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5.4.3 Slices of $(\infty ,2)$-Categories

The slice and coslice constructions of §4.3 provide many examples of interior fibrations of $(\infty ,2)$-categories.

Proposition 5.4.3.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the projection maps

\[ \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}} \]

are interior fibrations.

Warning 5.4.3.2. In the situation of Proposition 5.4.3.1, the projection maps

\[ \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}\quad \quad \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}} \]

are generally not inner fibrations of simplicial sets.

Remark 5.4.3.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then axioms $(3)$ and $(4)$ of Definition 5.4.1.3 can be stated as follows:

  • Let $X$ be any vertex of $\operatorname{\mathcal{C}}$ and let $q: \operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Then every degenerate edge of $\operatorname{\mathcal{C}}_{/X}$ is $q$-cocartesian.

  • Let $X$ be any vertex of $\operatorname{\mathcal{C}}$ and let $q': \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Then every degenerate edge of $\operatorname{\mathcal{C}}_{X/}$ is $q'$-cartesian.

Note that $(3')$ and $(4')$ appear as special cases of the conclusion of Proposition 5.4.3.1.

Corollary 5.4.3.4. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. Then the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are $(\infty ,2)$-categories.

Proof. Combine Proposition 5.4.3.1 with Proposition 5.4.2.8. $\square$

Corollary 5.4.3.5. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category. For every pair of objects $X$ and $Y$, the pinched morphism spaces $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ and $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ of Construction 4.6.6.1 are $\infty $-categories.

Proof. By definition, the left-pinched morphism space $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}(X,Y)$ is the fiber over $Y$ of the projection map $\pi : \operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$. Since $\pi $ is an interior fibration (Proposition 5.4.3.1), each of its fibers is an $\infty $-category (Remark 5.4.2.5). A similar argument shows that $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$ is an $\infty $-category. $\square$

Warning 5.4.3.6. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category containing objects $X$ and $Y$. Then the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$ of Construction 4.6.1.1 is generally not an $\infty $-category (see Example 4.6.1.12).

Remark 5.4.3.7. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category containing $X$ and $Y$. We will see later that the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}( X, Y)$ is naturally equivalent to the opposite of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}( X, Y)$ (Proposition ). When $\operatorname{\mathcal{C}}$ is the Duskin nerve of a $2$-category, we can do better: the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{L}}( X, Y)$ is isomorphic to the opposite of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}^{\mathrm{R}}(X,Y)$; see Example 4.6.6.12).

We will deduce Proposition 5.4.3.1 from the following more precise result:

Proposition 5.4.3.8. Let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets and let $f_0: K_0 \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then every lifting problem

5.14
\begin{equation} \label{equation:slice-interior-fibration-more-precise} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{C}}_{/f_0} } \end{gathered} \end{equation}

admits a solution provided that $m \geq 2$ and one of the following additional conditions is satisfied:

$(a)$

The simplicial set $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, $i=0$, and the composition

\[ \Delta ^{1} \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) \subseteq \Lambda ^{m}_{i} \xrightarrow { \sigma _0} \operatorname{\mathcal{C}}_{/f} \]

is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

$(b)$

The integer $i$ satisfies $0 < i < m$ and the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{m} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(c)$

The integer $i$ is equal to $m$ and, for every vertex $x \in K$, the composite map

\[ \Delta ^2 \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \star \{ x\} \hookrightarrow \Lambda ^{m}_{i} \star K \xrightarrow {\sigma } \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. Unwinding the definitions, we can identify the diagram (5.14) with a morphism of simplicial sets

\[ \overline{f}: ( \Lambda ^{m}_{i} \star K ) \coprod _{ (\Lambda ^{m}_{i} \star K_0 ) } ( \Delta ^{m} \star K_0 ) \rightarrow \operatorname{\mathcal{C}}, \]

and we wish to show that $\overline{f}$ can be extended to a morphism $\Delta ^{m} \star K \rightarrow \operatorname{\mathcal{C}}$. Let $P$ be the collection of all pairs $(L,g)$, where $L$ is a simplicial subset of $K$ containing $K_0$ and $g: \Delta ^{m} \star L \rightarrow \operatorname{\mathcal{C}}$ is a morphism satisfying

\[ g|_{ \Delta ^{m} \star K_0} = \overline{f} |_{ \Delta ^{m} \star K_0} \quad \quad g|_{ \Lambda ^{m}_{i} \star L} = \overline{f} |_{ \Lambda ^{m}_{i} \star L}. \]

