Notation. By a 2-category, we will always mean a strict 2-category. By a 2-functor, we will mean a strict 2-functor unless specified otherwise. We will denote strict 2-categories by blackboard bold letters, e.g. $\mathbb {D}$ .

A 2-category $\mathbb {D}$ has three duals, determined by reversing the direction of 1-morphisms, 2-morphisms, or both, respectively. In this work, we will primarily make use of the 1-morphism dual. To better accord with the notation used in (scaled) simplicial sets, we will denote the 1-morphism dual simply by $\mathbb {D}^{\operatorname {op}}$ . Where needed, we denote the 2-morphism dual by $\mathbb {D}^{(-,{\operatorname {op}})}$ , and the dual which reverses both 1- and 2-morphisms by $\mathbb {D}^{({\operatorname {op}},{\operatorname {op}})}$ . We will denote the 1-category of 2-categories and strict 2-functors by $2\!\operatorname {Cat}$ .

Notation. We denote by $\operatorname {Cat}_\Delta ^+$ the category of ${{\operatorname {Set}}_{\Delta }^+}$ -enriched categories.

Proof. This is a special case of [Reference Lurie23, A.3.2.4].

Notation. We will sometimes make use of subscripts to denote scalings. In particular, $(X,X_2)$ will be will be denoted by $X_\sharp $ , and $(X,\operatorname {deg}(X_2))$ will be denoted by $X_\flat $ . Similarly, we will sometimes use superscripts to denote markings on a simplicial set: $(X,X_1)=X^\sharp $ , and $(X,\operatorname {deg}(X_1))=X^\flat $ .

Proof. This is [Reference Lurie25, Thm A.3.2.4]. The characterisation of fibrant objects is[Reference Gagna, Harpaz and Lanari11, Thm 5.1].

Notation. Let $(X,E_X, T_X \subseteq C_X)$ be a mb simplicial set. Suppose that the collection $E_X$ consist only of degenerate edges. Then we fix the notation $(X,E_X, T_X \subseteq C_X)=(X,\flat ,T_X \subseteq E_X)$ and do similarly for the collection $T_X$ . If $C_X$ consists only of degenerate triangles we fix the notation $(X,E_X, T_X \subseteq C_X)=(X,E_X, \flat )$ . In an analogous fashion we wil use the symbol “ $\sharp $ ” to denote a collection containing all edges (resp. all triangles). Finally suppose that $T_X=C_X$ then we will employ the notation $(X,E_X,T_X)$ .

Proof. This is [Reference Abellán García and Stern5, Cor. 3.17].

Proof. See Proposition 3.38 in [Reference Abellán García and Stern5].

Proof. See [Reference Lurie25, Rmk 3.1.4].

Proof. The proof follows exactly as in [Reference Abellán García and Stern5, Lem. 3.7].

Proof. Every case is, mutatis mutandis, the same as the corresponding case in the proof of [Reference Abellán García and Stern5, Prop. 3.10].

Proof. It is immediate that $(-)^\flat $ preserves cofibrations. Suppose that $f:(X,T_X)\to (Y,T_Y)$ is a weak equivalence. Let $(Z,E_Z,T_Z)$ be a fibrant object in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ . It is easy to see that $G(Z,E_Z,T_Z)=(Z,T_Z)$ is a fibrant object in ${\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ . We can then note that, by definition, there is an isomorphism of mapping scaled simplicial sets

$$\begin{align*}\operatorname{Map}_{\mathbf{sc}}((X,T_X),(Z,T_Z))\cong \operatorname{Map}_{\mathbf{ms}}((X,\flat,T_X),(Z,E_Z,T_Z)). \end{align*}$$

Thus, since f induces a bicategorical equivalence

$$\begin{align*}\operatorname{Map}_{\mathbf{sc}}((Y,T_Y),(Z,T_Z))\to \operatorname{Map}_{\mathbf{sc}}((X,T_X),(Z,T_Z)) \end{align*}$$

we see that the map

$$\begin{align*}\operatorname{Map}_{\mathbf{ms}}((Y,\flat,T_Y),(Z,E_Z,T_Z))\to \operatorname{Map}_{\mathbf{ms}}((X,\flat,T_X),(Z,E_Z,T_Z)) \end{align*}$$

induced by $(f)^\flat $ is also an equivalence. We therefore see that $(f)^\flat $ is a weak equivalence in ${{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ , as desired.

