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8.1.1 The Twisted Arrow Construction

We now describe an $\infty $-categorical generalization of Construction 8.1.0.1.

Construction 8.1.1.1 (Twisted Arrows in Simplicial Sets). Let $\operatorname{{\bf \Delta }}$ denote the simplex category (Definition 1.1.1.2) and let $\operatorname{\mathcal{C}}$ be a simplicial set. We let $\operatorname{Tw}(\operatorname{\mathcal{C}}): \operatorname{{\bf \Delta }}^{\operatorname{op}} \rightarrow \operatorname{Set}$ denote the functor given by the construction

\[ (J \in \operatorname{{\bf \Delta }}^{\operatorname{op}} ) \mapsto \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{N}_{\bullet }( J^{\operatorname{op}} \star J ), \operatorname{\mathcal{C}}). \]

We will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the simplicial set of twisted arrows of $\operatorname{\mathcal{C}}$.

Remark 8.1.1.2. For every integer $n \geq 0$, there is a unique isomorphism of simplicial sets $\operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \simeq \Delta ^{2n+1}$. It follows that, for every simplicial set $\operatorname{\mathcal{C}}$, we can identify $n$-simplices $\sigma $ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with $(2n+1)$-simplices $\overline{\sigma }$ of $\operatorname{\mathcal{C}}$. In terms of these identifications, the face and degeneracy operators of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ are given explicitly by the formulae

\[ \overline{d_ i \sigma } = d_{n-i} d_{n+1+i} \overline{\sigma } \quad \quad \overline{s_ i \sigma } = s_{n-i} s_{n+1+i} \overline{\sigma }. \]

Remark 8.1.1.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set. We will generally use Remark 8.1.1.2 to identify vertices of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ with edges $f: X \rightarrow Y$ of $\operatorname{\mathcal{C}}$. More generally, it will be useful to think of $n$-simplices of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as encoding diagrams

\[ \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^{f_0} & X_1 \ar [l] \ar [d]^{f_1} & X_2 \ar [l] \ar [d]^{f_2} & \cdots \ar [l] \ar [d] & X_ n \ar [l] \ar [d]^{f_ n} \\ Y_0 \ar [r] & Y_1 \ar [r] & Y_2 \ar [r] & \cdots \ar [r] & Y_ n. } \]

Remark 8.1.1.4. The construction $\operatorname{\mathcal{C}}\mapsto \operatorname{Tw}(\operatorname{\mathcal{C}})$ determines a functor from the category of simplicial sets to itself, which preserves all limits and colimits (this follows from Remark 8.1.1.2, since limits and colimits in the category $\operatorname{Set_{\Delta }}$ are computed levelwise).

Notation 8.1.1.5 (Projection Maps). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is equipped with projection maps

\[ \lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \quad \quad \lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}. \]

Here $\lambda _{+}$ carries an $n$-simplex $\sigma $ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to the $n$-simplex of $\operatorname{\mathcal{C}}$ given by the composition

\[ \Delta ^ n = \operatorname{N}_{\bullet }( [n] ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}}, \]

while $\lambda _{-}$ carries $\sigma $ to the $n$-simplex of $\operatorname{\mathcal{C}}^{\operatorname{op}}$ given by the composite map

\[ (\Delta ^ n)^{\operatorname{op}} = \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} ) \hookrightarrow \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow {\sigma } \operatorname{\mathcal{C}}. \]

Concretely, $\lambda _{-}$ and $\lambda _{+}$ are given on vertices by the formulae $\lambda _{-}( f: X \rightarrow Y ) = X$ and $\lambda _{+}( f: X \rightarrow Y ) = Y$.

Remark 8.1.1.6 (Duality). Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there is a canonical isomorphism of simplicial sets $\rho : \operatorname{Tw}(\operatorname{\mathcal{C}}) \simeq \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}} )$, given on $n$-simplices by precomposition with the unique isomorphism

\[ \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] )^{\operatorname{op}} \simeq \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ). \]

The isomorphism $\rho $ interchanges the projection maps $\lambda _{-}$ and $\lambda _{+}$ of Notation 8.1.1.5.

