'$, so that Proposition 1.1.4.12 supplies a pushout diagram of simplicial sets

where the horizontal maps are monomorphisms. It follows that the front and back faces of the diagram

are categorical pushout squares (Proposition 4.5.5.11). Consequently, to show that $f_{S'}$ is categorically $n$-connective, it will suffice to show that $f_{S''}$, $u$, and $v$ are categorically $n$-connective (Proposition 4.8.8.12). In the first two cases, this follows from our inductive hypothesis. We may therefore replace $S'$ by the coproduct $\coprod _{i \in I} \Delta ^{k}$, and thereby reduce to the case where $S'$ is a coproduct of simplices. Using Remark 4.8.8.9, we can further reduce to the case where $S' \simeq \Delta ^{k}$ is a standard simplex, in which case the desired result follows from our hypothesis on $f$. $\square$

Proof. Combine Corollary 4.8.8.13 with Remark 4.8.8.7. $\square$

Proof. The equivalences $(1) \Leftrightarrow (2) \Leftrightarrow (3)$ follow from Remarks 4.8.5.24 and 4.8.8.3, and the implication $(3) \Rightarrow (4)$ is immediate. We will complete the proof by showing that $(4)$ implies $(3)$. Assume that condition $(4)$ is satisfied, and let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an $n$-faithful isofibration of $\infty $-categories. We wish to show that, for every simplicial set $B'$ and every simplicial subset $A' \subseteq B'$, every lifting problem

admits a solution. Unwinding the definitions, we can rewrite (4.92) as a lifting problem

The existence of a solution follows from $(4)$, since $V'$ is also an $n$-faithful isofibration of $\infty $-categories (Corollary 4.8.4.14 and Proposition 4.4.5.1). $\square$

Proof. Using Proposition 1.1.4.12, we can choose a simplicial subset $A' \subseteq \operatorname{sk}_{n}(A)$ which contains the $(n-1)$-skeleton of $A$, such that $f$ restricts to an isomorphism of $A'$ with the $n$-skeleton of $B$. It follows from Example 4.8.8.16 that $f|_{A'}$ is categorically $n$-connective, and that the inclusion map $A' \hookrightarrow A$ is categorically $(n-1)$-connective. Applying Remark 4.8.8.6, we deduce that $f$ is categorically $n$-connective. $\square$

Proof. Without loss of generality, we may assume that $m \leq n$.If $m = 0$, then $\iota _{m,n}$ can be identified with the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$, which is categorically $(n-1)$-connective by virtue of Example 4.8.8.16. We may therefore assume that $m \geq 1$, so that $n \geq 1$. We proceed by induction on $n$. If $n > 1$, then we can choose an integer $0 < i < n$, so that $\operatorname{\partial \Delta }^{n}$ contains an inner horn $\Lambda ^{n}_{i}$. In this case, Lemma 1.5.7.5 guarantees that the composite map

is inner anodyne, and therefore categorically $(m+n-1)$-connective. Consequently, to prove that $\iota _{m,n}$ is categorically $(m+n-1)$-connective, it will suffice to show that $\rho $ is categorically $(m+n-2)$-connective (Remark 4.8.8.6). This follows from Proposition 4.8.8.12, since $\rho $ is a pushout of $\iota _{m,n-1}$ (which is categorically $(m+n-2)$-connective by our inductive hypothesis).

It remains to treat the case $m = n = 1$. Let $K$ be a simplicial subset of $\Delta ^1 \times \Delta ^1$ obtained by removing a single nondegenerate $2$-simplex. In this case, we observe that $\iota _{m,n}$ factors as a composition

where the map on the left is inner anodyne (it is a pushout of the inner horn inclusion $\Lambda ^{2}_{1} \hookrightarrow \Delta ^2$). We are therefore reduced to proving that $\iota '$ is categorically $1$-connective, which is a special case of Example 4.8.8.16 (since $K$ contains the $1$-skeleton of $\Delta ^1 \times \Delta ^1$). $\square$

Proof. Without loss of generality, we may assume that $f$ is a monomorphism. Let us temporarily regard $f$ as fixed, and say that a morphism $g: X \rightarrow Y$ is good if $\nu _{f,g}$ is categorically $(m+n-1)$-connective. We wish to show that if $g$ is categorically $(n-1)$-connective, then it is good. Our proof will make use of the following observations:

$(a)$

Suppose we are given a commutative diagram

\[ \xymatrix { X \ar [r]^{g} \ar [d] & Y \ar [d] \\ X' \ar [r]^{ g' } & Y' } \]

where the vertical maps are categorical equivalences. Since $f$ is a monomorphism, we deduce that the vertical maps in the diagram

\[ \xymatrix { (B \times X) \coprod _{ (A \times X)} (A \times Y) \ar [r]^{\nu _{f,g}} \ar [d] & B \times Y \ar [d] \\ (B \times X') \coprod _{ (A \times X')} (A \times Y') \ar [r]^{\nu _{f,g'}} & B \times Y' } \]

are categorical equivalences. It follows that $g$ is good if and only if $g'$ is good (Remark 4.8.8.4).