Proof. If, for any $\infty $ -bicategory $(Z,T_Z)$ , there exists a set $E_Z$ of marked edges for Z such that $(Z,E_Z,T_Z)$ is a fibrant marked-scaled simplicial set, then this follows from the characterisation in terms of mapping $\infty $ -bicategories.

To see that this is the case, let $(Z,T_Z)$ be an $\infty $ -bicategory. Then $Z^{\operatorname {th}}$ is an $\infty $ -category, and so we can define a marking $E_Z$ on Z by declaring an edge to be marked if it lies in the maximal Kan complex in $Z^{\operatorname {th}}$ . From the definition, it is immediate that $(Z,E_Z,T_Z)$ has the extension property with respect to (MS1), (MS2),(MS3),(MS5),(MS6) and (MSE).

It follows from [Reference Abellán García and Stern5, Cor 4.20] and [Reference Abellán García and Stern5, Cor 4.23] that $Z\to \Delta ^0$ is a 2-Cartesian fibration in which the strongly Cartesian edges are precisely the equivalences, and so we see that $(Z,E_Z,T_Z)$ has the extension property with respect to (MS4), (MS7) and (MS8) as well.

Proof. It is immediate from the definitions that $(Y^{\operatorname {th}},E_Y)$ is a fibrant marked simplicial set, and the lemma follows.

Proof. By [Reference Abellán García and Stern5, Thm 3.39], the composite adjunction $D\circ (-)^\flat \dashv G\circ R$ is a Quillen equivalence. It thus suffices for us to check that the adjunction $(-)^{\flat }\dashv G$ is a Quillen equivalence. We will check explicitly that the derived adjunction unit and counit are equivalences.

First, let $(X,T_X)\in {\operatorname {Set}}_\Delta ^{\mathbf {sc}}$ . The derived adjunction unit on $(X,T_X)$ is the composite

$$\begin{align*}(X,T_X) \to G(X,\flat,T_X) \to G((X,\flat,T_X)^{\operatorname{fib}}) \end{align*}$$

where the superscript $\operatorname {fib}$ denotes fibrant replacement. The first of these maps is the identity (since $G(X,\flat ,T_X)=(X,T_X)$ ) and the latter is the image under G of an equivalence of marked-scaled simplicial sets. By Lemma 2.50, this is an equivalence.

Now, let $(Y,E_Y,T_Y)\in {{\operatorname {Set}}^{\mathbf {ms}}_{\Delta }}$ be a fibrant object. The derived adjunction counit on $(Y,E_Y,T_Y)$ is the composite

Since every scaled simplicial set is cofibrant, the first map is an isomorphism, leaving us to check that the usual adjunction counit $\eta _Y$ is an equivalence. Note that $\eta _Y$ is simply the inclusion $(Y,\flat ,T_Y)\to (Y,E_Y,T_Y)$ .

We have a pushout square

and, by Lemma 2.51 the morphism $\psi $ is a morphism in $\mathbf {MS}$ of type (MSE). Thus, $\eta _Y$ is $\mathbf {MS}$ -anodyne, and is a weak equivalence.

Proof. All three statements hold on the level of $\operatorname {St}_\phi ^+$ , and so the proof amounts to checking scalings. We prove (1), and leave the other two statements to the reader.

It follows from the definition that

preserves colimits. Since colimits in functor categories are computed pointwise, it will thus suffice to show that, given a diagram

$$\begin{align*}D \colon I\to ({\operatorname{Set}}_\Delta^{\mathbf{mb}})_{/S} \end{align*}$$

the scalings on $\operatorname *{\mathrm {colim}}_I {\mathbb {S}\text {t}}_\phi (D(i))$ and ${\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i))$ coincide. Indeed, applying the universal property, it will suffice to show that the map

$$\begin{align*}{\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)) \to \operatorname*{\mathrm{colim}}_I {\mathbb{S}\text{t}}_\phi(D(i)) \end{align*}$$

which is the identity on underlying marked simplicial sets preserves the scalings.

