Kerodon

Proof. Suppose we are given integers $n \geq 3$, $0 < i < n$, and a morphism of simplicial sets $\tau _0: \Lambda ^{n}_{i} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ for which the restriction $\tau _0|_{ \operatorname{N}_{\bullet }( \{ i-1 < i < i+1 \} ) }$ is the $2$-simplex $\sigma $. We wish to show that $\tau _0$ can be extended to an $n$-simplex of $\operatorname{\mathcal{C}}$. Let $\operatorname{Path}[n]_{\bullet }$ be the simplicial category described in Notation 2.4.3.1, and let us identify $\operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet }$ with the simplicial subcategory of $\operatorname{Path}[n]_{\bullet }$ described in Proposition 2.4.5.8. Then $\tau _0$ can be identified with a simplicial functor $F_0: \operatorname{Path}[ \Lambda ^{n}_{i} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$, and we wish to show that $\tau _0$ can be extended to a simplicial functor $F: \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$.

For $0 \leq j \leq n$, let $C_ j$ denote the object of $\operatorname{\mathcal{C}}$ given by $F_0(j)$. For $1 \leq j \leq n$, let $u_{j}: C_{j-1} \rightarrow C_ j$ be the morphism in $\operatorname{\mathcal{C}}$ obtained by applying $F_0$ to the unique vertex of $\operatorname{Hom}_{ \operatorname{Path}[ \Lambda ^{n}_{i} ] }( j-1, j)$, so that we have a chain of composable morphisms

in the simplicial category $\operatorname{\mathcal{C}}$. Let $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the simplicial cube of dimension $(n-1)$ and let ${\boldsymbol {\sqcap }}^{n-1}_{i} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the hollow cube of Notation 2.4.5.5, so that Remark 2.4.5.4 and Proposition 2.4.5.8 supply isomorphisms

Let $\lambda _0$ denote the composite map

By virtue of Corollary 2.4.5.10, it will suffice to show that $\lambda _0$ can be extended to a morphism of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$.

Let $I$ denote the set $\{ 1, 2, \ldots , i-1, i+1, \cdots , n-1 \} $, so that we can identify $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ with the product $\Delta ^1 \times \operatorname{\raise {0.1ex}{\square }}^{I}$. Under this identification, ${\boldsymbol {\sqcap }}^{n-1}_{i}$ corresponds to the pushout

Let $v \in \operatorname{\raise {0.1ex}{\square }}^{I}$ be the initial vertex (corresponding to the empty subset of $I$), and let $e$ be the edge of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$ given by the composite map

Unwinding the definitions, we see that $e$ is the image of $\mu $ under the morphism of simplicial sets

and is therefore an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_0, C_ n )_{\bullet }$. Note that every simplex of $\operatorname{\raise {0.1ex}{\square }}^{I}$ which is not contained in the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{I}$ has initial vertex $v$. The existence of the desired extension $\lambda $ now follows from Proposition 4.4.5.8. $\square$

Proof. Assertion $(a)$ is immediate from Theorem 2.4.4.10. To prove the remaining assertions, fix an integer $m \geq 0$. Using Lemma 2.4.4.16, we see that $\operatorname{Path}[ \Delta ^ n ]_{m}$ can be identified with the path category $\operatorname{Path}[G]$ of a directed graph $G$ which can be described concretely as follows:

• The vertices of $G$ are the elements of the set $[n] = \{ 0 < 1 < \cdots < n \} $.

• For $0 \leq i < j \leq n$, an edge of $G$ with source $j$ and target $k$ is a chain of subsets

  \[ \{ i, i+1, \ldots , j-1, j\} \supseteq I_0 \supseteq \cdots \supseteq I_ m = \{ i, j \} \]

Using Theorem 2.4.4.10, we see that $\operatorname{Path}[ \Lambda ^ n_ n ]_{m}$ can be identified with the path category of the directed subgraph $G' \subseteq G$ having the same vertices, where an edge $\overrightarrow {I} = (I_0 \supseteq \cdots \supseteq I_ m)$ of $G$ belongs to $G'$ if and only if the subset $I_0 \subseteq [n]$ corresponds to a simplex of $\Delta ^ n$ which belongs to the horn $\Lambda ^{n}_{n}$: that is, if and only if $[n-1] \nsubseteq I_{0}$. We now argue as follows:

• For $(0, n-1) \neq (i,j) \neq (0,n)$, every path from $i$ to $j$ in the graph $G$ is also a path in the graph $G'$. This proves $(b)$.

• Let $\tau $ be a morphism from $0$ to $n-1$ in the category $\operatorname{Path}[ n]_{m}$, which we identify with a chain of subsets

  \[ [n-1] \supseteq I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m} \supseteq \{ 0, n-1 \} . \]

  Then $\tau $ belongs to $\operatorname{Path}[ \Lambda ^{n}_{n} ]_{m}$ if and only if $I_0 \neq [n-1]$ or $I_ m \neq \{ 0, n-1 \} $: that is, if and only if $\tau $ corresponds to an $m$-simplex of the cube $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-2}$. This proves $(c)$.

