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4.2.6 Slicing $\infty $-Categories

We now specialize Construction 4.2.5.1 to the setting of $\infty $-categories.

Proposition 4.2.6.1. Let $K$ be a simplicial set, let $\operatorname{\mathcal{C}}$ be an $\infty $-category, and let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a diagram. Then the projection map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets, and the projection map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}$ is a right fibration of simplicial sets. In particular, the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are $\infty $-categories (see Remark 4.1.1.4).

Remark 4.2.6.2. In the special case where $\operatorname{\mathcal{C}}$ is (the nerve of) an ordinary category, Proposition 4.2.6.1 follows from Corollary 4.2.5.17; in fact, both of the simplicial sets $\operatorname{\mathcal{C}}_{f/}$ and $\operatorname{\mathcal{C}}_{/f}$ are (the nerves of) ordinary categories.

We begin with some elementary remarks.

Construction 4.2.6.3. Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. Using Remark remark:join-of-monomorphism, we see that the induced maps

\[ A \star B' \xrightarrow { f \star \operatorname{id}_{B'} } A' \star B' \xleftarrow { \operatorname{id}_{A'} \star g } A' \star B \]

are also monomorphisms. Moreover, the intersection of their images is the image of the monomorphism $(f \star g): A \star B \hookrightarrow A' \star B'$. We therefore obtain a monomorphism of simplicial sets

\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B', \]

which we will refer to as the pushout-join of $f$ and $g$.

We will deduce Proposition 4.2.6.1 from the following property of Construction 4.2.6.3:

Proposition 4.2.6.4. Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets. If $f$ is right anodyne or $g$ is left anodyne, then the pushout-join

\[ (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B' \]

is an inner anodyne morphism of simplicial sets.

Proposition 4.2.6.4 implies the following stronger version of Proposition 4.2.6.1:

Corollary 4.2.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. Then the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a right fibration, and the restriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ is a left fibration.

Proof. We will prove that the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a right fibration; the proof of the analogous assertion $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ follows by a similar argument. By virtue of Proposition 4.1.2.5, it will suffice to show that for every right anodyne morphism $i: A \hookrightarrow A'$, every lifting problem

\[ \xymatrix { A \ar [d]^{i} \ar [r] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{/f_0} } \]

admits a solution. Unwinding the definitions, this is equivalent to solving an associated lifting problem

\[ \xymatrix { (A \star K) \coprod _{ A \star K_0 } (A' \star K_0) \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ A' \star K \ar@ {-->}[ur] \ar [r] & \Delta ^0, } \]

where the left vertical morphism is the pushout-join of Construction 4.2.6.3. Proposition 4.2.6.4 guarantees that this morphism is inner anodyne, so that the desired extension exists by virtue of our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category (Proposition 1.4.6.6). $\square$

Proof of Proposition 4.2.6.1. Apply Corollary 4.2.6.5 in the special case $K_0 = \emptyset $. $\square$

Proposition 4.2.6.4 also gives a useful variant of Corollary 4.2.6.5:

Corollary 4.2.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $f: K \rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets, and let $f_0 = f|_{K_0}$ be the restriction of $f$ to a simplicial subset $K_0 \subseteq K$. If the inclusion $K_0 \hookrightarrow K$ is left anodyne, then the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a trivial Kan fibration. If the inclusion $K_0 \hookrightarrow K$ is right anodyne, then the resriction map $\operatorname{\mathcal{C}}_{f/} \rightarrow \operatorname{\mathcal{C}}_{f_0/}$ is a trivial Kan fibration.

Proof. We proceed as in the proof of Corollary 4.2.6.5. Assume that the inclusion $K_0 \hookrightarrow K$ is left anodyne; we will show that the restriction map $\operatorname{\mathcal{C}}_{/f} \rightarrow \operatorname{\mathcal{C}}_{/f_0}$ is a trivial Kan fibration (the proof of the second assertion is similar). For this, it will suffice to show that if $i: A \hookrightarrow A'$ is any monomorphism of simplicial sets, then any lifting problem

\[ \xymatrix { A \ar [d]^{i} \ar [r] & \operatorname{\mathcal{C}}_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] \ar [r] & \operatorname{\mathcal{C}}_{/f_0} } \]

admits a solution. Unwinding the definitions, this is equivalent to solving an associated lifting problem

\[ \xymatrix { (A \star K) \coprod _{ A \star K_0 } (A' \star K_0) \ar [r] \ar [d] & \operatorname{\mathcal{C}}\ar [d] \\ A' \star K \ar@ {-->}[ur] \ar [r] & \Delta ^0, } \]

where the left vertical morphism is the pushout-join of Construction 4.2.6.3. Since the left vertical map is inner anodyne (Proposition 4.2.6.4), the desired solution exists by virtue of our assumption that $\operatorname{\mathcal{C}}$ is an $\infty $-category (Proposition 1.4.6.6). $\square$

We now turn to the proof of Proposition 4.2.6.4.

Lemma 4.2.6.8. Let $p,q \geq 0$ be nonnegative integers. Then:

  • Assume $p > 0$. Then, for $0 \leq i \leq p$, the pushout-join monomorphism

    \[ (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

    of Construction 4.2.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$.