Fix , we will first show that the map

$$\begin{align*}f_s\colon ({\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)))(s) \to (\operatorname*{\mathrm{colim}}_I {\mathbb{S}\text{t}}_\phi(D(i)))(s) \end{align*}$$

preserves the scalings inherited from $P_{X^\triangleright }^s$ . To this end, suppose given a thin simplex $\sigma $ in $P_{({\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i)))_S}^s$ which does not come from a lean simplex in the colimit. Tracing through the definition, we note that there must be a simplex $\eta :\Delta ^n\to \operatorname *{\mathrm {colim}}_I(D(i))$ and a simplex $\mu :=\{S_0\cup \{n+1\}\to S_1\cup \{n+1\}\to S_2\cup \{n+1\}\}$ in ${\mathfrak {C}}^{\operatorname {sc}}[\Delta ^{n+1}](0,n+1)$ with $\max (S_i)=\max (S_j)$ for some $i,j=0,1,2$ such that $\sigma $ is the image of $\mu $ under the canonical map

$$\begin{align*}g_1\colon{\mathfrak{C}}^{\operatorname{sc}}[\Delta^{n+1}](0,n+1)\to ({\mathbb{S}\text{t}}_\phi(\operatorname*{\mathrm{colim}}_I D(i)))(s) \end{align*}$$

is not scaled.

By the construction of colimits in simplicial sets, this means that there is a $k\in I$ and a simplex $\hat {\eta }:\Delta ^n\to D(k)$ such that $\eta $ factors through the canonical map $D(k)\to \operatorname *{\mathrm {colim}}_I D(i)$ as $\hat {\eta }$ . We can then note that $\hat {\eta }$ will yield a map

$$\begin{align*}g_2\colon{\mathfrak{C}}^{\operatorname{sc}}[\Delta^{n+1}](0,n+1)\to {\mathbb{S}\text{t}}_\phi(D(k))(s)\to (\operatorname*{\mathrm{colim}}_I{\mathbb{S}\text{t}}_\phi(D(i)))(s) \end{align*}$$

such that the diagram

commutes. We thus see that $g_2(\mu )=f_s(g_1(\mu ))=f_s(\sigma )$ is scaled, as desired. The same argument holds, mutatis mutandis, for $\sigma \in P_{({\mathbb {S}\text {t}}_\phi (\operatorname *{\mathrm {colim}}_I D(i)))_S}^s$ coming from a lean 2-simplex in the colimit.

We can now easily check that the full scalings $T_{X_S}^s$ are preserved by $f_s$ by simply noting that the diagram

commutes.

To show (2) and (3), we again note that the statements are immediate if we replace ${\mathbb {S}\text {t}}_\phi $ with $\operatorname {St}_\phi ^+$ (cf. [Reference Lurie25, Rmk 3.5.16] and [Reference Lurie23, Prop 3.2.1.4]). A similar check to the above assures us that the scalings coincide.

Proof. This follows from the first part in Proposition 3.8 using the adjoint functor theorem.

Proof. The existence of the morphism is clear from the definitions. Moreover, if T is given by the minimal lean-scaling the map is in fact an isomorphism.

Suppose that we are given a thin 2-simplex $\sigma :S_0 \subseteq S_1 \subseteq S_2$ in $\mathcal {L}^n_{T}(i)$ . As before, we adopt the convention that $s_i:=\max (S_i)$ . We will show that $\sigma $ can be scaled by taking pushouts along $\mathbf {MS}$ -anodyne morphisms. First let us consider the 3-simplex

We immediately observe that all of its faces are scaled in $\mathbb {S}\!{\operatorname {t}}_{\Delta ^n_\flat }\!( \Delta ^n)_T(s)$ except the face missing . It follows we can scale the remaining face using a pushout along a $\mathbf {MS}$ -anodyne map of the type described in Lemma 2.42. Now we consider another 3-simplex

Again we observe that all of its faces are scaled except possibly the face missing which is precisely $\sigma $ . The conclusion easily follows from Lemma 2.42

Proof. The existence of the natural transformation is automatic since both functors only differ on the scaling. It is clear that both functors preserve colimits and that they satisfy base change with respect to morphisms of simplicial sets $S \to T$ (see Proposition 3.8). In addition, it is routine to verify that both functors respect cofibrations. A standard argument then shows that it suffices to check that the natural transformation is an equivalence (1) when $S=(\Delta ^n)^{\flat }$ with $n\geqslant 0$ and $X \to S$ is the identity morphism, and (2) on $(\Delta ^1)^{\sharp } \to \Delta ^1$ when $S=\Delta ^1$ . This is a direct consequence of Lemma 3.12.

