5.4.6 The Four-out-of-Five Property
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. Recall that the collection of isomorphisms in $\operatorname{\mathcal{C}}$ has the “two-out-of-three” property: if $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ are composable morphisms of $\operatorname{\mathcal{C}}$ and any two of the morphisms $f$, $g$, and $g \circ f$ is an isomorphism, then so is the third (Remark 1.4.6.3). This can be regarded as a special case of a more general closure property.
Definition 5.4.6.1. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$. We will say that $W$ has the two-out-of-six property if it satisfies the following condition:
Let $\sigma $ be a $3$-simplex of $\operatorname{\mathcal{C}}$ and, for every pair of integers $0 \leq i < j \leq 3$, let $\sigma _{ji}$ denote the edge of $\operatorname{\mathcal{C}}$ given by $\sigma |_{ \operatorname{N}_{\bullet }( \{ i < j \} )}$. If the edges $\sigma _{20}$ and $\sigma _{31}$ belong to $W$, then the edges $\sigma _{10}$, $\sigma _{21}$, $\sigma _{32}$, and $\sigma _{30}$ also belong to $W$.
Exercise 5.4.6.2. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $W$ be a collection of edges of $\operatorname{\mathcal{C}}$ which has the two-out-of-six property. Show that $W$ has the two-out-of-three property. That is, for any $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$, if any two of the faces $d^{2}_0(\sigma )$, $d^{2}_1(\sigma )$, and $d^{2}_2(\sigma )$ belong to $W$, then so does the third.
Proposition 5.4.6.5 (Two-out-of-Six). Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be the collection of isomorphisms in $\operatorname{\mathcal{C}}$. Then $W$ has the two-out-of-six property.
Proof.
By definition, a morphism $f$ of $\operatorname{\mathcal{C}}$ is an isomorphism if and only if its homotopy class $[f]$ is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}$ (Definition 1.4.6.1). By virtue of Remark 5.4.6.4, we can replace $\operatorname{\mathcal{C}}$ by the nerve $\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}} )$ and thereby reduce to the case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}' )$ for some category $\operatorname{\mathcal{C}}'$. Let $\sigma $ be a $3$-simplex of $\operatorname{\mathcal{C}}$, corresponding to a triple of morphisms
\[ A \xrightarrow {f} B \xrightarrow {g} C \xrightarrow {h} D \]
in $\operatorname{\mathcal{C}}'$, and suppose that $g \circ f$ and $h \circ g$ are isomorphisms. Then $g \circ f$ admits an inverse $u: C \rightarrow A$. It follows that $g \circ (f \circ u) = (g \circ f) \circ u = \operatorname{id}_{C}$, so that $g$ admits a right inverse. A similar argument shows that $g$ also admits a left inverse, and is therefore an isomorphism (Remark 1.4.6.7). Applying the two-out-three property, we deduce that $f$ and $h$ are also isomorphisms. Since the collection of isomorphisms is closed under composition, it also follows that $h \circ g \circ f$ is an isomorphism.
$\square$
Proposition 5.4.6.5 admits a converse:
Proposition 5.4.6.6. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $W$ be a collection of morphisms of $\operatorname{\mathcal{C}}$ which has the two-out-of-six property. If $W$ contains every identity morphism of $\operatorname{\mathcal{C}}$, then it contains every isomorphism of $\operatorname{\mathcal{C}}$.
In other words, the collection of isomorphisms in an $\infty $-category $\operatorname{\mathcal{C}}$ is the smallest collection of morphisms which contains all identity morphisms and has the two-out-of-six property.
Warning 5.4.6.7. The analogue of Proposition 5.4.6.6 for the two-out-of-three property is false in general. For example, if $\operatorname{\mathcal{C}}$ is the nerve of a category, then the collection of identity morphisms of $\operatorname{\mathcal{C}}$ has the two-out-of-three property, but usually does not contain all the isomorphisms of $\operatorname{\mathcal{C}}$.
Proof of Proposition 5.4.6.6.
Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $f: X \rightarrow Y$ be an isomorphism in $\operatorname{\mathcal{C}}$. Then $f$ admits a homotopy inverse $g: Y \rightarrow X$. Let $\sigma $ be a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $\operatorname{id}_{X}$ as a composition $f$ and $g$, and let $\sigma '$ be a $2$-simplex of $\operatorname{\mathcal{C}}$ which witnesses $\operatorname{id}_{Y}$ as a composition of $g$ and $f$. Then the triple $(\sigma ', s^{1}_0(f), \bullet , \sigma )$ can be regarded as a morphism of simplicial sets $\tau _0: \Lambda ^{3}_{2} \rightarrow \operatorname{\mathcal{C}}$. Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, we can extend $\tau _0$ to a $3$-simplex $\tau : \Delta ^3 \rightarrow \operatorname{\mathcal{C}}$, whose restriction to the $1$-skeleton of $\Delta ^3$ is indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [r]^-{g} \ar [drr]_{\operatorname{id}_ Y} & X \ar [dr]^{ f} & \\ X \ar [ur]^{f} \ar [urr]_{ \operatorname{id}_{X} } \ar [rrr]^{f} & & & Y. } \]
It follows that if $W$ is a collection of morphisms of $\operatorname{\mathcal{C}}$ which contains $\operatorname{id}_{X}$, $\operatorname{id}_{Y}$, and has the two-out-of-six property, then $W$ also contains the isomorphism $f$.
