4.6.6 Digression: Diagrams in Slice $\infty $-Categories
For some applications, it will be useful to combine the slice and coslice constructions introduced in ยง4.3.
Notation 4.6.6.1. Let $K_{-}$, $K_{+}$, and $\operatorname{\mathcal{C}}$ be simplicial sets, and suppose we are given a morphism $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$. Set $f_{-} = f_{\pm } |_{ K_{-} }$ and $f_{+} = f_{\pm } |_{ K_{+} }$, and let $\pi : \operatorname{\mathcal{C}}_{ / f_{+} } \rightarrow \operatorname{\mathcal{C}}$ denote the projection map. Then $f_{\pm }$ determines a morphism of simplicial sets $\widetilde{f}_{-}: K_{-} \rightarrow \operatorname{\mathcal{C}}_{ / f_{+} }$ for which the diagram
\[ \xymatrix { & \operatorname{\mathcal{C}}_{ / f_{+} } \ar [dr]_{ \pi } & \\ K_{-} \ar [ur]_{ \widetilde{f}_{-} } \ar [rr]^{ f_{-} } & & \operatorname{\mathcal{C}}} \]
is commutative. In this situation, we let $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }$ denote the coslice simplicial set $( \operatorname{\mathcal{C}}_{ / f_{+} } )_{ \widetilde{f}_{-} / }$.
It follows that the simplicial set $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } = ( \operatorname{\mathcal{C}}_{ / f_{+} } )_{ \widetilde{f}_{-} / }$ can also be identified with the slice simplicial set $(\operatorname{\mathcal{C}}_{ f_{-} / })_{ /\widetilde{f}_{+} }$.
Warning 4.6.6.3. In the situation of Notation 4.6.6.1, the simplicial set $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }$ depends on the morphism $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$, and not only on the morphisms $f_{-} = f_{\pm }|_{ K_{-} }$ and $f_{+} = f_{\pm } |_{ K_{+} }$ indicated in the notation.
In the situation of Notation 4.6.6.1, suppose that the simplicial set $\operatorname{\mathcal{C}}$ is an $\infty $-category. Applying Proposition 4.3.6.1 twice, we deduce that the simplicial set $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }$ is also an $\infty $-category. We now exploit the relationship between slice constructions and oriented fiber products (Theorem 4.6.4.17) to give an alternative description of $\operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }$.
Construction 4.6.6.4. Let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and set $f_{-} = f_{\pm }|_{K_{-}}$ and $f_{+} = f_{\pm }|_{ K_{+} }$. Let $K$ be another simplicial set, set $M = \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} } )$. Let $\operatorname{ev}: M \times K \rightarrow \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} }$ be the evaluation map, and let $\pi _{-}: M \times K_{-} \rightarrow K_{-}$ and $\pi _{+}: M \times K_{+} \rightarrow K_{+}$ be given by projection onto the second factor. Then the composition
\begin{eqnarray*} M \times (K_{-} \star K \star K_{+} ) & \rightarrow & (M \times K_{-} ) \star (M \times K) \star ( M \times K_{+} ) \\ & \xrightarrow {\pi _{-} \star \operatorname{ev}\star \pi _{+}} & K_{-} \star \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } \star K_{+} \\ & \rightarrow & \operatorname{\mathcal{C}}\end{eqnarray*}
classifies a morphism of simplicial sets $M \rightarrow \operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}})$, whose composition with the restriction map $\operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}})$ is the constant map taking the value $f_{\pm }$. We therefore obtain a comparison map
\[ \theta : \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} } ) \rightarrow \operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}}) } \{ f_{\pm } \} . \]
Theorem 4.6.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then, for any simplicial set $K$, the comparison map
\[ \theta : \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, /f_{+} } ) \rightarrow \operatorname{Fun}( K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}}) } \{ f_{\pm } \} \]
of Construction 4.6.6.4 is an equivalence of $\infty $-categories.