Exercise 8.1.1.7 (Slices of Twisted Arrows). Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $f: X \rightarrow Y$ be an edge of $\operatorname{\mathcal{C}}$, which we regard as a vertex of the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Show that there is a canonical isomorphism of simplicial sets

\[ \operatorname{Tw}(\operatorname{\mathcal{C}})_{ / f } \simeq \operatorname{Tw}( \operatorname{\mathcal{C}}_{ X/ \, / Y} ). \]

Here $\operatorname{\mathcal{C}}_{X/ \, /Y}$ denotes the simplicial set $(\operatorname{\mathcal{C}}_{X/ })_{/Y} \simeq ( \operatorname{\mathcal{C}}_{/Y} )_{X/}$, obtained either by promoting $Y$ to a vertex of $\operatorname{\mathcal{C}}_{X/}$ or $X$ to a vertex of $\operatorname{\mathcal{C}}_{/Y}$ by means of the edge $f$.

Warning 8.1.1.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set. Then there is a tautological map $T: \operatorname{\mathcal{C}}\rightarrow \operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}})$, which carries an $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$ to the $n$-simplex $T(\sigma )$ of $\operatorname{Tw}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}})$ given by the composition

\[ \operatorname{N}_{\bullet }( [n]^{\operatorname{op}} \star [n] ) \xrightarrow { \sigma ^{\operatorname{op}} \star \sigma } \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}. \]

If $\operatorname{\mathcal{D}}$ is another simplicial set, then precomposition with $T$ induces a comparison map

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}, \operatorname{\mathcal{D}}) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Tw}(\operatorname{\mathcal{C}}^{\operatorname{op}} \star \operatorname{\mathcal{C}}), \operatorname{Tw}(\operatorname{\mathcal{D}}) ) \xrightarrow { \circ T} \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\mathcal{C}}, \operatorname{Tw}(\operatorname{\mathcal{D}}) ). \]

Beware that, in general, this map is not a bijection. However, it is a bijection whenever $\operatorname{\mathcal{C}}$ is isomorphic to the nerve of a linearly ordered set $Q$. To prove this, we can write $Q$ as a filtered colimit of its finite subsets and thereby reduce to the case where $Q$ is finite. In this case, the linearly ordered set $Q$ is either empty (in which case the desired result is obvious) or isomorphic to $[n]$ for some integer $n \geq 0$ (in which case the desired result follows from the definition of the $\operatorname{Tw}(\operatorname{\mathcal{D}})$).

We now show that Construction 8.1.1.1 can be regarded as a generalization of Construction 8.1.0.1.

Proposition 8.1.1.9. Let $\operatorname{\mathcal{C}}$ be a category. Then there is a canonical isomorphism of simplicial sets $T: \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$, which is uniquely determined by the following requirements:

$(1)$

For every morphism $f: C \rightarrow D$ in the category $\operatorname{\mathcal{C}}$, the map $T$ carries $f$ (regarded as an object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$) to itself (regarded as a vertex of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$).

$(2)$

The diagram

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \ar [d] \ar [r]^-{T} & \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [d]^{ ( \lambda _{-}, \lambda _{+}) } \\ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \ar [r]^-{\sim } & \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})^{\operatorname{op}} \times \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]

commutes, where the right vertical map is given by Notation 8.1.1.5 and the left vertical map is the nerve of the forgetful functor

\[ \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\quad \quad (f: X \rightarrow Y) \mapsto (X,Y). \]

