Kerodon

Proof. We will prove that $(1) \Leftrightarrow (2) \Leftrightarrow (3)$; the proof that $(1) \Leftrightarrow (4) \Leftrightarrow (5)$ follows by applying the same argument to the opposite $(\infty ,2)$-category $\operatorname{\mathcal{C}}^{\operatorname{op}}$. The implication $(1) \Rightarrow (2)$ follows from Proposition 5.4.3.8, and the implication $(2) \Rightarrow (3)$ is immediate (see Remark 5.1.3.3). For each integer $n \geq 3$, consider the following weaker version of condition $(1)$:

$(1_ n)$

For every integer $0 < i < n$ and every morphism of simplicial sets $\mu _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$ for which the composition

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1\} ) \hookrightarrow \Lambda ^{n}_{i} \xrightarrow {\mu _0} \operatorname{\mathcal{C}} \]

is equal to $\sigma $, there exists a map $\mu : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ extending $\mu _0$.

Note that $\sigma $ satisfies condition $(1)$ if and only if it satisfies condition $(1_ n)$ for each $n \geq 3$. We will complete the proof by showing that $(3) \Rightarrow (1_ n)$, using a fairly elaborate induction on $n$.

Assume that $\sigma $ is locally $q$-cartesian when viewed as a morphism in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}_{/Z}$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, we can choose a thin $2$-simplex $\sigma '$ satisfying $d^{2}_0(\sigma ') = d^{2}_0(\sigma )$ and $d^{2}_2(\sigma ') = d^{2}_2(\sigma )$, which we represent as a diagram

The implication $(1) \Rightarrow (3)$ shows that $\sigma '$ is also locally $q$-cartesian when viewed as an edge of the simplicial set $\operatorname{\mathcal{C}}_{/Z}$. Let us regard the edge $u$ as a morphism of simplicial sets $\Delta ^{1} \rightarrow \operatorname{\mathcal{C}}$, and let $\operatorname{\mathcal{E}}$ denote the fiber product $\Delta ^{1} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z}$. Since $q$ is an interior fibration, it follows from Remark 5.4.2.4 and Example 5.4.2.2 that the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is an inner fibration. Moreover, we can identify $\sigma $ and $\sigma '$ with $\pi $-cartesian edges of $\operatorname{\mathcal{E}}$ having nondegenerate images under $\pi $. Applying Remark 5.1.3.8, we see that there exists a $2$-simplex of $\operatorname{\mathcal{E}}$ which exhibits $\sigma '$ as a composition of $\sigma $ with an isomorphism in $\operatorname{\mathcal{E}}$. The image of this $2$-simplex under the projection map $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_{/Z}$ can be identified with a $3$-simplex $\rho $ of $\operatorname{\mathcal{C}}$ such that $d^{3}_0(\rho ) = \sigma $, $d^{3}_1(\rho ) = \sigma '$, and $d^{3}_3(\rho ) = s^{1}_0(u)$ is left-degenerate; the restriction of $\rho $ to the $1$-skeleton of $\Delta ^3$ we can represent by the diagram

By construction, the remaining face $\sigma '' = d^{3}_2(\rho )$ is an isomorphism when viewed as a morphism in the $\infty $-category $\operatorname{Hom}^{\mathrm{R}}_{\operatorname{\mathcal{C}}}(X,Z) = \{ X\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z}$, and is therefore locally $q$-cartesian (Example 5.1.3.6). In particular, our inductive hypothesis guarantees that the simplex $\sigma ''$ satisfies condition $(1_{m})$ for $3 \leq m < n$.

Fix a morphism of simplicial sets $\mu _0: \Lambda ^{n}_{i} \rightarrow \operatorname{\mathcal{C}}$ as in condition $(1_ n)$; we wish to show that $\mu _0$ can be extended to an $n$-simplex $\mu $ of $\operatorname{\mathcal{C}}$. Let $\delta ^{i-1}_{n+1}: \Delta ^{n} \hookrightarrow \Delta ^{n+1}$ denote the inclusion of the $(i-1)$st face, given on vertices by the formula

We will construct an $(n+1)$-simplex $\nu : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ which satisfies the following conditions:

• The composite map

  \[ \Lambda ^{n}_{i} \hookrightarrow \Delta ^{n} \xrightarrow { \delta ^{i-1}_{n+1} } \Delta ^{n+1} \xrightarrow {\nu } \operatorname{\mathcal{C}} \]

  is equal to $\mu _0$.

• The composite map

  \[ \Delta ^3 \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 < i+2 \} ) \hookrightarrow \Delta ^{n+1} \xrightarrow {\nu } \operatorname{\mathcal{C}} \]

  is equal to the $3$-simplex $\rho $.