Proof. The generators of the class of cofibrations of marked biscaled simplicial sets are given by

1. (C1) $\left (\partial \Delta ^n,\flat ,\flat \right ) \rightarrow \left ( \Delta ^n,\flat ,\flat \right )$ .

2. (C2) $\left (\Delta ^1,\flat ,\flat \right ) \rightarrow \left (\Delta ^1,\sharp ,\flat \right ) $ .

3. (C3) $\left (\Delta ^2,\flat ,\flat \right ) \rightarrow \left (\Delta ^2,\flat ,\flat \subset \sharp \right )$ .

4. (C4) $\left (\Delta ^2,\flat ,\flat \subset \sharp \right ) \rightarrow \left ( \Delta ^2,\flat ,\sharp \right )$ .

Note that (C4) and (S2) are the same morphism. Therefore using standard arguments it will suffice to check our claim on those generators.

Let $i:A \to B$ be a cofibration. As stated above it will suffice to check in the case where i is one of the generating cofibrations. Furthermore we can use Proposition 3.8 to reduce to the case where S is the underlying scaled simplicial set of B, and $\phi $ is $\operatorname {id}:{\mathfrak {C}}^{{\operatorname {sc}}}[S]\to {\mathfrak {C}}^{{\operatorname {sc}}}[S]$ .

Let be a trivial fibration in $\left ({\operatorname {Set}}^{\mathbf {ms}}_{\Delta }\right )^{\mathfrak {C}[\mathcal {C}]^{\operatorname {op}}}$ . Then in particular for every $c \in \mathcal {C}$ the map $F(c) \to G(c)$ is a trivial fibration of marked-scaled simplicial sets and thus detects scalings and markings. This settles the cases (C2)-(C4). For the remaining case, we observe that

$$\begin{align*}{\mathbb{S}\text{t}}_{\Delta^n_\flat}(\partial\Delta^n)(i) \to {\mathbb{S}\text{t}}_{\Delta^n_\flat}(\Delta^n)(i) \end{align*}$$

is an isomorphism unless $i=0$ , in which case the map is a cofibration. The claim follows immediately.

Proof. The map $\xi ^r_C$ is clearly surjective on vertices. Moreover, given a morphism $U\to K$ in $\mathcal {E}_C$ , choose a marked morphism $U\to U^r$ to a rigid chain in $\mathcal {E}_C$ . Then for every , the object $K\cup \{(a,b)\}$ will lie in $\mathcal {E}_C$ over the same element of $\mathcal {U}_C$ as K. We thus obtain a morphism $U^r\to \hat {K}$ lying over the original morphism in $\mathcal {U}_C$ , showing that $\xi ^r_C$ is surjective on morphisms, and thus on higher simplices.

Moreover $\xi ^r_C$ detects and preserves marked edges and thin simplices. It will therefore suffice to show that $\xi _C^r$ is injective. Let $K_i \in \mathcal {E}^r_C$ for $i=1,2$ such that $\xi ^r_C(K_1)=\xi ^r_C(K_2)$ . Let us denote for $i=1,2$ . Without loss of generality let us assume that we have some $(a^1_s,b^1_s)$ such that this pair is not an element in $K_2$ . However, note that since $K_i \subset C$ for $i=1,2$ then there exists a map $K_2 \to \hat {K}_2$ where $\hat {K}_2$ is obtained from $K_2$ by appending the element $(a^1_s,b^1_s)$ . By construction it follows that $\xi ^r_C(K_2)=\xi ^r_C(\hat {K}_2)$ since $K_2$ is rigid it follows that $\hat {K}_2=K_2$ and therefore $K_1=K_2$ .

