Proof. Let $i\,:\, A\to B$ be a trivial cofibration in $\textrm {Set}_{\Delta }^{+}$ . We consider a commutative diagram of solid arrows

(3.1)

We need to show that there is a dotted arrow $h\,:\, B\to X^{\natural }$ which makes the whole diagram commute.

We have a commutative diagram

of Kan complexes. The vertical arrows are trivial Kan fibrations by [Reference Lurie15, Lemma 3.1.3.6] and the definition of weak equivalences in $\textrm {Set}_{\Delta }^{+}$ (cf. [Reference Lurie15, Proposition 3.1.3.3]). Thus, we obtain a homotopy equivalence of Kan complexes

(3.2) \begin{equation} (i^*,p^{\natural }_*)\,:\, \textrm {Map}^{\sharp }(B,X^{\natural }) \stackrel {\simeq }{\longrightarrow } \textrm {Map}^{\sharp }(A,X^{\natural }) \times _{\textrm {Map}^{\sharp }(A,Y^{\natural })} \textrm {Map}^{\sharp }(B,Y^{\natural }). \end{equation}

Commutative diagram (3.1) determines a vertex $(f,g)$ on the right-hand side of (3.2). By homotopy equivalence (3.2), we can take $k\in \textrm {Map}^{\sharp }(B,X^{\natural })$ and a homotopy $K=(K_1,K_2)\,:\, \Delta ^1\to \textrm {Map}^{\sharp }(A,X^{\natural }) \times _{\textrm {Map}^{\sharp }(A,Y^{\natural })} \textrm {Map}^{\sharp }(B,Y^{\natural })$ from $(f,g)$ to $(i^*,p^{\natural }_*)(k)$ . By using $k$ and $K$ , we obtain a commutative diagram of solid arrows

(3.3)

where $M$ is a subobject of $B\times (\Delta ^1)^{\sharp }$ given by $(A\times (\Delta ^1)^{\sharp })\coprod _{A\times \Delta ^{\{1\}}} (B\times \Delta ^{\{1\}})$ , and $\overline {K}_1$ is a morphism whose restrictions are given by $\overline {K}_1|_{A\times (\Delta ^1)^{\sharp }}=K_1$ and $\overline {K}_1|_{B\times \Delta ^{\{1\}}}=k$ .

Combining [Reference Lurie15, Proposition 3.1.2.3] with the fact that $\Delta ^{\{1\}}\to (\Delta ^1)^{\sharp }$ is a marked anodyne morphism by [Reference Lurie15, Definition 3.1.1], the inclusion map $M\to B\times (\Delta ^1)^{\sharp }$ is marked anodyne. We can verify that $p^{\natural }$ satisfies the conditions in [Reference Lurie15, Proposition 3.1.1.6] by using the fact that $p\,:\, X\to Y$ is a categorical fibration of $\infty$ -categories, the dual of [Reference Lurie15, Corollary 2.4.6.5], and [Reference Lurie15, Proposition 2.4.1.5]. Hence, $p^{\natural }$ has the right lifting property with respect to all marked anodyne morphisms, and we can construct a lifting $H\,:\, B\times (\Delta ^1)^{\sharp }\to X^{\natural }$ which makes whole diagram (3.3) commute. Then $h=H|_{B\times \Delta ^{\{0\}}}$ gives the desired lifting of commutative diagram (3.1).

Proof of Proposition 3.5. The induced map $\textrm {FUN}(B,X)\to \textrm {FUN}(A,X)$ is a fibration of fibrant objects in $\textrm {Set}_{\Delta }^{\textrm {sc}}$ since the model category $\textrm {Set}_{\Delta }^{\textrm {sc}}$ is Cartesian closed. The fact that $u_1$ is a right Quillen functor implies that $\textrm {Fun}(B,X)\to \textrm {Fun}(A,X)$ is a fibration of fibrant objects in $\textrm {Set}_{\Delta }^{\textrm {Joy}}$ . The lemma follows from Lemma 3.6.

Proof. We notice that all thin triangles in $\Delta ^n_{\dagger }$ are contained in $\Lambda ^n_{M,\dagger }$ by the assumption that $|M|\le n-2$ and $\Delta ^{[n]-M}$ is not thin when $|M|=n-2$ . We prove the lemma by induction on the cardinality $|M|$ of $M$ .

First, we consider the case $|M|=1$ . We write $M=\{m\}$ with $0\lt m\lt n$ . In this case we have $0\le s=m-1$ and $t=m\lt n$ . By the assumption, $\Delta ^{\{m-1,m,m+1\}}$ is thin in $\Delta ^n_{\dagger }$ . There is a pushout diagram

\begin{equation*} \begin{array}{ccc} (\Lambda ^n_m,\{\Delta ^{\{m-1,m,m+1\}}\} \cup \{\textrm {degenerate}\}) & \longrightarrow & (\Delta ^n, \{\Delta ^{\{m-1,m,m+1\}}\} \cup \{\textrm {degenerate}\})\\ \bigg \downarrow & & \bigg \downarrow \\ \Lambda ^n_{m,\dagger } & \longrightarrow & \Delta ^n_{\dagger }\\ \end{array}\end{equation*}

of scaled simplicial sets. Since the top horizontal arrow is a scaled anodyne map, so is the bottom horizontal arrow.