Example 4.6.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Applying Theorem 4.6.6.5 in the special case $K = \Delta ^0$, we obtain an equivalence of $\infty $-categories
\[ \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } \rightarrow \operatorname{Fun}( K_{-} \star \Delta ^0 \star K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}}) } \{ f_{\pm } \} . \]
We begin by proving Theorem 4.6.6.5 in the special case where $K_{-}$ is empty.
Lemma 4.6.6.7. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{+}: K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then, for every simplicial set $K$, the comparison map
\[ \theta : \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ /f_{+} } ) \rightarrow \operatorname{Fun}( K \star K_{+}, \operatorname{\mathcal{C}}) \times _{ K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} \]
of Construction 4.6.6.4 is an equivalence of $\infty $-categories.
Proof.
Let $c: K \diamond K_{+} \rightarrow K \star K_{+}$ be as in Notation 4.5.8.3. We then have a commutative diagram
\[ \xymatrix { \operatorname{Fun}( K \star K_{+}, \operatorname{\mathcal{C}}) \ar [r]^{ \circ c} \ar [d] & \operatorname{Fun}( K \diamond K_{+}, \operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}( K_{+}, \operatorname{\mathcal{C}}) \ar@ {=}[r] & \operatorname{Fun}( K_{+}, \operatorname{\mathcal{C}}) } \]
where the vertical maps are isofibrations (Corollary 4.4.5.3). Since $c$ is a categorical equivalence of simplicial sets (Theorem 4.5.8.8), the upper horizontal map is an equivalence of $\infty $-categories. Applying Corollary 4.5.2.32, we conclude that composition with $c$ induces an equivalence of $\infty $-categories
\begin{eqnarray*} \operatorname{Fun}( K \star K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} & \xrightarrow {\theta '} & \operatorname{Fun}( K \diamond K_{+}, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}(K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} \\ & \simeq & \operatorname{Fun}( K, \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} ). \end{eqnarray*}
It will therefore suffice to show that $\theta ' \circ \theta $ is an equivalence of $\infty $-categories. We conclude by observing that $\theta ' \circ \theta $ is given by postcomposition with the slice diagonal $\operatorname{\mathcal{C}}_{ /f_{+} } \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} $, which is an equivalence of $\infty $-categories by virtue of Theorem 4.6.4.17.
$\square$
Example 4.6.6.8. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{+}: K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Applying Lemma 4.6.6.7 in the special case $K = \Delta ^0$, we obtain an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{ / f_{+} } \rightarrow \operatorname{Fun}( K_{+}^{\triangleleft }, \operatorname{\mathcal{C}}) \times _{ \operatorname{Fun}( K_{+}, \operatorname{\mathcal{C}}) } \{ f_{+} \} $.
Proof of Theorem 4.6.6.5.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Set $f_{-} = f_{\pm } |_{ K_{-} }$ and $f_{+} = f_{\pm } |_{ K_{+} }$, so that $f_{\pm }$ can be identified with a morphism $\widetilde{f}_{-}: K_{-} \rightarrow \operatorname{\mathcal{C}}_{ / f_{+} }$. We then have a commutative diagram of simplicial sets
\[ \xymatrix { \operatorname{Fun}(K, \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} } ) \ar [r] \ar [d] & \operatorname{Fun}(K_{-} \star K, \operatorname{\mathcal{C}}_{ / f_{+} } ) \ar [r] \ar [d] & \operatorname{Fun}(K_{-} \star K \star K_{+}, \operatorname{\mathcal{C}}) \ar [d] \\ \{ \widetilde{f}_{-} \} \ar [r] & \operatorname{Fun}( K_{-}, \operatorname{\mathcal{C}}_{ / f_{+} } ) \ar [r] \ar [d] & \operatorname{Fun}( K_{-} \star K_{+} , \operatorname{\mathcal{C}}) \ar [d] \\ & \{ f_{+} \} \ar [r] & \operatorname{Fun}( K_{+}, \operatorname{\mathcal{C}}), } \]
where the vertical maps are isofibrations (Corollary 4.4.5.3). It follows from Lemma 4.6.6.7 (and Proposition 4.5.2.26) that the lower right square and the outer right rectangle are categorical pullback squares, so the upper right corner is also a categorical pullback square (Proposition 4.5.2.18). Similarly, the dual of Lemma 4.6.6.7 guarantees that the upper left corner is a categorical pullback square. Applying Proposition 4.5.2.18, we conclude that the outer rectangle on the top of the diagram is a categorical pullback square, which is a restatement of Theorem 4.6.6.5 (Proposition 4.5.2.26).