• For every integer $0 \leq j < i-1$, the $2$-simplex

  \[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ j < i-1 < i \} ) \hookrightarrow \Delta ^{n+1} \xrightarrow {\nu } \operatorname{\mathcal{C}} \]

  is right-degenerate (in particular, it is thin).

• For every integer $i+2 < j \leq n+1$, the $2$-simplex

  \[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < j \} ) \hookrightarrow \Delta ^{n+1} \xrightarrow {\nu } \operatorname{\mathcal{C}} \]

  is left-degenerate (in particular, it is thin).

Assuming that this construction is possible, we complete the proof by observing that $\mu = \nu \circ \delta ^{i-1}_{n+1}$ provides the desired extension of $\mu _0$ (by virtue of assumption $(a)$).

The construction of the $(n+1)$-simplex $\nu $ will take place in several steps. We define simplicial subsets

and maps $\nu _ j: K_{j} \rightarrow \operatorname{\mathcal{C}}$ as follows:

• Let $K_0 \subseteq \Delta ^{n+1}$ be the image of the horn $\Lambda ^ n_{i}$ under $\delta ^{i-1}_{n+1}$, so that $\delta ^{i-1}_{n+1}$ induces an isomorphism $\Lambda ^{n}_{i} \xrightarrow {\sim } K_0$. It follows that there is a unique morphism of simplicial sets $\nu _0: K_0 \rightarrow \operatorname{\mathcal{C}}$ satisfying $\mu _0 = \nu _{0} \circ \delta ^{i-1}_{n+1}|_{ \Lambda ^{n}_{i} }$. By construction, the map $\nu _0$ satisfies condition $(a)$.

• Let $K_1 \subseteq \Delta ^{n+1}$ be the union of $K_0$ with the $3$-simplex $\operatorname{N}_{\bullet }( \{ i-1 < i < i+1 < i+2 \} )$. It follows from the identity $d^{3}_0(\rho )= \sigma $ that $\nu _0$ extends uniquely to a map $\nu _1: K_1 \rightarrow \operatorname{\mathcal{C}}$ satisfying condition $(b)$.

• Let $K_2$ be the simplicial subset of $\Delta ^{n+1}$ obtained by removing those nondegenerate simplices which contain all of the vertices $\{ 0 < 1 < \cdots < i-2 < i+2 < i+3 < \cdots < n+1 \} $ and at least one of the vertices $\{ i-1, i \} $. We will prove below that $\nu _1$ can be extended to a map $\nu _2: K_2 \rightarrow \operatorname{\mathcal{C}}$ which satisfies conditions $(c)$ and $(d)$.

• Let $\delta ^{i}_{n+1}: \Delta ^{n} \hookrightarrow \Delta ^{n+1}$ denote the inclusion of the $i$th face, given on vertices by the formula

  \[ \delta ^{i}_{n+1}(j) = \begin{cases} j & \textnormal{ if } j < i \\ j+1 & \textnormal{ if } j \geq i. \end{cases} \]

  Let $K_3$ be the union of $K_2$ with the image of $\delta ^{i}_{n+1}$. Note that $\delta ^{i}_{n+1}$ determines a pushout diagram of simplicial sets

  \[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r] \ar [d] & K_2 \ar [d] \\ \Delta ^{n} \ar [r]^-{\delta ^{i}_{n+1}} & K_3. } \]

  Let $\alpha _0$ denote the composite map $\Lambda ^{n}_{i} \xrightarrow { \delta ^{i}_{n+1} } K_2 \xrightarrow { \nu _2 } \operatorname{\mathcal{C}}$. Since $\nu _1$ satisfies condition $(b)$, $\alpha _0$ carries $\operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} )$ to the thin $2$-simplex $\sigma '$ of $\operatorname{\mathcal{C}}$, and can therefore be extended to an $n$-simplex $\alpha $ of $\operatorname{\mathcal{C}}$. It follows that $\nu _2$ extends uniquely to a morphism of simplicial sets $\nu _3: K_3 \rightarrow \operatorname{\mathcal{C}}$ satisfying $\nu _3 \circ \delta ^{i}_{n+1} = \alpha $.