Next, we suppose that $|M|\gt 1$ . We let $m=\min M$ and set $M'=M-\{m\}$ . When we regard $M'$ as a subset of $[n]-\{n\}$ , we can verify that $M'$ satisfies the assumptions on the lemma if $M$ does. By the hypothesis of induction, $\Lambda ^{[n]}_{M',\dagger }\to \Delta ^{[n]}_{\dagger }$ is scaled anodyne. Thus, it suffices to show that the inclusion map $\Lambda ^{[n]}_{M,\dagger }\to \Lambda ^{[n]}_{M',\dagger }$ is scaled anodyne. There is a pushout diagram

\begin{equation*} \begin{array}{ccc} \Lambda ^{[n]-\{m\}}_{M',\dagger }& \longrightarrow & \Delta ^{[n]-\{m\}}_{\dagger }\\ \bigg \downarrow & & \bigg \downarrow \\[3mm] \Lambda ^{[n]}_{M,\dagger }&\longrightarrow & \Lambda ^{[n]}_{M',\dagger }\\ \end{array}\end{equation*}

of scaled simplicial sets. We have an isomorphism $\theta \,:\, [n]-\{m\}\cong [n-1]$ of finite ordered sets. We can verify that the subset $\theta (M')\subset [n-1]-\{n-1\}$ satisfies the assumptions on the lemma if $M\subset [n]-\{n\}$ does. Thus, the top horizontal arrow is scaled anodyne by the hypothesis of induction, and so is the bottom horizontal arrow.

Notation 4.2. For a subcomplex $K$ of $\Delta ^n\star \Delta ^{n,\textrm {op}}$ , we denote by $K_{\S }$ the scaled simplicial set whose underlying simplicial set is $K$ equipped with the induced scaling from $Q(n)$ .

Proof. The first part follows by Proposition 3.5. If $A^{\bullet }$ satisfies the co-Segal condition, then we see that $\textrm {Map}^{\textrm {sc}}(A^{\bullet },X)$ satisfies the Segal condition.

Proof. The first part follows by observing that the $n$ th latching object of $Q(\!\bullet\! )$ is isomorphic to the subcomplex $\cup _{i=0}^n(\Delta ^{[n]-\{i\}}\star \Delta ^{[n]-\{i\},\textrm {op}})_{\S }$ of $Q(n)$ . The second part follows by Lemma 4.6 since $\textrm {TW}^r(Z)=\textrm {Map}^{\textrm {sc}}(Q(\!\bullet\! ),Z)$ .

Proof. Let $(K^n_i)_{\S }$ be a subcomplex of $Q(n)$ defined in [Reference García and Stern1, Construction 2.7], that is, $(K^n_i)_{\S }=Q(\Lambda ^n_i)\cup \Delta ^n_{\sharp } \cup \Delta ^{n,\textrm {op}}_{\sharp }$ . By [Reference García and Stern1, Lemmas 2.10.1 and 2.12.1], the inclusion map $(K^n_i)_{\S }\to Q(n)$ is scaled anodyne, and hence, it suffices to show that $Q(\Lambda ^n_i)\to (K^n_i)_{\S }$ is scaled anodyne. This follows by observing that $Q(\Lambda ^n_i)\cap (\Delta ^n_{\sharp }\cup \Delta ^{n,\textrm {op}}_{\sharp })= \Lambda ^n_{i,\sharp }\cup \Lambda ^{n,\textrm {op}}_{i,\sharp }$ and $\Lambda ^n_{i,\sharp }\cup \Lambda ^{n,\textrm {op}}_{i,\sharp } \to \Delta ^n_{\sharp }\cup \Delta ^{n,\textrm {op}}_{\sharp }$ is scaled anodyne by [Reference Lurie17, Remark 3.1.5].