Proof. Let $\iota :\mathcal {E}_C^r \to \mathcal {E}_C$ denote the obvious inclusion. Using Lemma 3.21 we can construct a map $s_C=\iota \circ (\xi ^r_C)^{-1}$ . It is clear that $\xi _C \circ s_C=\operatorname {id}$ . Given K, observe that by construction $s_C\circ \xi _C(K)$ is rigid. Let $K \to K^r$ be a marked edge where $K^r$ is rigid. Since the restriction of $\xi _C$ to rigid objects is injective it follows that $s_C \circ \xi _C(K)= K^r$ . This yields a marked homotopy from $s_C \circ \xi _C$ to the identity and the result follows.

Proof. Immediate.

Proof. First let us observe that the map $\xi ^{n,k}_{(i,j)}$ is an isomorphism if either n or k is equal to $0$ . Using an inductive argument it will suffice to show that the map $\xi ^{n,k}_{(0,0)}$ is an equivalence. Note that we can cover $\mathbb {E}^{n,k}_{(i,j)}$ with the subsimplicial sets $\mathcal {E}_C$ where C is a chain of maximal length. Since $\xi ^{n,k}_{(0,0)}$ is surjective, $\mathbb {S}^{n,k}_{(0,0)}$ is covered by the subsimplicial sets $\mathcal {U}_C$ . Applying [Reference Abellán García, Dyckerhoff and Stern4, Lemma 3.2.13], we express $\mathbb {E}^{n,k}_{(0,0)}$ and $\mathbb {S}^{n,k}_{(0,0)}$ as the colimit over the same diagram of two homotopy cofibrant diagrams. We can now identify $\xi ^{n,k}_{(0,0)}$ as the map induced by the natural transformation whose components are $\xi _C$ . Therefore using Lemma 3.22 it follows that $\xi ^{n,k}_{(0,0)}$ is a weak equivalence.

Proof of Theorem 3.15.

Let us commence the proof by observing that since the morphism ${\mathfrak {C}}^{\operatorname {sc}}[A \times B] \to {\mathfrak {C}}^{\operatorname {sc}}[A] \times {\mathfrak {C}}^{\operatorname {sc}}[B]$ is a weak equivalence we can use [Reference Lurie23, Proposition A.3.3.8.] and (3) in Proposition 3.8 to reduce our problem to showing that the map

is a pointwise weak equivalence. As in [Reference Lurie23, 3.2.1.13], it suffices to check this in the special case when $X_A\to A$ and $X_B\to B$ are identity morphisms on underlying simplicial sets, and both A and B are one of the following cases

• • The scaled 2-simplex $\Delta ^2_\sharp $ .

• • The unscaled n-simplex $\Delta ^n_\flat $ .

In the case where $A=\Delta ^n_\flat $ and $B=\Delta ^k_\flat $ , the morphism

$$\begin{align*}\mathbb{S}\!{\operatorname{t}}_{A \times B}(\Delta^n\times \Delta^k)(i,j) \to \phi^*\left(\mathbb{S}\!{\operatorname{t}}_{\Delta^n}(\Delta^n)\boxtimes \mathbb{S}\!{\operatorname{t}}_{\Delta^k}(\Delta^k)\right)(i,j) \end{align*}$$

is precisely the morphism

$$\begin{align*}\xi^{n,k}_{(i,j)}\colon\mathbb{E}^{n,k}_{(i,j)} \to \mathbb{S}^{n,k}_{(i,j)} \end{align*}$$

and thus is an equivalence of marked-scaled simplicial sets by Proposition 3.24. Each other case is a pushout of some $\xi _{(i,j)}^{n,k}$ by a cofibration, and thus is also an equivalence.

Proof. It is clear that for every $0 \leqslant s \leqslant n$ the map in the statement is an isomorphism on the underlying simplicial sets since its just the restriction of the isomorphism described at the beginning of Lemma 3.12 to the corresponding horn. We further note that the proof of Lemma 3.12 still holds in this setting. Consequently, the claim follows.

Proof. We use the order from Definition 3.29 to add simplices to $\Lambda \mathcal {R}^n_0$ . We will add simplices in reverse order, i.e. for any path $\sigma $ , we denote by $X^{\geqslant \sigma }$ the scaled simplicial subset of $\mathcal {R}^n$ obtained by adding to $\Lambda \mathcal {R}^n_0$ all paths $\theta $ such that $\theta \geqslant \sigma $ .