Proof. Let $\sigma $ be an $n$-simplex of the simplicial set $\operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$, which we identify with a diagram

\[ (f_0: X_0 \rightarrow Y_0) \xrightarrow { (u_1,v_1) } (f_1: X_1 \rightarrow Y_1) \xrightarrow { (u_2,v_2)} \cdots \xrightarrow { (u_ n, v_ n) } ( f_ n: X_ n \rightarrow Y_ n ) \]

in the category $\operatorname{Tw}(\operatorname{\mathcal{C}})$. Here each $f_ i: X_ i \rightarrow Y_ i$ denotes a morphism in $\operatorname{\mathcal{C}}$, and each $(u_ i, v_ i)$ is a pair of morphisms in $\operatorname{\mathcal{C}}$ which determine a commutative diagram

\[ \xymatrix@R =50pt@C=50pt{ X_{i-1} \ar [d]^{f_{i-1} } & X_{i} \ar [l]_-{ u_ i } \ar [d]^{f_ i} \\ Y_{i-1} \ar [r]^-{ v_ i } & Y_{i}. } \]

In this case, we can regard the chain of morphisms

8.1
\begin{equation} \begin{gathered}\label{equation:twisted-arrow-canonical-isomorphism} \xymatrix@R =50pt@C=50pt{ X_0 \ar [d]^-{f_0} & X_1 \ar [l]_-{u_1} & X_2 \ar [l]_-{u_2} & \cdots \ar [l] & X_ n \ar [l]_-{u_ n} \\ Y_0 \ar [r]^-{v_1} & Y_1 \ar [r]^-{v_2} & Y_2 \ar [r] & \cdots \ar [r]^-{v_ n} & Y_ n } \end{gathered} \end{equation}

as a $(2n+1)$-simplex of $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$, which we identify with an $n$-simplex $T(\sigma )$ of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. The construction $\sigma \mapsto T(\sigma )$ then determines a morphism of simplicial sets $T: \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \rightarrow \operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$, which satisfies conditions $(1)$ and $(2)$ by construction.

We now claim that $T$ is an isomorphism of simplicial sets. Let $\tau $ be an $n$-simplex of $\operatorname{Tw}( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$; we wish to show that there is a unique $n$-simplex $\sigma $ of $\operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) )$ satisfying $T(\sigma ) = \tau $. Let us identify $\tau $ with a diagram of the form (8.1) in the category $\operatorname{\mathcal{C}}$. We wish to show that there is a unique collection of morphisms $\{ f_{i}: X_{i} \rightarrow Y_{i} \} _{1 \leq i \leq n}$ satisfying the identities $f_{i} = v_{i} \circ f_{i-1} \circ u_{i}$, which follows immediately by induction on $i$.

We now complete the proof by establishing the uniqueness of $T$. Suppose that $T': \operatorname{N}_{\bullet }( \operatorname{Tw}(\operatorname{\mathcal{C}}) ) \xrightarrow {\sim } \operatorname{Tw}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) )$ is another morphism of simplicial sets satisfying conditions $(1)$ and $(2)$. Then $T^{-1} \circ T'$ determines a functor $F$ from the twisted arrow category $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. Because $T$ and $T'$ both satisfy condition $(1)$, the functor $F$ carries each object of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. Since the forgetful functor $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ is faithful, condition $(2)$ guarantees that $F$ also carries each morphism of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ to itself. It follows that $F$ is the identity functor, so that $T' = T$. $\square$

Let $\operatorname{\mathcal{C}}$ be a simplicial set. It follows from Proposition 8.1.1.9 that if $\operatorname{\mathcal{C}}$ is isomorphic to the nerve of a category, then the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is also isomorphic to the nerve of a category. Moreover, the projection maps of Notation 8.1.1.5 determine a left covering map $\operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ (see Remark 8.1.0.2). This observation has an $\infty $-categorical counterpart:

Proposition 8.1.1.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the projection maps of Notation 8.1.1.5 determine a left fibration of simplicial sets

\[ (\lambda _{-}, \lambda _{+}): \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}. \]

Corollary 8.1.1.11. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the simplicial set $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is also an $\infty $-category.

In the situation of Corollary 8.1.1.11, we will refer to $\operatorname{Tw}(\operatorname{\mathcal{C}})$ as the twisted arrow $\infty $-category of $\operatorname{\mathcal{C}}$.