Proof. We prove the lemma by induction on $n$ . When $n=2$ , the map $Q(\Lambda ^2_1)\to Q(2)$ is scaled anodyne by Lemma 4.8. Now, we suppose that $n\ge 3$ and assume that the lemma holds for less than $n$ . We show that $Q(\Lambda ^n_{\{1,\ldots ,i\}})\to Q(n)$ is scaled anodyne by induction of $i$ . When $i=1$ , it holds by Lemma 4.8. We suppose that $2\le i\lt n$ and assume that it holds for less than $i$ . Since $\Lambda ^{n}_{\{1,\ldots ,i\}}\cap \Delta ^{[n]-\{i\}}= \Lambda ^{[n]-\{i\}}_{\{1,\ldots ,i-1\}}$ , there is a pushout diagram of scaled simplicial sets

By using the isomorphism $Q(\Delta ^{[n]-\{i\}})\cong Q(n-1)$ and the hypothesis of induction, the top horizontal arrow is scaled anodyne and hence so is the bottom horizontal arrow. By the hypothesis of induction, $Q(\Lambda ^n_{\{1,\ldots ,i-1\}})\to Q(n)$ is scaled anodyne. Thus, the composite $Q(\Lambda ^n_{\{1,\ldots ,i\}})\to Q(\Lambda ^n_{\{1,\ldots ,i-1\}})\to Q(n)$ is also scaled anodyne.

Proof of Proposition 4.5. It suffices to show that $Q(\!\bullet\! )$ satisfies the co-Segal condition. For $0\le i\le j\le n$ , we define a subcomplex $\textrm {sp}_{[i,j]}$ of $\Delta ^n$ by

\begin{equation*} \textrm {sp}_{[i,j]}=\Delta ^{\{i,i+1\}} \coprod _{\Delta ^{\{i+1\}}}\cdots \coprod _{\Delta ^{\{j-1\}}} \Delta ^{\{j-1,j\}}. \end{equation*}

We show that the co-Segal map $Q(\textrm {sp}_{[0,n]})\to Q(n)$ is scaled anodyne by induction on $n$ . When $n=0,1$ , it is trivial. We suppose $n\ge 2$ and assume that it holds for less than $n$ . We define a subcomplex $s(i,n)$ of $\Delta ^n$ by

\begin{equation*} s(i,n)=\Delta ^{\{0,\ldots ,n-1\}}\coprod _{\Delta ^{\{i,\ldots ,n-1\}}} \Delta ^{\{i,\ldots ,n\}}\end{equation*}

for $0\le i \lt n$ . By setting $S(i,n)=Q(s(i,n))$ , we obtain a filtration of scaled simplicial sets $Q(\textrm {sp}_{[0,n]})\to S(n-1,n)\to S(n-2,n)\to \cdots \to S(0,n)=Q(n)$ . The inclusion map $Q(\textrm {sp}_{[0,n]})\to S(n-1,n)$ is scaled anodyne since it is obtained as a pushout of $Q(\textrm {sp}_{[0,n-1]})\to Q(\Delta ^{\{0,\ldots ,n-1\}})$ , which is scaled anodyne by the hypothesis of induction, along the inclusion map $Q(\textrm {sp}_{[0,n-1]})\to Q(\textrm {sp}_{[0,n]})$ . Therefore, it suffices to show that $S(i+1,n)\to S(i,n)$ is scaled anodyne for $0\le i\lt n-1$ . Since $s(i+1,n)\cap \Delta ^{\{i,\ldots ,n\}}= \Lambda ^{\{i,\ldots ,n\}}_{\{i+1,\ldots ,n-1\}}$ , there is a pushout diagram

The top horizontal arrow is scaled anodyne by Lemma 4.9, and hence so is the bottom horizontal arrow.

Proof. Suppose that $M\in \mathrm{TW}^r(Z)_1^{\textrm {eq}}$ . By applying $q_1^{\textrm {eq}}$ to $M$ , we see that $M(\Delta ^{\{00,01\}})$ and $M(\Delta ^{\{11,10\}})$ are equivalence $1$ -morphisms in $Z$ .

We shall show that $M$ extends to a map from $(\Delta ^1\star \Delta ^{1,\textrm {op}})_{\sharp }$ . For this purpose, it suffices to show that $M(\Delta ^{\{00,10,11\}})$ is thin by [Reference Lurie17, Remark 3.1.4]. By the assumption, there are $L,R\in \mathrm{TW}^r(Z)_2$ such that $d_0(L)=M, d_1(L)=s_0(N_1)$ and $d_1(R)=s_0(N_0),d_2(R)=M$ . By using Lemma 4.8 for $n=3$ and $i=2$ , we obtain $A\in \mathrm{TW}^r(Z)_3$ such that $d_0(A)=R$ , $d_3(A)=L$ , which is given by a map $A\,:\, Q(3)\to Z$ of scaled simplicial sets. We notice that $A(\Delta ^{\{00,10,12\}})$ and $A(\Delta ^{\{01,11,13\}})$ are thin triangles since $d_1d_3(A)=d_1(L)=s_0(N_1)$ and $d_1d_0(A)=d_1(R)=s_0(N_0)$ are degenerate. Furthermore, we see that the edges $A(\Delta ^{\{00,01\}})$ and $A(\Delta ^{\{10,11\}})$ are equivalences by applying $q_3$ to $A$ .