  • Assume $q> 0$. Then, for $0 \leq j \leq q$, the pushout-join monomorphism

    \[ (\operatorname{\partial \Delta }^ p \star \Delta ^ q) \coprod _{ ( \operatorname{\partial \Delta }^ p \star \Lambda ^{q}_{j} ) } ( \Delta ^ p \star \Lambda ^ q_ j) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

    of Construction 4.2.6.3 is isomorphic to the horn inclusion $\Lambda ^{p+1+q}_{p+1+j} \hookrightarrow \Delta ^{p+1+q}$.

Proof. We will prove the first assertion; the second follows by symmetry. We begin by observing that there is a unique isomorphism of simplicial sets $u: \Delta ^{p} \star \Delta ^{q} \simeq \Delta ^{p+1+q}$ (Example 4.2.3.20). Let $\sigma $ be an $n$-simplex of the join $\Delta ^{p} \star \Delta ^{q}$; we wish to show that $u(\sigma )$ belongs to the horn $\Lambda ^{p+1+q}_{i}$ if and only if $\sigma $ belongs to the union of the simplicial subsets

\[ \Lambda ^{p}_{i} \star \Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q} \supseteq \Delta ^{p} \star \operatorname{\partial \Delta }^{q}. \]

We consider three cases (see Remark 4.2.3.15):

  • The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{p} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Delta ^{p} \star \operatorname{\partial \Delta }^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$.

  • The simplex $\sigma $ belongs to the simplicial subset $\Delta ^{q} \subseteq \Delta ^{p} \star \Delta ^{q}$. In this case, $\sigma $ is contained in $\Lambda ^{p}_{i} \star \Delta ^{q}$ and $u(\sigma )$ is contained in $\Lambda ^{p+1+q}_{i}$ (since $p > 0$).

  • The simplex $\sigma $ factors as a composition

    \[ \Delta ^{n} = \Delta ^{p' + 1 + q'} \simeq \Delta ^{p'} \star \Delta ^{q'} \xrightarrow { \sigma _{-} \star \sigma _{+} } \Delta ^{p} \star \Delta {q}. \]

    Let us abuse notation by identifying $\sigma _{-}$ and $\sigma _{+}$ with nondecreasing functions $[p'] \rightarrow [p]$ and $[q'] \rightarrow [q]$, and $u(\sigma )$ with the nondecreasing function $[n] \rightarrow [p+1+q]$ given by their join. In this case, $\sigma $ fails to belong to the union $(\Lambda ^{p}_{i} \star \Delta ^{q}) \cup ( \Delta ^{p} \star \operatorname{\partial \Delta }^{q} )$ if and only if both of the following conditions are satisfied:

    • The image of the nondecreasing function $\sigma _{-}: [p'] \rightarrow [p]$ contains $[p] \setminus \{ i\} $.

    • The nondecreasing function $\sigma _{+}: [q'] \rightarrow [q]$ is surjective.

    Together, these are equivalent to the assertion that the image of the nondecreasing function $u(\sigma ): [n] \rightarrow [p+1+q]$ contains $[p+1+q] \setminus \{ i\} $: that is, it fails to belong to the horn $\Lambda ^{p+1+q}_{i} \subseteq \Delta ^{p+1+q}$.

$\square$

Proof of Proposition 4.2.6.4. For every pair of morphisms of simplicial sets $f: A \rightarrow A'$ and $g: B \rightarrow B'$, let

\[ \theta _{f,g}: (A \star B') \coprod _{ (A \star B) } (A' \star B) \rightarrow A' \star B' \]

denote their pushout join. We will show that, if $f$ is right anodyne and $g$ is a monomorphism, then $\theta _{f,g}$ is inner anodyne (the analogous assertion for the case where $g$ is left anodyne follows by a similar argument). Let us first regard $f$ as fixed, and let $T$ be the collection of all morphisms $g$ of simplicial sets for which $\theta _{f,g}$ is inner anodyne. Then $T$ weakly saturated (in the sense of Definition 1.4.4.15). We wish to prove that $T$ contains every monomorphism of simplicial sets. By virtue of Proposition 1.4.5.12, we are reduced to proving that the morphism $\theta _{f,g}$ is inner anodyne in the special case where $g$ is the boundary inclusion $\operatorname{\partial \Delta }^{q} \hookrightarrow \Delta ^{q}$ for some $q \geq 0$.

Let us now regard $g: \operatorname{\partial \Delta }^ q \hookrightarrow \Delta ^ q$ as fixed, and let $S$ denote the collection of all morphisms of simplicial sets for which $\theta _{f,g}$ is inner anodyne. To complete the proof, we must show that $S$ contains every right anodyne morphism of simplicial sets. As before, we note that $S$ is weakly saturated. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{p}_{i} \hookrightarrow \Delta ^ p$ for $0 < i \leq p$ (see Variant 4.1.2.2). In other words, we are reduced to checking that the pushout-join

\[ \theta _{f,g}: (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

is inner anodyne. This is clear, since $\theta _{f,g}$ can be identified with the inner horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$ by virtue of Lemma 4.2.6.8. $\square$