• Let $K_{4}$ denote the horn $\Lambda ^{n+1}_{i-1} \subsetneq \Delta ^{n}$. Note that $K_4$ can be written as the union of $K_3$ with the image of the face inclusion $\delta ^{i+1}_{n+1}: \Delta ^{n} \hookrightarrow \Delta ^{n+1}$, given on vertices by the formula

  \[ \delta ^{i+1}_{n+1}(j) = \begin{cases} j & \textnormal{ if } j \leq i \\ j+1 & \textnormal{ if } j > i. \end{cases} \]

  Moreover, we have a pushout diagram

  \[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i-1} \ar [r] \ar [d] & K_3 \ar [d] \\ \Delta ^{n} \ar [r]^-{\delta ^{i+1}_{n+1}} & K_4. } \]

  Let $\beta _0$ denote the composite map

  \[ \Lambda ^{n}_{i-1} \xrightarrow { \delta ^{i+1}_{n+1} } K_3 \xrightarrow { \nu _3 } \operatorname{\mathcal{C}}. \]

  If $i > 1$, then condition $(c)$ guarantees that the restriction $\beta _0|_{\operatorname{N}_{\bullet }( \{ i-2 < i-1 < i \} )}$ is a right-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. If $i=1$, then condition $(d)$ guarantees that the restriction $\beta _0|_{\operatorname{N}_{\bullet }( \{ 0 < 1 < n \} )}$ is a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. In either case, our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category guarantees that $\beta _0$ can be extended to an $n$-simplex $\beta $ of $\operatorname{\mathcal{C}}$, so that $\nu _3$ can be extended uniquely to a map $\nu _4: K_4 \rightarrow \operatorname{\mathcal{C}}$ satisfying $\nu _{4} \circ \delta ^{i+1}_{n+1} = \beta $.

• If $i > 1$, then condition $(c)$ guarantees that the map $\nu _{4}: \Lambda ^{n+1}_{i-1} \rightarrow \operatorname{\mathcal{C}}$ carries $\operatorname{N}_{\bullet }( \{ i-2 < i-1 < i \} )$ to a right-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. If $i=1$, then condition $(d)$ guarantees that $\nu _{4}$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 < n+1 \} )$ to a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. In either case, our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category guarantees that we can extend $\nu _{4}$ to an $(n+1)$-simplex $\nu : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$, thereby completing the proof of Theorem 5.4.4.1.

It remains to show that $\nu _1$ admits an extension $\nu _2: K_2 \rightarrow \operatorname{\mathcal{C}}$ which satisfies conditions $(c)$ and $(d)$. Let us say that a simplex $\tau : \Delta ^{m} \rightarrow K_2$ is free if it is nondegenerate, not contained in $K_1$, and there exists an integer $0 \leq j \leq m$ satisfying $\tau (j) = i$. Note that in this case, we automatically have $j > 0$ and $\tau (j-1) = i-1$ (otherwise, $\tau $ would be contained in $K_1$). Moreover, if $\tau $ is any nondegenerate $m$-simplex of $K_2$ which is not contained in $K_1$, then $\tau $ is either free or can be realized uniquely as a face of a free $(m+1)$-simplex $\tau ': \Delta ^{m+1} \rightarrow K_2$ (obtained by adjoining $i$ to the image of $\tau $).

Let $\{ \tau _1, \tau _2, \cdots , \tau _ t \} $ be an enumeration of the collection of all free simplices of $K_{2}$, chosen so $\dim ( \tau _1 ) \leq \dim ( \tau _2) \leq \cdots \leq \dim ( \tau _ t )$. For $0 \leq s \leq t$, let $K_{2}(s)$ denote the union of $K_{1}$ with the images of the maps $\{ \tau _1, \tau _2, \cdots , \tau _{s} \} $, so that we have inclusions of simplicial sets

We will complete the proof by inductively constructing a compatible sequence of maps $\nu _{2}(s): K_2(s) \rightarrow \operatorname{\mathcal{C}}$ satisfying $\nu _{2}(0) = \nu _1$ together with the following translation of conditions $(c)$ and $(d)$:

$(\ast _{s})$

If the simplex $\tau _{s}$ has dimension $2$, then the $2$-simplex $\nu _{2} \circ \tau _{s}$ of $\operatorname{\mathcal{C}}$ is left-degenerate if $\tau _ s(1) =i$ and right-degenerate if $\tau _{s}(2) = i$.

Assume that $s > 0$ and that the map $\nu _{2}(s-1)$ has already been constructed. Set $\tau = \tau _{s}: \Delta ^{m} \rightarrow K_2$, so that there is a unique integer $1 \leq j \leq m$ satisfying $\tau (j) = i$. Note that for $0 \leq k \leq m$ with $k \neq j$, the face $d^{m}_ k(\tau )$ is either free or belongs to $K_1$; in either case, it belongs to $K_2(s-1)$. Moreover, the face $d^{m}_ j(\tau )$ is neither free, nor contained in $K_1$, nor contained as a face of any other free $m$-simplex of $K_2$. It follows that $\tau $ determines a pushout diagram of simplicial sets