We would like to show that $M(\Delta ^{\{00,10,11\}})=A(\Delta ^{\{01,11,12\}})$ is thin. We consider the restriction of $A$ to the $3$ -simplex $\Delta ^{\{00,01,11,12\}}$ . In order to show that $A(\Delta ^{\{01,11,12\}})$ is thin, it suffices to show that $A(\Delta ^{\{00,11,12\}})$ is thin by [Reference Gagna, Harpaz and Lanari6, Proposition 3.4(1)] since the edge $A(\Delta ^{\{00,01\}})$ is an equivalence and the triangle $A(\Delta ^{\{00,01,11\}})$ is thin.

First, we will show that $A(\Delta ^{\{00,11,13\}})$ is thin. We consider the restriction of $A$ to the $3$ -simplex $\Delta ^{\{00,01,11,13\}}$ . Since $A(\Delta ^{\{01,11,13\}}), A(\Delta ^{\{00,01,13\}}), A(\Delta ^{\{00,01,11\}})$ are thin, so is $A(\Delta ^{\{00,11,13\}})$ by [Reference Lurie17, Remark 3.1.4]. Next, we consider the restriction of $A$ to the $4$ -simplex $\Delta ^{\{00,10,11,12,13\}}$ . We notice that the trivial cofibrations are invariant under taking opposites of simplicial sets since the bicategorical equivalences are invariant (cf. [Reference Gagna, Harpaz and Lanari6, Remark 1.32]). A lifting against the opposite of An2, which is a trivial cofibration, implies that $A(\Delta ^{\{00,10,11\}})$ is thin. Finally, we consider the restriction of $A$ to the $3$ -simplex $\Delta ^{\{00,10,11,12\}}$ . Since $A(\Delta ^{\{00,10,12\}}), A(\Delta ^{\{00,10,11\}}), A(\Delta ^{\{10,11,12\}})$ are thin and the edge $A(\Delta ^{\{10,11\}})$ is an equivalence, $A(\Delta ^{\{00,11,12\}})$ is thin by [Reference Gagna, Harpaz and Lanari6, Proposition 3.4(2)].

Proof. It suffices to construct $L,R\in \mathrm{TW}^r(Z)_2$ such that $d_0(L)=M, d_1(L)=s_0(N_1)$ and $d_1(R)=s_0(N_0), d_2(R)=M$ . We recall that we have a Cartesian fibration $p\,:\, \textrm {Tw}^r(Z)\to u_1Z\times u_1Z^{\textrm {op}}$ of $\infty$ -categories by [Reference García and Stern1, Theorem 0.1]. Using the fact that the set of $n$ -simplices of $\textrm {Tw}^r(Z)$ is $\textrm {Hom}_{\textrm {Set}_{\Delta }^{\textrm {sc}}}(Q(n),Z)$ , we regard a $0$ -simplex $X\in \mathrm{TW}^r(Z)_n=\textrm {Map}^{\textrm {sc}}(Q(n),Z)$ as an $n$ -simplex $\overline {X}\in \textrm {Tw}^r(Z)_n$ . In particular, we regard $M\in \mathrm{TW}^r(Z)_1=\textrm {Map}^{\textrm {sc}}(Q(1),Z)$ as a morphism $\overline {M}$ of $\textrm {Tw}^r(Z)$ . By [Reference García and Stern1, Theorem 2.6], a morphism in $\textrm {Tw}^r(Z)$ is $p$ -Cartesian if and only if it extends to a map from $(\Delta ^1\star \Delta ^{1,\textrm {op}})_{\sharp }$ . Thus, $\overline {M}$ is a $p$ -Cartesian morphism of $\textrm {Tw}^r(Z)$ . The assumption that $M(\Delta ^{\{00,01\}})$ and $M(\Delta ^{\{11,10\}})$ are equivalence $1$ -morphisms in $Z$ implies that $\overline {M}$ is an equivalence in $\textrm {Tw}^r(Z)$ by [Reference Lurie15, Proposition 2.4.1.5]. Hence there exist $\overline {L},\overline {R}\in \textrm {Tw}^r(Z)_2$ such that $d_0(\overline {L})=\overline {M}, d_1(\overline {L})=s_0(\overline {N}_1)$ and $d_1(\overline {R})=s_0(\overline {N}_0), d_2(\overline {R})=\overline {M}$ . By using the isomorphism of simplicial sets $i_1^*\mathrm{TW}^r(Z)\cong \textrm {Tw}^r(Z)$ , we obtain the desired $L,R\in \mathrm{TW}^r(Z)_2$ .

Proof of Proposition 4.11. The proposition follows from Lemmas 4.12 and 4.13.

Proof. This follows from the fact that it is obtained by iterated pushouts along scaled anodyne maps of type An1.