We regard $P$ as a partially ordered set, with $(L, g) \leq (L', g')$ if $L$ is contained in $L'$ and $g = g'|_{\Delta ^ m \star L}$. The partially ordered set $P$ satisfies the hypotheses of Zorn's lemma and therefore admits a maximal element $(L_{\mathrm{max}}, g_{\mathrm{max}} )$. We will complete the proof by showing that $L_{\mathrm{max}} = K$ (so that $g_{\mathrm{max}}$ is the desired extension of $\overline{f}$). Suppose otherwise. Then there is some nondegenerate simplex $\rho : \Delta ^{n} \rightarrow K$ which is not contained in $L_{\mathrm{max}}$. Choosing $\rho $ so that $n$ is as small as possible, we may assume without loss of generality that $\rho $ carries the boundary $\operatorname{\partial \Delta }^ n$ into $L_{\mathrm{max}}$. Let $L' \subseteq K$ be the simplicial subset given by the union of $L_{\mathrm{max}}$ together with the image of $\rho $, so that $\rho $ determines a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & L_{\mathrm{max}} \ar [d] \\ \Delta ^{n} \ar [r] & L'. } \]

We will show that $g_{\mathrm{max}}$ can be extended to a morphism of simplicial sets $g': \Delta ^{m} \star L' \rightarrow \operatorname{\mathcal{C}}$ satisfying $g'|_{ \Lambda ^{m}_{i} \star L'} = \overline{f}|_{ \Lambda ^{m}_{i} \star L'}$; thereby contradicting the maximality of $(L_{\mathrm{max}}, g_{\mathrm{max}} )$ and completing the proof of Proposition 5.4.3.8. Note that the composite maps

\[ \Lambda ^{m}_{i} \star \Delta ^{n} \xrightarrow { \operatorname{id}\star \rho } \Lambda ^{m}_{i} \star K \xrightarrow { \overline{f} } \operatorname{\mathcal{C}} \]

\[ \Delta ^{m} \star \operatorname{\partial \Delta }^ n \xrightarrow { \operatorname{id}\star \rho } \Delta ^{m} \star L_{\mathrm{max}} \xrightarrow { g_{\mathrm{max}}} \operatorname{\mathcal{C}} \]

can be amalgamated to a morphism of simplicial sets

\[ \tau _0: (\Lambda ^{m}_{i} \star \Delta ^{n}) \coprod _{ (\Lambda ^{m}_{i} \star \operatorname{\partial \Delta }^ n)} (\Delta ^ m \star \operatorname{\partial \Delta }^ n) \rightarrow \operatorname{\mathcal{C}}, \]

whose source can be identified with the horn $\Lambda ^{m+1+n}_{i} \subseteq \Delta ^{m+1+n}$ (Lemma 4.3.6.11). We wish to show that $\tau _0$ can be extended to a map

\[ \tau : \Delta ^{m} \star \Delta ^{n} \simeq \Delta ^{m+1+n} \rightarrow \operatorname{\mathcal{C}}. \]

If $0 < i \leq m$, the desired extension exists because the composite map

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

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(b)$ when $i < m$ or $(c)$ in the case $i = m$). If $i=0$, then the desired extension exists because assumption $(a)$ guarantees that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category and the $2$-simplex

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ 0 < 1 < m+1+n\} ) \subseteq \Lambda ^{m+1+n}_{i} \xrightarrow {\tau _0} \operatorname{\mathcal{C}} \]

is left-degenerate. $\square$

Proof of Proposition 5.4.3.1. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets. We will show that the projection map $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration; the analogous assertion for the coslice simplicial set $\operatorname{\mathcal{C}}_{f/}$ follows by a similar argument. Let $m \geq 2$ and suppose that we are given a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{m}_{i} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ \Delta ^{m} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur] & \operatorname{\mathcal{C}}. } \]

We wish to show that a solution exists under any of the following additional assumptions:

$(a)$

The integer $i$ is equal to zero and the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < 1 \} ) }$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

$(b)$

The integer $i$ satisfies $0 < i < m$ and the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) \subseteq \Delta ^{m} \xrightarrow { \overline{\sigma } } \operatorname{\mathcal{C}}_{/f_0} \rightarrow \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

$(c)$

The integer $i$ is equal to $m$ and the restriction $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ m-1 < m \} ) }$ is a degenerate edge of $\operatorname{\mathcal{C}}_{/f}$.