Corollary 8.1.1.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Then the projection maps $\lambda _{-}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\lambda _{+}: \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ are cocartesian fibrations of $\infty $-categories. Moreover, a morphism $f$ of $\operatorname{Tw}(\operatorname{\mathcal{C}})$ is $\lambda _{-}$-cocartesian if and only if $\lambda _{+}(f)$ is an isomorphism, and $\lambda _{+}$-cocartesian if and only if $\lambda _{-}(f)$ is an isomorphism.

Proof. Let $\pi _{-}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}^{\operatorname{op}}$ and $\pi _{+}: \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{C}}$ denote the projection maps. Then $\pi _{-}$ and $\pi _{+}$ are cocartesian fibrations of simplicial sets. Moreover, a morphism $(e_{-}, e_{+})$ of $\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}$ is $\pi _{-}$-cocartesian if and only if $e_{+}$ is an isomorphism in $\operatorname{\mathcal{C}}$, and $\pi _{+}$-cocartesian if and only if $e_{-}$ is an isomorphism in $\operatorname{\mathcal{C}}^{\operatorname{op}}$ (this follows immediately from Remark 5.1.4.6 and Example 5.1.1.4). Corollary 8.1.1.12 now follows by applying Proposition 8.1.1.10 to left and right sides of the diagram

\[ \xymatrix@R =50pt@C=50pt{ & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [dl]_{ \lambda _{-} } \ar [dr]^{ \lambda _{+} } \ar [d]^{ ( \lambda _{-}, \lambda _{+} ) } & \\ \operatorname{\mathcal{C}}^{\operatorname{op}} & \operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}\ar [l]_-{\pi _{-}} \ar [r]^-{ \pi _{+} } & \operatorname{\mathcal{C}}, } \]

since the vertical map in the center is a left fibration (Proposition 8.1.1.10). $\square$

Proposition 8.1.1.10 is a special case of the following more general assertion:

Proposition 8.1.1.13. Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be an inner fibration of simplicial sets. Then the projection maps of Notation 8.1.1.5 determine a left fibration of simplicial sets

\[ \operatorname{Tw}(\operatorname{\mathcal{C}}) \rightarrow (\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \times _{ ( \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) } \operatorname{Tw}(\operatorname{\mathcal{D}}). \]

Proof. Fix a pair of integers $0 < i \leq n$; we wish to show that every lifting problem

8.2
\begin{equation} \begin{gathered}\label{equation:twisted-arrow-left-fibration} \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{n-i} \ar [r] \ar [d] & \operatorname{Tw}(\operatorname{\mathcal{C}}) \ar [d] \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & (\operatorname{\mathcal{C}}^{\operatorname{op}} \times \operatorname{\mathcal{C}}) \times _{ ( \operatorname{\mathcal{D}}^{\operatorname{op}} \times \operatorname{\mathcal{D}}) } \operatorname{Tw}(\operatorname{\mathcal{D}}) } \end{gathered} \end{equation}

admits a solution.

For each nonempty subset $S \subseteq [2n+1] = \{ 0 < 1 < \cdots < 2n+1 \} $, let $\sigma _{S}$ denote the corresponding nondegenerate simplex of $\Delta ^{2n+1}$. Let us say that $S$ is basic if it satisfies one of the following conditions:

$(a)$

The set $S$ is contained in $\{ 0 < 1 < \cdots < n \} $.

$(b)$

The set $S$ is contained in $\{ n+1 < n_2 < \cdots < 2n+1 \} $.

$(c)$

There exists an integer $j \neq i$ such that $0 \leq j \leq n$ and $S \cap \{ j, 2n+1- j \} = \emptyset $.

Let $K_0 \subseteq \Delta ^{2n+1}$ be the simplicial subset whose nondegenerate simplices have the form $\sigma _{S}$, where $S$ is basic. Unwinding the definitions, we can rewrite (8.2) as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ K_0 \ar [d] \ar [r] & \operatorname{\mathcal{C}}\ar [d]^{U} \\ \Delta ^{2n+1} \ar [r] \ar@ {-->}[ur] & \operatorname{\mathcal{D}}. } \]

Since $U$ is an inner fibration, it will suffice to show that the inclusion $K_0 \hookrightarrow \Delta ^{2n+1}$ is an inner anodyne map of simplicial sets.