Proof. There are canonical maps $Q(1)\to P(1)\to \widetilde {P}(1)$ . We denote by $\widetilde {d}{}^0\,:\, Q(0)\to \widetilde {P}(1)$ the composite of these maps with $d^0\,:\, Q(0)\to Q(1)$ . In order to prove the lemma, it suffices to show that $\widetilde {d}^0$ is a trivial cofibration since the composite $\widetilde {s}^0\widetilde {d}^0$ is the identity of $Q(0)$ .

We set

\begin{equation*} \textrm {Sp}_1(\widetilde {P})= \Delta ^0_{\sharp }\coprod _{\Delta ^{\{00,01\}}_{\sharp }} \textrm {Sp}_1(P) \coprod _{\Delta ^{\{10,11\}}_{\sharp }}\Delta ^0_{\sharp }.\end{equation*}

By Lemma 4.14, we obtain a scaled anodyne map $\textrm {Sp}_1(\widetilde {P})\to \widetilde {P}(1)$ . The desired result follows from the fact that the map $d^0$ induces an isomorphism $Q(0)\cong \textrm {Sp}_1(\widetilde {P})$ of scaled simplicial sets.

Proof of Theorem 4.10. We would like to show that (4.2) is a pullback diagram in the $\infty$ -category of $\infty$ -groupoids. First, we note that the map $q_n\,:\, \mathrm{TW}^r(Z)_n\to (v_1Z)_n\times (v_1^{\textrm {rev}}Z)_n$ is a Kan fibration between Kan complexes for any $n\ge 0$ by Proposition 3.5. By using the fact that $W_1^{\textrm {eq}}$ is a full subspace of the Kan complex $W_1$ for a Segal space $W$ , we see that $q_1^{\textrm {eq}}\,:\, \mathrm{TW}^r(Z)_1^{\textrm {eq}}\to (v_1Z)_1^{\textrm {eq}}\times (v_1^{\textrm {rev}}Z)_1^{\textrm {eq}}$ is also a Kan fibration between Kan complexes. Thus, in order to prove that (4.2) is a pullback diagram, it suffices to show that the induced map $\mathrm{TW}^r(Z)_{0,(x,y)}\to \mathrm{TW}^r(Z)_{1,(s_0(x),s_0(y))}^{\textrm {eq}}$ on fibers is an equivalence for any $(x,y)\in (v_1Z)_0\times (v_1^{\textrm {rev}}Z)_0$ .

The map $\widetilde {s}^0\,:\, \widetilde {P}(1)\to Q(0)$ induces a map $s_0\,:\, \mathrm{TW}^r(Z)_0\to \textrm {Map}^{\textrm {sc}}(\widetilde {P}(1),Z)$ , which is an equivalence by Lemma 4.15. By composition with the inclusion map $\Delta ^0_{\sharp }\coprod \Delta ^0_{\sharp } \to \widetilde {P}(1)$ , we obtain a Kan fibration between Kan complexes $\widetilde {q}_1\,:\, \textrm {Map}^{\textrm {sc}}(\widetilde {P}(1),Z) \to (v_1Z)_0\times (v_1^{\textrm {rev}}Z)_0$ by Proposition 3.5, which makes the following diagram commute

By taking fibers over $(x,y)\in (v_1Z)_0\times (v_1^{\textrm {rev}}Z)_0$ , we obtain the desired equivalence.

Proof. This follows by observing that the $n$ th latching object is isomorphic to $\Omega (\Delta ^1\times \partial \Delta ^n)_{\S }$ .

Proof. First, we show that the inclusion map $\Lambda ^n_iT\to \overline {\Lambda }{}^n_iT$ is scaled anodyne. There is a pushout diagram

of scaled simplicial sets. We note that there is a natural isomorphism of scaled simplicial sets $\Omega (\partial \Delta ^1\times K)_{\S }\cong K_{\sharp }\coprod K_{\sharp }^{\textrm {op}}$ for any subcomplex $K$ of $\Delta ^n$ . Since the top horizontal arrow is scaled anodyne by [Reference Lurie17, Remark 3.1.5], so is the bottom horizontal arrow.

Next, we show that the inclusion map $\overline {\Lambda }{}^n_iT \to T(n)$ is scaled anodyne. For $0\le r\le n$ , we recall that $\sigma (r)$ is the $(n+1)$ -dimensional simplex $\Delta ^{\{00,\ldots ,0r,1r,\ldots ,1n\}}$ of $\Delta ^n\star \Delta ^{n,\textrm {op}}$ . For $0\le s\le n$ , we define a scaled simplicial set $X(s)$ , which is a subcomplex of $\Omega ^n$ equipped with the induced scaling from $T(n)$ , by

\begin{equation*} X(s)= \overline {\Lambda }{}^n_iT \cup \bigcup _{s\le r\le n}\ \sigma (r)_{\S }.\end{equation*}

By definition, we obtain a filtration

\begin{equation*} \overline {\Lambda }{}^n_iT= X(n+1)\to X(n)\to X(n-1)\to \cdots \to X(0) = T(n)\end{equation*}

of scaled simplicial sets, which consists of subcomplexes of $T(n)$ . In order to prove that $\overline {\Lambda }{}^n_iT \to T(n)$ is scaled anodyne, it suffices to show that the inclusion map $X(s+1)\to X(s)$ is scaled anodyne for any $0\le s\le n$ .