• Let $\tau $ be a morphism from $0$ to $n$ in the category $\operatorname{Path}[n]_ m$, which we identify with a chain of subsets

  \[ [n] \supseteq I_0 \supseteq I_1 \supseteq \cdots \supseteq I_{m} \supseteq \{ 0, n \} . \]

  Then $\tau $ belongs to $\operatorname{Path}[ \Lambda ^{n}_{n} ]_{m}$ if and only if $I_0 \neq [n]$ or $\{ 0, n \} \neq I_ m \neq \{ 0, n-1, n\} $: that is, if and only if $\tau $ corresponds to an $m$-simplex of the hollow cube ${\boldsymbol {\sqcup }}^{n-1}_{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$. This proves $(d)$.

Proof. By virtue of Corollary 2.4.6.13, we can identify $\theta $ with a pullback of the restriction map

It will therefore suffice to show that $\theta '$ is bijective. Let us identify $\Delta ^{n-1}$ with a simplicial subset of $\Delta ^ n$ (via the map which is the identity on vertices), so that the boundary $\operatorname{\partial \Delta }^{n-1}$ is contained in the horn $\Lambda ^{n}_{n}$. Let $\tau _0$ denote the restriction of $\sigma _0$ to $\operatorname{\partial \Delta }^{n-1}$, let $\mu _0$ denote the $\lambda _0$ to the simplicial subset $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2} \times \{ 0\} \subseteq \boldsymbol {\sqcup }^{n-1}_{n-1}$. Note that $\mu _0$ can be written as a composition

where $\nu _0$ is determined by $\tau _0$. Using the identifications

we can identify $\theta '$ with composition

Here the first map is bijective by virtue of Corollary 2.4.6.13, and the second by virtue of our assumption that $e$ is an isomorphism in the simplicial category $\operatorname{\mathcal{C}}$. $\square$

Proof of Theorem 5.4.8.1. Let $\operatorname{\mathcal{C}}$ be a simplicial category with the property that, for every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)_{\bullet }$ is an $\infty $-category. Using Example 5.4.8.3, we immediately deduce that every degenerate $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is thin, and that every morphism $\Lambda ^{2}_{1} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ can be extended to a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. We will complete the proof that $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is an $(\infty ,2)$-category by showing that, if $n \geq 3$ and $\sigma _0: \Lambda ^{n}_{n} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ is a morphism of simplicial sets for which the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} )}$ is right-degenerate, then $\sigma _0$ can be extended to an $n$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ (the dual assertion regarding extension of maps $\Lambda ^{n}_{0} \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ follows by the same argument, applied to the opposite simplicial category $\operatorname{\mathcal{C}}^{\operatorname{op}}$). Let us identify $\sigma _0$ with a simplicial functor $F: \operatorname{Path}[ \Lambda ^{n}_{n} ]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$, carrying each element $i \in [n]$ to an object $C_ i \in \operatorname{\mathcal{C}}$.

Let $\operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the simplicial cube of dimension $(n-1)$ and let ${\boldsymbol {\sqcup }}^{n-1}_{n-1} \subseteq \operatorname{\raise {0.1ex}{\square }}^{n-1}$ denote the hollow cube of Notation 2.4.5.5, so that Remark 2.4.5.4 and Proposition 5.4.8.4 supply isomorphisms

Let $\lambda _0$ denote the composite map

Note that our degeneracy assumption on $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} )}$ guarantees that the functor $F$ induces an isomorphism $C_{n-1} \simeq C_{n}$ in the category $\operatorname{\mathcal{C}}$. By virtue of Corollary 5.4.8.5, it will suffice to show that $\lambda _0$ can be extended to a morphism of simplicial sets $\lambda : \operatorname{\raise {0.1ex}{\square }}^{n-1} \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}C_0, C_ n)_{\bullet }$.

Let us identify ${\boldsymbol {\sqcup }}^{n-1}_{n-1}$ with the pushout

Let $v$ be the final vertex of the cube $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2}$ (corresponding to the set $\{ 1, 2, \ldots , n-2 \} $, regarded as a subset of itself). Our assumption that the $2$-simplex $\sigma _0|_{ \operatorname{N}_{\bullet }( \{ 0 < n-1 < n \} ) }$ is right-degenerate guarantees that the composite map

is a degenerate edge of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }$; in particular, it is an isomorphism of $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_0, C_ n)_{\bullet }$. Note that every simplex of $\operatorname{\raise {0.1ex}{\square }}^{n-2}$ which is not contained in the boundary $\operatorname{\partial \raise {0.1ex}{\square }}^{n-2}$ has final vertex $v$. The existence of the desired extension $\lambda $ now follows by applying Proposition 4.4.5.8. $\square$

Proof. It follows from Theorem 5.4.8.1 that the simplicial sets $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$ are $(\infty ,2)$-categories. We will show that the morphism $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ is a functor by verifying the criterion of Proposition 5.4.7.9. Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms in the category $\operatorname{\mathcal{C}}$ (or, equivalently, in the $(\infty ,2)$-category $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$). Then $f$ and $g$ determine a $2$-simplex of the nerve $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we identify with $2$-simplex $\sigma $ of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ (see Remark 2.4.3.8). By virtue of Example 5.4.8.3, $\sigma $ is a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$ and its image $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)(\sigma )$ is a thin $2$-simplex of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{D}})$. $\square$

Proof. It follows from Proposition 5.4.8.2 that if $\mu $ is an isomorphism, then $\sigma $ is thin. Conversely, assume that $\sigma $ is thin; we wish to show that $\mu $ is an isomorphism. Define a strict $2$-category $\operatorname{\mathcal{E}}$ as follows:

• The objects of $\operatorname{\mathcal{E}}$ are the objects of $\operatorname{\mathcal{C}}$.