We now introduce two more collections of subsets of $[2n+1]$.

  • We say that a subset $S \subseteq [2n+1]$ is primary if it is not basic, the intersection $S \cap \{ 0, 1, \cdots , i-1\} $ is empty, and $2n+1-i \in S$.

  • We say that a subset $S \subseteq [2n+1]$ is secondary if it is not basic, the intersection $S \cap \{ 0, 1, \cdots , i-1 \} $ is nonempty, and $i \in S$.

Let $\{ S_1, S_2, \cdots , S_ m \} $ be an ordering of the collection of all subsets of $[2n+1]$ which are either primary or secondary, satisfying the following conditions:

  • The sequence of cardinalities $| S_1 |, |S_2 |, \cdots , |S_ m |$ is nondecreasing. That is, for $1 \leq p \leq q \leq m$, we have $|S_ p | \leq | S_{q} |$.

  • If $|S_ p| = |S_ q|$ for $p \leq q$ and $S_{q}$ is primary, then $S_ p$ is also primary.

For $1 \leq q \leq m$, let $\sigma _{q} \subseteq \Delta ^{2n+1}$ denote the simplex spanned by the vertices of $S_ q$, and let $K_{q}$ denote the union of $K_0$ with the simplices $\{ \sigma _1, \sigma _2, \cdots , \sigma _ q \} $. We have inclusion maps

\[ K_0 \hookrightarrow K_1 \hookrightarrow K_{2} \hookrightarrow \cdots \hookrightarrow K_{m}. \]

Note that we have $\sigma _{m} = K_{m} = \Delta ^{2n+1}$ (since the set $[2n+1]$ is secondary). It will therefore suffice to show that for $1 \leq q \leq m$, the inclusion map $K_{q-1} \hookrightarrow K_{q}$ is inner anodyne.

In what follows, we regard $q$ as fixed. Let $d$ be the dimension of the simplex $\sigma _{q}$. Let us abuse notation by identifying $\sigma _{q}$ with a morphism of simplicial sets $\Delta ^{d} \rightarrow K_{q} \subseteq \Delta ^{2n+1}$, and set $L = \sigma _{q}^{-1} K_{q-1} \subseteq \Delta ^{d}$. We then have a pushout diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ L \ar [r] \ar [d] & K_{q-1} \ar [d] \\ \Delta ^{d} \ar [r]^-{ \sigma _ q} & K_{q}. } \]

We will complete the proof by showing that $L$ is an inner horn of $\Delta ^{d}$.

We first consider the case where the set $S_{q} = \{ j_0 < j_1 < \cdots < j_ d \} $ is primary, so that we have $j_0 \geq i$ and $j_ k = 2n+1-i$ for some $0 \leq k \leq d$. Note that we must have $k > 0$ (otherwise $S_ q$ satisfies condition $(b)$) and $k < d$ (otherwise, $S_ q$ satisfies condition $(c)$, since it is disjoint from $\{ 0, 2n+1\} $). In this case, we will show that $L$ coincides with the inner horn $\Lambda ^{d}_{k} \subset \Delta ^{d}$. This can be restated as follows:

$(\ast )$

Let $j$ be an element of $S_{q}$, and set $S' = S_{q} \setminus \{ j\} $. Then $\sigma _{S'}$ is contained in $K_{q-1}$ if and only if $j \neq 2n+1-i$.

Assume first that $j \neq 2n+1-i$. Then the set $S'$ contains $2n+1-i$ and satisfies $S' \cap \{ 0, 1, \cdots , i-1 \} = \emptyset $. Consequently, the set $S'$ is either primary (and therefore coincides with $S_{q'}$ for some $q' < q$) or basic. In either case, the simplex $\sigma _{S'}$ belongs to the simplicial subset $K_{q-1} \subseteq \Delta ^{2n+1}$.