$\square$
Corollary 4.6.6.9. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f_{\pm }: K_{-} \star K_{+} \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then, for any inclusion of simplicial sets $A \hookrightarrow B$, the diagram of $\infty $-categories
4.59
\begin{equation} \begin{gathered}\label{equation:double-slice-consequence} \xymatrix { \operatorname{Fun}(B, \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }) \ar [r] \ar [d] & \operatorname{Fun}( K_{-} \star B \star K_{+},\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}(A, \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+}} ) \ar [r] & \operatorname{Fun}( K_{-} \star A \star K_{+}, \operatorname{\mathcal{C}}) } \end{gathered} \end{equation}
is a categorical pullback square.
Proof.
We can identify (4.59) with the upper half of a commutative diagram
\[ \xymatrix { \operatorname{Fun}(B, \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+} }) \ar [r] \ar [d] & \operatorname{Fun}( K_{-} \star B \star K_{+},\operatorname{\mathcal{C}}) \ar [d] \\ \operatorname{Fun}(A, \operatorname{\mathcal{C}}_{ f_{-} / \, / f_{+}} ) \ar [r] \ar [d] & \operatorname{Fun}( K_{-} \star A \star K_{+}, \operatorname{\mathcal{C}}) \ar [d] \\ \{ f_{\pm } \} \ar [r] & \operatorname{Fun}( K_{-} \star K_{+}, \operatorname{\mathcal{C}}). } \]
By virtue of Proposition 4.5.2.18, it will suffice to show that the lower half and outer rectangle of the diagram are categorical pullback squares, which follows from Theorem 4.6.6.5.
$\square$
We conclude this section by recording a thematically related result, which characterizes slices of functor $\infty $-categories (rather than functors into a slice $\infty $-category).
Variant 4.6.6.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram, which we identify with an object $F$ of the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$. Then the functors
\[ \operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \quad \quad \operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{F/} \times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{\mathcal{C}} \]
of Exercise 4.6.4.16 are equivalences of $\infty $-categories.
Proof.
We will show that the slice diagonal $\delta _{/f}$ induces an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F}$; the analogous assertion for coslice $\infty $-categories follows by a similar argument. By virtue of Theorem 4.6.4.17, it will suffice to show that the inclusion map
\[ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \hookrightarrow \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \]
is an equivalence of $\infty $-categories. By construction, this map fits into a commutative diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}\times _{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \ar [r] \ar [d]^{\iota } & \operatorname{Fun}(K, \operatorname{\mathcal{C}})_{/F} \ar [d]^{U} \\ \operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \ar [r] \ar [d] & \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \operatorname{\vec{\times }}_{ \operatorname{Fun}(K, \operatorname{\mathcal{C}}) } \{ F \} \ar [d]^{V} \\ \operatorname{\mathcal{C}}\ar [r] & \operatorname{Fun}(K, \operatorname{\mathcal{C}}), } \]
where the upper square and lower square are both pullback diagrams. Note that the morphisms $V$ and $V \circ U$ are both right fibrations (Propositions 4.6.4.11 and 4.3.6.1), and therefore isofibrations (Example 4.4.1.11). Using Propositions 4.5.2.26 and 4.5.2.18, we see that the upper square is a categorical pullback. Theorem 4.6.4.17 guarantees that $U$ is an equivalence of $\infty $-categories, so that $\iota $ is an equivalence of $\infty $-categories by virtue of Proposition 4.5.2.21.
$\square$