The procedure yields a filtration

$$\begin{align*}\Lambda \mathcal{R}^n_0=X^{\geqslant \sigma_0} \to X^{\geqslant \sigma_1} \to X^{\geqslant \sigma_2} \to \cdots \to \mathcal{R}^n \end{align*}$$

where we have labeled our paths $\sigma _i$ so that $\sigma _i>\sigma _{i+1}$ . The proof proceeds by showing that $X^{\geqslant \sigma _{i-1}}\to X^{\geqslant \sigma _{i}}$ is scaled anodyne for any i.

The proof proceeds by cases. We fix the notation that $S_{\sigma _i}=\{a_k\}_{k=1}^n$ .

1. 1. Suppose that $a_1\neq 1$ . We prove this case by showing that the top horizontal map in the pullback diagram

   

   is itself scaled anodyne.

   We see that $A_{\sigma _i}$ is the union of the following faces of $\sigma _i$ :

   1. a) The face $d_0(\sigma _i)$ , since we will have $0\leqslant 1<a_1$ in each $S_k$ (see ii) in Definition 3.25).

   2. b) The face $d_n(\sigma _i)$ , since this face will always be missing $a_{n}\neq 0$ (see i) in Definition 3.25).

   3. c) The face $d_j(\sigma _i)$ for every j such that $a_{j+1}>a_j$ . This is because this will, equivalently, be the j ^th face of the (greater) simplex with vertex list

      $$\begin{align*}\{a_1,\ldots, a_{j-1},a_{j+1},a_j,a_{j+2},\ldots, a_n\}. \end{align*}$$

   To see that the inclusion of $A_{\sigma _i}\to \Delta ^n$ is scaled anodyne, we first note that, by necessity, there is at least one face of $\Delta ^n$ not contained in $A_{\sigma _i}$ . We then choose the biggest $t\in [n]$ such that $d_t(\Delta ^n)$ is not contained in $A_{\sigma _i}$ and note that $t<n$ . We wish to apply Lemma 2.36 with $\mathcal {A}$ given by the subsets of the form $\{0\},\{n\}$ and $\{j\}$ as in c) above, and where the pivot point is t. Note that by construction $\{t+1\} \in \mathcal {A}$ . We conclude after noting that $\Delta ^{\{s,t,t+1\}}$ is scaled for every $\{s\} \in \mathcal {A}$ , since $\max (S_t)=\max (S_{t+1})$ by maximality of t. Consequently, we see that $\alpha _i$ is scaled anodyne, as desired.

2. 2. Now suppose that $a_1=1$ , and $S_{\sigma _i}\neq \{1,2,\ldots ,n-1,n\}$ . We now must instead consider the pullback diagram

   
   as above, we see that $B_{\sigma _i}$ consists of the faces

   1. a) $d_n(\sigma _i)$

   2. b) $d_j(\sigma _i)$ for each j such that $a_{j+1}>a_j$ .

   Since $S_{\sigma _i}\neq \{1,2,\ldots ,n-1,n\}$ , there exists some $1<t<n$ such that $B_{\sigma _i}$ does not contain the t ^th face of $\Delta ^n\coprod _{\Delta ^{\{0,1\}}}\Delta ^0$ . We can apply again Lemma 2.36 with pivot point given the biggest t satisfying the conditions above and where $\mathcal {A}$ is given by $\{n\}$ and $\{j\}$ for elements j as in b).

3. 3. If $S_{\sigma _i}=\{1,2,\ldots ,n-1,n\}$ , then the map $X^{\geqslant \sigma _{i-1}}\to X^{\geqslant \sigma _i}=\mathcal {R}^n$ is obtained as a pushout along the inclusion

   $$\begin{align*}\Lambda^n_0\coprod_{\Delta^{\{0,1\}}}\Delta^0 \to \Delta^n\coprod_{\Delta^{\{0,1\}}}\Delta^0 \end{align*}$$
   where $\Delta ^{\{0,1,n\}}$ is scaled.