We now prove $(\ast )$ in the case $j = 2n+1-i$. Since $S_{q}$ does not satisfy conditions $(b)$ or $(c)$, the set $S'$ also does not satisfy conditions $(b)$ or $(c)$. It also cannot satisfy condition $(a)$: if $S'$ were contained in the set $\{ 0, 1, \cdots , n \} $, then $S_{q}$ would be contained in the set $\{ i, i+1, \cdots , n, 2n+1-i \} $, and would therefore satisfy condition $(c)$. It follows that $S'$ is not basic. Assume, for a contradiction, that $\sigma _{S'}$ is contained in $K_{q-1}$. We then have $\sigma _{S'} \subseteq \sigma _{q'}$ for some $q' < q$. Since $S'$ is neither primary nor secondary, this must be a proper inclusion: that is, we must have

\[ \dim ( \sigma _{q} ) - 1 = \dim ( \sigma _{S'} ) < \dim ( \sigma _{q'} ) \leq \dim ( \sigma _{q} ). \]

It follows that the second inequality must be an equality: that is, we have $|S_{q'}| = |S_{q} |$ and therefore $S_{q'}$ is also primary. In particular, the set $S_{q'}$ contains $2n+1-i$, and therefore contains the union $S' \cup \{ 2n+1-i\} = S_{q}$. Since $S_{q}$ and $S_{q'}$ have the same cardinality, it follows that $S_{q} = S_{q'}$ and therefore $q = q'$, contradicting our assumption that $q' < q$.

We now consider the case where $S_{q} = \{ j_0 < j_1 < \cdots < j_ d \} $ is secondary, so that we have $j_0 < i$ and $j_ k = i$ for some $0 < k \leq d$. Note that we must have $k < d$ (otherwise, $S_ q$ satisfies condition $(a)$). In this case, we will show that $L$ coincides with the inner horn $\Lambda ^{d}_{k} \subset \Delta ^{d}$. This can be restated as follows:

$(\ast ')$

Let $j \in S_ q$ and set $S' = S_ q \setminus \{ j \} $. Then the simplex $\sigma _{S'}$ is contained in $K_{q-1}$ if and only if $j = i$.

We first treat the case where $j \neq i$, so that $i \in S'$. If $S'$ is basic, then $\sigma _{S'} \subseteq K \subseteq K_{q-1}$. We may therefore assume that $S'$ is not basic. If the intersection $S' \cap \{ 0, 1, \cdots , i-1 \} $ is nonempty, then $S'$ is secondary and has smaller cardinality than $S_{q}$. It follows that $S' = S_{q'}$ for some $q' < q$, so that $\sigma _{S'} \subseteq K_{q'} \subseteq K_{q-1}$. We may therefore assume that the intersection $S' \cap \{ 0, 1, \cdots , i-1\} $ is empty. In this case, the union $S' \cup \{ 2n+1 - i \} $ is a primary set of cardinality $\leq |S_ q |$, and therefore has the form $S_{q'}$ for some $q' < q$. From this, we again conclude that $\sigma _{S'} \subseteq K_{q'} \subseteq K_{q-1}$.

We now prove $(\ast ')$ in the case $j=i$. Since $S_{q}$ does not satisfy conditions $(a)$ or $(c)$, it follows that $S'$ also does not satisfy conditions $(a)$ or $(c)$. The set $S'$ also does not satisfy condition $(b)$, since the intersection $S' \cap \{ 0, \cdots , i-1 \} $ is nonempty. It follows that $S'$ is not basic. Assume, for a contradiction, that $\sigma _{S'}$ is contained in $K_{q-1}$. We then have $\sigma _{S'} \subseteq \sigma _{q'}$ for some $q' < q$. Since the intersection $S_{q'} \cap \{ 1, \cdots , i-1 \} $ is nonempty, the set $S_{q'}$ cannot be primary and is therefore secondary. In particular, the set $S_{q'}$ contains the element $i$ and therefore contains the union $S' \cup \{ i\} = S_{q}$. Combining this observation with the inequality $| S_{q'} | \leq | S_{q} |$, we deduce that $S_{q'} = S_{q}$ and therefore $q' = q$, contradicting our assumption that $q' < q$. $\square$