We fix $s$ with $0\le s\le n$ and show that $X(s+1)\to X(s)$ is scaled anodyne. We set $I(s)=\{00,\ldots ,0s,1s,\ldots ,1n\}$ . Then we have $\sigma (s)=\Delta ^{I(s)}$ . The intersection of $\sigma (s)$ with the underlying simplicial set of $X(s+1)$ has the form $\Lambda ^{I(s)}_{M(s)}$ , where

\begin{equation*} M(s)=\left \{\begin{array}{ll} \{1i\} & (s=0),\\ \{0s,1i\} & (0\lt s\lt i),\\ \{0i\} & (s=i),\\ \{0i,0s\} & (i\lt s\le n).\\ \end{array} \right . \end{equation*}

Hence, there is a pushout diagram

Thus, it suffices to show that the inclusion map $\Lambda ^{I(s)}_{M(s),\S } \to \Delta ^{I(s)}_{\S }$ is scaled anodyne. This follows from Lemma 3.7.

Proof. It suffices to show that the co-Segal map

\begin{equation*} T(1)\coprod _{T(0)}\cdots \coprod _{T(0)}T(1)\longrightarrow T(n) \end{equation*}

is scaled anodyne for all $n\ge 0$ . This follows by induction on $n$ together with Lemma 5.5.

Proof. This follows from Lemmas 4.6, 5.4, and 5.6.

Proof. The map $T(\!\bullet\! )\to Q(\!\bullet\! )$ of cosimplicial scaled simplicial sets induces a map of Segal spaces $\mathrm{TW}^r(Z) \to \mathbb{T}\mathrm{W}^r(Z)$ , which is a levelwise weak equivalence by Proposition 5.8. The theorem follows from the fact that $\mathrm{TW}^r(Z)$ is a complete Segal space by Theorem4.10.

Proof of Proposition 5.8. For $0\le k\le n$ , we define a scaled simplicial set $U(k)$ , which is a subcomplex of $\Delta ^n\star \Delta ^{n,\textrm {op}}$ equipped with the induced scaling from $Q(n)$ , by

\begin{equation*} U(k)=T(n)\cup \bigcup _{k\le i\le n} \tau (i,i+1)_{\S }. \end{equation*}

By definition, we obtain a filtration $T(n)= U(n)\to \cdots \to U(0)=Q(n)$ of scaled simplicial sets consisting of subcomplexes of $Q(n)$ . In order to prove Proposition 5.8, it suffices to show that $U(k+1)\to U(k)$ is scaled anodyne for all $0\le k\lt n$ .

For this purpose, we introduce a further filtration. For $k+1\le l\le n+1$ , we define a scaled simplicial set $V(k,l)$ , which is a subcomplex of $\Delta ^n\star \Delta ^{n,\textrm {op}}$ equipped with the induced scaling from $Q(n)$ , by

\begin{equation*} V(k,l)=U(k+1)\cup \bigcup _{l\le j\le n+1}\tau (k,j)_{\S }. \end{equation*}

By definition, we obtain a filtration $U(k+1)=V(k,n+1)\to \cdots \to V(k,k+1)=U(k)$ of scaled simplicial sets consisting of subcomplexes of $U(k)$ . In order to prove that $U(k+1)\to U(k)$ is scaled anodyne, it suffices to show that $V(k,j+1)\to V(k,j)$ is scaled anodyne for all $k+1\le j\le n$ .

We set $J(k,j)=\{00,\ldots ,0k,0j,\ldots ,0n,1k,\ldots ,1n\}$ . Then we have $\tau (k,j)=\Delta ^{J(k,j)}$ . The intersection of $\tau (k,j)$ with the underlying simplicial set of $V(k,j+1)$ has the form $\Lambda ^{J(k,j)}_{M(k,j)}$ , where

\begin{equation*} M(k,j)=\left \{\begin{array}{ll} \{00,\ldots ,0k, 1(k+1),\ldots ,1n\}&(j=n),\\\{00,\ldots ,0k, 1(k+1),\ldots ,1n, 0(j+1),\ldots ,0n\}&(k+1\le j\lt n).\\\end{array}\right .\end{equation*}

Hence, there is a pushout diagram of scaled simplicial sets

\begin{equation*} \begin{array}{ccc} \Lambda ^{J(k,j)}_{M(k,j),\S } & \longrightarrow & \Delta ^{J(k,j)}_{\S }\\ \bigg \downarrow & & \bigg \downarrow \\ V(k,j+1) & \longrightarrow & V(k,j).\\ \end{array}\end{equation*}

Since the top horizontal arrow is scaled anodyne by Lemma 3.7, so is the bottom horizontal arrow.