In cases $(a)$ and $(b)$, this follows immediately from Proposition 5.4.3.8. In case $(c)$, we observe that for every vertex $x \in K$, the composite map

\[ \Delta ^{2} \simeq \operatorname{N}_{\bullet }( \{ m-1 < m \} ) \star \{ x\} \hookrightarrow \Lambda ^{m}_{i} \star K \xrightarrow { \sigma _0} \operatorname{\mathcal{C}} \]

is a left-degenerate $2$-simplex of $\operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category, this degenerate $2$-simplex is thin, so that existence of the desired extension again follows from Proposition 5.4.3.8. $\square$

In the situation of Proposition 5.4.3.1, the interior fibration $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ behaves like a cartesian fibration (with the caveat that it need not be an inner fibration).

Proposition 5.4.3.9. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ be the projection map. Let $Y$ be an object of the $(\infty ,2)$-category $\operatorname{\mathcal{C}}_{/f}$, and let $\overline{u}: \overline{X} \rightarrow q(Y)$ be a morphism in the $(\infty ,2)$-category $\operatorname{\mathcal{C}}$. Then $\overline{u}$ can be lifted to a morphism $u: X \rightarrow Y$ of $\operatorname{\mathcal{C}}_{/f}$ with the following property:

$(\ast )$

For every vertex $z \in K$, the image of $u$ in $\operatorname{\mathcal{C}}_{/f(z)}$ is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Remark 5.4.3.10. In the situation of Proposition 5.4.3.9, condition $(\ast )$ guarantees that that $u$ is a $q$-cartesian morphism of $\operatorname{\mathcal{C}}_{/f}$ (this follows immediately from Proposition 5.4.3.8). In §5.4.4, we will prove the converse: every $q$-cartesian morphism of $\operatorname{\mathcal{C}}_{/f}$ satisfies condition $(\ast )$ (Corollary 5.4.4.2).

Proposition 5.4.3.9 is a special case of the following more general assertion:

Proposition 5.4.3.11. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0: K_0 \rightarrow \operatorname{\mathcal{C}}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Let $q: \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ denote the projection map, and suppose we are given a lifting problem

5.15
\begin{equation} \label{equation:slice-interior-fibration-cartesian} \begin{gathered} \xymatrix@R =50pt@C=50pt{ \{ 1\} \ar [r]^-{\sigma _0} \ar [d] & \operatorname{\mathcal{C}}_{/f} \ar [d]^{q} \\ \Delta ^{1} \ar [r]^-{\overline{\sigma }} \ar@ {-->}[ur]^{\sigma } & \operatorname{\mathcal{C}}_{/f_0} } \end{gathered} \end{equation}

with the following property:

$(\ast _0)$

For every vertex $x \in K_0$, the composition

\[ \Delta ^2 \simeq \Delta ^1 \star \{ x\} \hookrightarrow \Delta ^1 \star K_0 \xrightarrow { \overline{\sigma }} \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Then there exists an edge $\sigma : \Delta ^1 \rightarrow \operatorname{\mathcal{C}}_{/f}$ which solves the lifting problem problem (5.15) and which satisfies the following stronger version of $(\ast )$:

$(\ast )$

For every vertex $x \in K$, the composition

\[ \Delta ^2 \simeq \Delta ^1 \star \{ x\} \hookrightarrow \Delta ^1 \star K \xrightarrow { \sigma } \operatorname{\mathcal{C}} \]

is a thin $2$-simplex of $\operatorname{\mathcal{C}}$.

Proof. Arguing as in the proof of Proposition 5.4.3.8, we can reduce to the case where $K = \Delta ^ n$ is a standard simplex and $K_0 = \operatorname{\partial \Delta }^ n$ is its boundary. In this case, the lifting problem (5.15) determines a morphism of simplicial sets

\[ \tau _0: ( \{ 1\} \star \Delta ^ n ) \coprod _{( \{ 1\} \star \operatorname{\partial \Delta }^ n) } (\Delta ^1 \star \operatorname{\partial \Delta }^ n) \rightarrow \operatorname{\mathcal{C}}, \]

whose source can be identified with the horn $\Lambda ^{n+2}_{1} \subseteq \Delta ^{n+2}$ (Lemma 4.3.6.11), and we wish to extend $\tau $ to an $(n+2)$-simplex of $\operatorname{\mathcal{C}}$. If $n > 0$, then the desired extension exists because $\tau _0$ carries $\operatorname{N}_{\bullet }( \{ 0 < 1 < 2 \} )$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$ (by virtue of assumption $(\ast _0)$). If $n = 0$, then our assumption that $\operatorname{\mathcal{C}}$ is an $(\infty ,2)$-category allows us to extend $\tau _0$ to a thin $2$-simplex of $\operatorname{\mathcal{C}}$. $\square$