$(b)$

The collection of good morphisms of simplicial sets is closed under filtered colimits. This follows from Proposition 4.8.8.10, since the construction $g \mapsto \nu _{f,g}$ preserves filtered colimits.

$(c)$

Let $g: X \rightarrow Z$ be a morphism of simplicial sets which factors as a composition $X \xrightarrow {g'} Y \xrightarrow {g''} Z$, where $g'$ is a monomorphism. Then $\nu _{f,g}$ factors as a composition

\[ (B \times X) \coprod _{ (A \times X)} (A \times Z) \rightarrow (B \times Y) \coprod _{ (A \times Y)} (A \times Z) \xrightarrow { \nu _{f,g''} } B \times Z, \]

where the map on the left is a pushout of the monomorphism $\nu _{f,g'}$. Using Proposition 4.8.8.12 and 4.8.8.6, we see that if $g'$ and $g''$ are good, then $g$ is good.

$(d)$

Let $g: X \hookrightarrow Y$ be a monomorphism of simplicial sets which is a pushout of a monomorphism $g_0: X_0 \hookrightarrow Y_0$. Then $\nu _{f,g}$ is a pushout of $\nu _{f,g_0}$. Using Proposition 4.8.8.12 we see that if $g_0$ is good, then $g$ is good.

Let $g: X \rightarrow Y$ be a categorically $(n-1)$-connective morphism of simplicial sets; we wish to prove that $g$ is good. Using Proposition 4.8.5.20, we can choose a categorical equivalence $Y \xrightarrow {e} Z$, where $Z$ is an $\infty $-category. Applying Corollary 4.5.3.24, we can factor $e \circ g$ as a composition $X \xrightarrow {w} X' \xrightarrow {u} Z$ where $w$ is a categorical equivalence and $u$ is an isofibration of $\infty $-categories. Since $g$ is categorically $(n-1)$-connective, $u$ is also categorically $(n-1)$-connective (Remark 4.8.8.4). Using Proposition 4.8.5.20, we can factor $u$ as a composition $X' \xrightarrow {g'} Y' \xrightarrow {u'} Z$, where $u'$ is a trivial Kan fibration and $g'$ is a monomorphism which is bijective on simplices of dimension smaller than $n$. Applying $(a)$ repeatedly, we see that $g$ is good if and only if $g'$ is good. We may therefore replace $g$ by $g'$ and thereby reduce to the case where $g$ is a monomorphism which is bijective on simplices of dimension smaller than $n$. Using $(b)$, we can reduce to the case where $Y$ is obtained from $X$ by adding finitely many nondegenerate simplices. Using $(c)$, we can reduce to the special case where $Y$ is obtained from $X$ by adjoining a single nondegenerate $n'$-simplex for $n' \geq n$. In this case, we can use $(d)$ (together with Proposition 1.1.4.12) to reduce to the case where $g$ is the inclusion map $\operatorname{\partial \Delta }^{n'} \hookrightarrow \Delta ^{n'}$, for some $n' \geq n$. Applying a similar argument (with the roles of $f$ and $g$ reversed), we may assume that $f$ is the inclusion map $\operatorname{\partial \Delta }^{m'} \hookrightarrow \Delta ^{m}$ for some $m' \geq m$. In this case, Lemma 4.8.8.18 guarantees that $\nu _{f,g}$ is categorically $(m'+n'-1)$-connective, and therefore also categorically $(m+n-1)$-connective (see Remark 4.8.5.7). $\square$

Proof. Assume that $U$ is an isofibration (the proof when $f$ is a monomorphism is similar). Using Exercise 3.1.8.11, we can factor $f$ as a composition $A \xrightarrow { \widetilde{f} } \widetilde{B} \xrightarrow {u} B$, where $\widetilde{f}$ is a monomorphism and $u$ is a trivial Kan fibration. We then have a commutative diagram

where the vertical maps are equivalences of $\infty $-categories. Consequently, to show that $U'$ is $(n-m)$-faithful, we may replace $f$ by $\widetilde{f}$ and thereby reduce to the case where $f$ is a monomorphism. In this case, $U'$ is an isofibration of $\infty $-categories (Proposition 4.4.5.1). By virtue of Proposition 4.8.4.5, it will suffice to show that for $k > n-m$, every lifting problem

has a solution. Unwinding the definitions, we can rewrite 4.93 as a lifting problem

The existence of a solution now follows from Proposition 4.8.8.15, since $U$ is an $n$-faithful isofibration and the left vertical map is categorically $n$-connective by virtue of Proposition 4.8.8.19. $\square$

Proof. Apply Proposition 4.8.8.20 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (see Example 4.8.3.14). $\square$