Proof. First, we show that $q_{\mathbb{T}}\,:\, \mathbb{T}\mathrm{W}^r(Z)\to v_1Z\times v_1^{\textrm {rev}}Z$ is a fibration between fibrant objects in $\textrm {bSet}_{\Delta }^{\textrm {CSS}}$ . Recall that the model category $\textrm {bSet}_{\Delta }^{\textrm {CSS}}$ is a left Bousfield localization of the Reedy model structure on $\textrm {bSet}_{\Delta }$ . Since $\mathbb{T}\mathrm{W}^r(Z)$ and $v_1Z\times v_1^{\textrm {rev}}Z$ are complete Segal spaces, in order to show that $q_{\mathbb{T}}$ is a fibration in $\textrm {bSet}_{\Delta }^{\textrm {CSS}}$ , it suffices to show that $q_{\mathbb{T}}$ is a Reedy fibration (cf. [Reference Hirschhorn11, Proposition 3.3.16]).

For a bisimplicial set $X$ , we denote by $M_nX$ the $n$ th matching object. We set $V= v_1Z\times v_1^{\textrm {rev}}Z$ for simplicity. We have to show that the map $f_n\,:\, \mathbb{T}\mathrm{W}^r(Z)_n\to M_n\mathbb{T}\mathrm{W}^r(Z) \times _{M_nV}V_n$ is a Kan fibration for all $n\ge 0$ . We observe that $f_n$ is isomorphic to the map $\textrm {Map}^{\textrm {sc}}(T(n),Z)\to \textrm {Map}^{\textrm {sc}}(\overline {\partial }T(n),Z)$ induced by the inclusion map $\overline {\partial }T(n)\to T(n)$ , where $\overline {\partial }T(n)$ is a subcomplex of the scaled simplicial set $T(n)$ given by

\begin{equation*} \overline {\partial }T(n)= \Omega ((\Delta ^1\times \partial \Delta ^n)\cup (\partial \Delta ^1\times \Delta ^n))_{\S }. \end{equation*}

By Proposition 3.5, we see that $f_n$ is a Kan fibration.

Thus, $q_{\mathbb{T}}$ is a fibration between fibrant objects in $\textrm {bSet}_{\Delta }^{\textrm {CSS}}$ . By applying the right Quillen equivalence $i_1^*\,:\, \textrm {bSet}_{\Delta }^{\textrm {CSS}}\to \textrm {Set}_{\Delta }^{\textrm {Joy}}$ , we see that $p_{\mathbb{T}}$ is a categorical fibration between $\infty$ -categories.

Proof. We recall that $\sigma (r)=\Delta ^{\{00,\ldots ,0r,1r,\ldots ,1n\}}$ for $0\le r\le n$ . For $0\le s\le n$ , we define a scaled simplicial set $X(s)_{\textrm {cart}}$ , which is a subcomplex of $\Omega ^n$ equipped with the induced scaling from $T(n)_{\textrm {cart}}$ , by

\begin{equation*} X(s)_{\textrm {cart}}=\overline {\Lambda }^n_nT_{\textrm {cart}}\cup \bigcup _{s\le r\le n} \sigma (r)_{\textrm {cart}}.\end{equation*}

By definition, we obtain a filtration $\overline {\Lambda }^n_nT_{\textrm {cart}} = X(n+1)_{\textrm {cart}} \to \cdots \to X(0)_{\textrm {cart}}=T(n)_{\textrm {cart}}$ of scaled simplicial sets consisting of subcomplexes of $T(n)_{\textrm {cart}}$ . In order to prove that $\overline {\Lambda }^n_nT_{\textrm {cart}}\to T(n)_{\textrm {cart}}$ is scaled anodyne, it suffices to show that $X(s+1)_{\textrm {cart}}\to X(s)_{\textrm {cart}}$ is scaled anodyne for all $0\le s\le n$ .

We use the notation in the proof of Lemma 5.5. We recall that $I(s)=\{00,\ldots ,0s,1s,\ldots ,1n\}$ for $0\le s\le n$ . The intersection of $\Delta ^{I(s)}$ with the underlying simplicial set of $X(s+1)_{\textrm {cart}}$ has the form $\Lambda ^{I(s)}_{N(s)}$ , where

\begin{equation*} N(s)= \left \{ \begin{array}{ll} \{1n\} & (s=0),\\\{0s,1n\} & (0\lt s\lt n),\\\{0n\}& (s=n).\\ \end{array}\right .\end{equation*}

Hence, there is a pushout diagram of scaled simplicial sets

By using the fact that $\Delta ^{\{0(n-1),0n,1n\}}$ and $\Delta ^{\{0i,1(n-1),1n\}}$ are thin for $0\le i\lt n$ , we see that the top horizontal arrow is scaled anodyne by Lemma 3.7. Hence, so is the bottom horizontal arrow.