Proof. The argument is nearly identical to the proof of Lemma 3.30. Using the same order as in that proof, we produce a filtration

$$\begin{align*}\Lambda \mathcal{T}^n_n=X^{\geqslant \sigma_{0}} \to X^{\geqslant \sigma_{1}}\to \cdots \to \mathcal{T}^n \end{align*}$$

and show each step is scaled anodyne.

As before, we set $S_{\sigma _i}=\{a_j\}_{j=1^n}$ , and consider the pullback diagram

The case distinction now rests on whether or not $d_n\sigma _i$ factors through $A_{\sigma _i}$ . The case when it does is formally identical to case (1) from Lemma 3.30.

If $d_n(\sigma _i)$ does not factor through $A_{\sigma _i}$ , then $a_n=n$ . There are again two cases, based on whether $S_{\sigma _i}=\{1,2\ldots ,n-1,n\}$ . The case $S_{\sigma _i}\neq \{1,2\ldots ,n-1,n\}$ is identical to the corresponding case in Lemma 3.30. In the case $S_{\sigma _i}=\{1,2\ldots ,n-1,n\}$ we find that $A_{\sigma _i}=\Lambda ^n_n$ , the last edge is marked, and $\Delta ^{\{0,n-1,n\}}$ is scaled. This is a morphism of type 2.41, and thus is $\mathbf {MS}$ -anodyne.

Proof. Let $S_\tau =\{1,2,\ldots ,i-1,i+1,\ldots ,n-1,n,i\}$ and denote by $\hat {\tau }$ the smallest maximal simplex such that $\hat {\tau }>\tau $ . We define a filtration as in Lemma 3.30 and Lemma 3.31 up until the stage $X^{\geqslant \hat {\tau }}$ , yielding

$$\begin{align*}\Lambda\mathcal{S}^n_i \to X^{\geqslant \sigma_1} \to \cdots \to X^{\geqslant \hat{\tau}} \to \mathcal{S}^n_i. \end{align*}$$

We will first prove that that each step of this factorisation is $\mathbf {MS}$ -anodyne, making a distinction into 2 cases.

We consider the map $X^{\geqslant \sigma _{k-1}}\to X^{\geqslant \sigma _k}$ , and set $S_{\sigma _k}:=\{a_j\}_{j=1}^n$ . We again form the pullback

We then have two cases.

1. 1. If $S_{\sigma _k}$ has as its last entry anything other than i, then $A_{\sigma _k}$ consists of

   • • The face $d_0(\sigma _k)$ .

   • • The face $d_n(\sigma _k)$ .

   • • The face $d_j(\sigma _k)$ for each j such that $a_{j+1}> a_j$ .

   The argument is then nearly identical to case (1) from Lemma 3.30.

2. 2. If the last entry of $S_{\sigma _k}$ is i, then $A_{\sigma _k}$ consists of

   • • The face $d_0(\sigma _k)$ .

   • • The face $d_j(\sigma _k)$ for each j such that $a_{j+1}>a_j$ .

   The remainder of the argument is nearly identical to case (2) of Lemma 3.30.

It now remains only for us to show that is $\mathbf {MS}$ -anodyne. For ease of notation, we set $Z:=X^{\geqslant \hat {\tau }}$ .

We now need to add the remaining simplices. Write $\Sigma ^\leqslant $ for the set of maximal simplices which are less than or equal to $\tau $ . Given $\theta \in \Sigma ^{\leqslant }$ , we write $S_\theta =\{b_j\}_{j=1}^n$ for the ordered vertex sequence, as usual. We further denote by $\widehat {S_\theta }$ the vertex sequence obtained by removing i. We will call a simplex $\theta \in \Sigma ^{\leqslant }$ disordered if $\widehat {S_\theta }>\widehat {S_{\tau }}$ . If $\widehat {S_\theta }=\widehat {S_{\tau }}$ , we will call $\theta $ calm.Footnote ³

Our first order of business is to add the disordered simplices in $\Sigma ^{\leqslant }$ . For each disordered $\theta $ , we define $Z^{\geqslant \theta }$ to be obtained from Z by adding all the disordered simplices $\sigma $ for which $\sigma \geqslant \theta $ . Applying the order induced on disordered simplices, we again obtain a filtration