Let $\xi _0 : \Lambda ^{m}_{j} \rightarrow \operatorname{\mathcal{C}}$ denote the composite map $\Lambda ^{m}_{j} \xrightarrow {\tau } K_{2}(s-1) \xrightarrow { \nu _{2}(s-1)} \operatorname{\mathcal{C}}$; we wish to show that $\xi _0$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$. If $m = 2$, then there is a unique such extension which satisfies condition $(\ast _ s)$ (since, by construction, the morphism $\nu _1$ carries $\operatorname{N}_{\bullet }( \{ i-1 < i \} )$ to the degenerate edge $\operatorname{id}_{X}$ of $\operatorname{\mathcal{C}}$). We may therefore assume that $m \geq 3$. We consider several cases:

• If $j=m$, then it follows from assumption $(\ast _{s'} )$ for $s' < s$ that $\xi _{0}$ carries $\operatorname{N}_{\bullet }( \{ 0 < m-1 < m \} )$ to a right-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$, so the desired extension exists by virtue of our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category.

• If $j < m$ and $\tau (j+1) = i+1$, then it follows from $(b)$ that $\xi _0$ carries $\operatorname{N}_{\bullet }( \{ j-1 < j < j+1 \} )$ to the left-degenerate $2$-simplex $d^{3}_3(\rho )$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, this $2$-simplex is thin so that $\xi _0$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.

• If $j < m$ and $\tau (j+1) > i+2$, then it follows from assumption $(\ast _{s'})$ for $s' < s$ that $\xi _0$ carries $\operatorname{N}_{\bullet }( \{ j-1 < j < j+1 \} )$ to a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, this $2$-simplex is thin so that $\xi _0$ can be extended to an $m$-simplex of $\operatorname{\mathcal{C}}$.

• If $j < m$ and $\tau (j+1) = i+2$, then it follows from $(b)$ that $\xi _{0}$ carries $\operatorname{N}_{\bullet }( \{ j-1 < j < j+1 \} )$ to the $2$-simplex $\sigma ''$ of $\operatorname{\mathcal{C}}$. In this case, our assumption that $\tau $ belongs to $K_2$ guarantees that $m < n$, so the existence of the desired extension follows the fact that $\sigma ''$ satisfies condition $(1_{m})$ (by virtue of our inductive hypothesis).

Proof. The implication $(1) \Rightarrow (2)$ follows from Proposition 5.4.3.8, and the implication $(2) \Rightarrow (3)$ is immediate (see Remark 5.1.3.3). We will show that $(3) \Rightarrow (1)$. Fix a vertex $z \in K$; we wish to show that the composite map

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$. Set $Z = f(z) \in \operatorname{\mathcal{C}}$, so that $q$ factors as a composition

By virtue of Theorem 5.4.4.1, it will suffice to show that the $q'( u )$ is a locally $q''$-cartesian morphism of the $(\infty ,2)$-category $\operatorname{\mathcal{C}}_{/Z}$.

Set $\overline{u} = q(u)$, which we regard as a morphism $\overline{q}: \overline{X} \rightarrow \overline{Y}$ in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}$. By virtue of Proposition 5.4.3.9, we can lift $\overline{u}$ to a morphism $u': X' \rightarrow Y$ in $\operatorname{\mathcal{C}}_{/f}$ which satisfies condition $(1)$ (and therefore also satisfies $(3)$). Regard $\overline{u}$ as a $1$-simplex of $\operatorname{\mathcal{C}}$ and let $\operatorname{\mathcal{E}}$ denote the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$. Since $q$ is an interior fibration (Proposition 5.4.3.1), the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is also an interior fibration (Remark 5.4.2.4) and therefore an inner fibration (Example 5.4.2.2). Let us abuse notation by identifying $u$ and $u'$ with morphisms in the $\infty $-category $\operatorname{\mathcal{E}}$ lying over the unique nondegenerate edge of $\Delta ^1$. Assumption $(3)$ then guarantees that $u$ and $u'$ are $\pi $-cartesian. Invoking Remark 5.1.3.8, we deduce that there exists a $2$-simplex $\rho : \Delta ^2 \rightarrow \operatorname{\mathcal{E}}$, which we display as a diagram

where $v$ is an isomorphism in the $\infty $-category $\{ 0\} \times _{\Delta ^1} \operatorname{\mathcal{E}}\simeq \{ \overline{X} \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/f}$. It follows that $q'(v)$ is an isomorphism in the $\infty $-category $\{ \overline{X} \} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{/Z}$. Since $u'$ satisfies condition $(1)$, Theorem 5.4.4.1 guarantees that $q'(u')$ is locally $q''$-cartesian. Invoking Remark 5.1.3.8 again, we deduce that $q'(u)$ is locally $q''$-cartesian, as desired. $\square$