Proof. When $n=2$ , it is the identity map. When $n\ge 3$ , the desired result follows by observing that $(\Delta ^{\{0i,0(n-1),1(n-1),1n\}},T) \to \Delta ^{\{0i,0(n-1),1(n-1),1n\}}_{\sharp }$ is scaled anodyne for $0\le i\lt n-1$ by [Reference Lurie17, Remark 3.1.4], where $T$ is the set of all $2$ -simplices of $\Delta ^{\{0i,0(n-1),1(n-1),1n\}}$ other than $\Delta ^{\{0i,1(n-1),1n\}}$ .

Proof. Let $M\in \mathbb{T}\mathrm{w}^r(Z)_1$ be a morphism of the $\infty$ -category $\mathbb{T}\mathrm{w}^r(Z)$ given by a map $\overline {M}\,:\, T(1)\to Z$ of scaled simplicial sets. We consider the following commutative diagram of simplicial sets

for $n\ge 2$ . By [Reference Lurie15, Remark 2.4.1.4], if there exists a lifting $\Delta ^n\to \mathbb{T}\mathrm{w}^r(Z)$ that makes the diagram commute, then $M$ is a $p_{\mathbb{T}}$ -Cartesian edge. Unwinding the definitions, we see that this is equivalent to the following lifting problem

in $\textrm {Set}_{\Delta }^{\textrm {sc}}$ .

If $\overline {M}$ factors through $T(1)_{\textrm {cart}}$ , then the map $\overline {\Lambda }^n_nT\to Z$ extends to a map $\overline {\Lambda }^n_nT'\to Z$ . We obtain the following commutative diagram

(5.1)

By Lemmas 5.12 and 5.13, the middle vertical arrow in (5.1) is scaled anodyne. Since $Z$ is a fibrant object in $\textrm {Set}_{\Delta }^{\textrm {sc}}$ , the dotted arrow in (5.1) exists that makes the right triangle commute. The desired lifting is obtained as the composite of $T(n)_{\textrm {cart}}\to Z$ with the inclusion map $T(n)\to T(n)_{\textrm {cart}}$ .

Proof of Proposition 5.11. We consider a lifting problem depicted by the following commutative diagram

(5.2)

in $\textrm {Set}_{\Delta }$ . Unwinding the definitions, we see that this is equivalent to the following lifting problem

in $\textrm {Set}_{\Delta }^{\textrm {sc}}$ . We would like to have a lifting $\overline {M}\,:\, T(1)\to Z$ which factors through $T(1)_{\textrm {cart}}$ .

Since the inclusion map $\overline {\Lambda }^1_1T\to \overline {\Lambda }^1_1T_{\textrm {cart}}$ is the identity, we can factor the map $\overline {\Lambda }^1_1T\to Z$ through $\overline {\Lambda }^1_1T_{\textrm {cart}}$ and we obtain the following commutative diagram

(5.3)

The middle vertical arrow in (5.3) is scaled anodyne by Lemma 5.12. Since $Z$ is a fibrant object in $\textrm {Set}_{\Delta }^{\textrm {sc}}$ , we obtain the dotted arrow $T(1)_{\textrm {cart}}\to Z$ in (5.3) that makes the right triangle commute. The composite $\overline {M}\,:\, T(1)\to T(1)_{\textrm {cart}}\to Z$ gives a lifting $M$ in (5.2). Since $\overline {M}$ factors through $T(1)_{\textrm {cart}}$ , the morphism $M$ is $p_{\mathbb{T}}$ -Cartesian by Corollary 5.14.

Proof. The commutative diagram is obtained from the inclusion maps $\Delta ^{\bullet }_{\sharp }\coprod \Delta ^{\bullet ,\textrm {op}}_{\sharp } \to T(\!\bullet\! )\to Q(\!\bullet\! )$ of cosimplicial scaled simplicial sets by applying $\textrm {Hom}_{\textrm {Set}_{\Delta }^{\textrm {sc}}}(-,Z)$ . The map $T(\!\bullet\! )\to Q(\!\bullet\! )$ induces an equivalence of complete Segal spaces $\mathrm{TW}^r(Z)\to \mathbb{T}\mathrm{W}^r(Z)$ by Proposition 5.8. By applying the right Quillen equivalence $i_1^*\,:\, \textrm {bSet}_{\Delta }^{\textrm {CSS}}\to \textrm {Set}_{\Delta }^{\textrm {Joy}}$ , we see that $\textrm {Tw}^r(Z)\to \mathbb{T}\mathrm{w}^r(Z)$ is an equivalence of $\infty$ -categories.