• For every pair of objects $A,B \in \operatorname{\mathcal{C}}$, we define $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,B)$ to be the homotopy category of the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet }$.

• For every triple of objects $A,B,C \in \operatorname{\mathcal{C}}$, we define the composition law

  \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(B,C) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,B) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}(A,C) \]

  to be the functor of homotopy categories induced by the composition law

  \[ \operatorname{Hom}_{\operatorname{\mathcal{C}}}(B,C)_{\bullet } \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet } \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,C) \]

  of the simplicial category $\operatorname{\mathcal{C}}$.

Let $\operatorname{\mathcal{D}}$ denote the simplicial category obtained by applying the construction of Example 2.4.2.8 to the strict $2$-category $\operatorname{\mathcal{E}}$: the simplicial category $\operatorname{\mathcal{D}}$ has the same objects as $\operatorname{\mathcal{C}}$, with simplicial morphism spaces given by

There is an evident functor of simplicial categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$, which is the identity on objects and which induces the unit map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)_{\bullet } \rightarrow \operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(A,B)}_{\bullet } )$ on simplicial morphism spaces. Invoking Proposition 5.4.8.6, we see that the induced map $\operatorname{N}_{\bullet }^{\operatorname{hc}}(F)$ carries $\sigma $ to a thin $2$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{\mathcal{D}})$, which we can identify with the Duskin nerve $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ of the $2$-category $\operatorname{\mathcal{E}}$ (Example 2.4.3.11). Using the description of the thin simplices of $\operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{E}})$ supplied by Theorem 2.3.2.5, we conclude that the homotopy class $[\mu ]$ is an isomorphism in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{E}}}( X,Z) = \mathrm{h} \mathit{\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)}_{\bullet }$, so that $\mu $ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)_{\bullet }$. $\square$

Proof. Let $\sigma $ be an $n$-simplex of the homotopy coherent nerve $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$, which we identify with a simplicial functor $F: \operatorname{Path}[n]_{\bullet } \rightarrow \operatorname{\mathcal{C}}$ carrying each $i \in [n]$ to an object $C_{i} \in \operatorname{\mathcal{C}}$. If $T \subseteq [n]$ is a nonempty subset having smallest element $i$ and largest element $k$, let us write $F(T)$ for the corresponding vertex of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. If $S \subseteq T$ is a subset containing $i$ and $k$, let us write $F(S \subseteq T): F(T) \rightarrow F(S)$ for the corresponding edge of the simplicial set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet }$. Let us abuse notation by identifying $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ with a simplicial subset of $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}})$. Unwinding the definitions, we see that $\sigma $ is contained in $\operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}')$ if and only if the following condition is satisfied:

$(1)$

For every inclusion $S \subseteq T$ of nonempty subsets of $[n]$ having the same smallest element $i$ and largest element $k$, the edge $F(S \subseteq T): F(T) \rightarrow F(S)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$.

Using the thinness criterion of Proposition 5.4.8.7, we see that $\sigma $ belongs to the pith $\operatorname{Pith}( \operatorname{N}_{\bullet }^{\operatorname{hc}}(\operatorname{\mathcal{C}}))$ if and only if the following a priori weaker condition is satisfied:

$(2)$

For every triple of elements $0 \leq i \leq j \leq k \leq n$, the edge

\[ F( \{ i,k \} \subseteq \{ i,j,k\} ): F( \{ i, j, k\} ) \rightarrow F( \{ i, k \} ) \]

is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(C_ i, C_ k)_{\bullet })$.

To complete the proof, it will suffice to show that $(2) \Rightarrow (1)$. Assume that $(2)$ is satisfied, and suppose that we are given nonempty subsets $S \subseteq T$ of $[n]$ having the same smallest element $i$ and largest element $k$. We wish to show that $F(S \subseteq T)$ is an isomorphism in the $\infty $-category $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( C_ i, C_ k)_{\bullet }$. Since the collection of isomorphisms contains all identity morphisms and is closed under composition (Remark 1.4.6.3), we may assume without loss of generality that the difference $T \setminus S$ contains exactly one element $j$. Set $S_{-} = \{ s \in S: s < j \} $ and $S_{+} = \{ s \in S: s > j \} $. Let $i'$ be the largest element of $S_{-}$, and let $k'$ denote the smallest element of $S_{+}$. Unwinding the definitions, we see that the edge $F(S \subseteq T)$ is the image of $F( \{ i',k' \} \subseteq \{ i',j,k'\} )$ under the functor

and is therefore an isomorphism by virtue of assumption $(2)$. $\square$