$\square$
Our goal in this section is to prove analogues of Propositions 5.4.6.5 and Proposition 5.4.6.6 in the setting of $(\infty ,2)$-categories, where we replace the set $W \subseteq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^1, \operatorname{\mathcal{C}})$ of isomorphisms with the set $T \subseteq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^2, \operatorname{\mathcal{C}})$ of thin $2$-simplices.
Definition 5.4.6.8. Let $\operatorname{\mathcal{C}}$ be a simplicial set and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. We say that $T$ has the four-out-of-five property if it satisfies the following condition:
Let $\sigma : \Delta ^{4} \rightarrow \operatorname{\mathcal{C}}$ be a $4$-simplex of $\operatorname{\mathcal{C}}$. For every triple of integers $0 \leq i < j < k \leq 4$, let $\sigma _{kji}$ denote the $2$-simplex of $\operatorname{\mathcal{C}}$ given by the restriction of $\sigma $ to $\operatorname{N}_{\bullet }( \{ i < j < k \} )$. If the $2$-simplices $\sigma _{310}$, $\sigma _{420}$, $\sigma _{321}$, and $\sigma _{432}$ belong to $T$, then the $2$-simplex $\sigma _{430}$ also belongs to $T$.
Warning 5.4.6.9. Definition 5.4.6.8 is not self-dual. Let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$ which satisfies the four-out-of-five property and let $T^{\operatorname{op}}$ denote the same set, regarded as a collection of $2$-simplices of the opposite simplicial set $\operatorname{\mathcal{C}}^{\operatorname{op}}$. Then $T^{\operatorname{op}}$ need not satisfy the four-out-of-five property.
Proposition 5.4.6.11 (Four-out-of-Five). Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $T$ be the collection of all thin $2$-simplices of $\operatorname{\mathcal{C}}$. Then $T$ has the four-out-of-five property.
Warning 5.4.6.12. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $\sigma : \Delta ^{4} \rightarrow \operatorname{\mathcal{C}}$ be a $4$-simplex of $\operatorname{\mathcal{C}}$. For $0 \leq i < j < k \leq 4$, let $\sigma _{kji}$ denote the restriction $\sigma |_{ \operatorname{N}_{\bullet }( \{ i < j < k \} ) }$. Proposition 5.4.6.11 asserts that, if the $2$-simplices $\sigma _{310}$, $\sigma _{420}$, $\sigma _{321}$, and $\sigma _{432}$ are thin, then $\sigma _{430}$ is also thin. Beware that the remaining $2$-simplices $\sigma _{210}$, $\sigma _{410}$, $\sigma _{320}$, $\sigma _{421}$, and $\sigma _{431}$ need not be thin.
Example 5.4.6.13. To get a feeling for the content of Proposition 5.4.6.11, let us consider the special case where $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }^{\operatorname{D}}(\operatorname{\mathcal{C}}')$ is the Duskin nerve of a strict $2$-category $\operatorname{\mathcal{C}}'$. Let $\sigma $ be a $4$-simplex of $\operatorname{\mathcal{C}}$, which we identify with a collection of objects $\{ X_ i \} _{0 \leq i \leq 4}$, $1$-morphisms $\{ f_{ji}: X_ j \rightarrow X_ i \} _{0 \leq i < j \leq 4}$, and $2$-morphisms $\{ \mu _{kji}: f_{kj} \circ f_{ji} \Rightarrow f_{ki} \} _{0 \leq i < j < k \leq 4}$ of $\operatorname{\mathcal{C}}$ satisfying the condition described in Proposition 2.3.1.9. Proposition 5.4.6.11 asserts that if the $2$-morphisms $\mu _{310}$, $\mu _{420}$, $\mu _{321}$, and $\mu _{432}$ are invertible, then the $2$-morphism $\mu _{430}$ is also invertible. This follows by inspecting the cubical diagram
\[ \xymatrix@R =50pt@C=40pt{ f_{43} \circ f_{32} \circ f_{21} \circ f_{10} \ar@ {=>}[rr]^{ \mu _{210} } \ar@ {=>}[dr]^{ \mu _{321} }_-{\sim } \ar@ {=>}[dd]^{ \mu _{432} } & & f_{43} \circ f_{32} \circ f_{20} \ar@ {=>}[dr]^{\mu _{320} } \ar@ {=>}[dd]^(.6){\mu _{432}}_(.6){\sim } & \\ & f_{43} \circ f_{31} \circ f_{10} \ar@ {=>}[dd]^(.6){ \mu _{431} } \ar@ {=>}[rr]^(.4){ \mu _{310} }_(.4){\sim } & & f_{43} \circ f_{30} \ar@ {=>}[dd]^{ \mu _{430} } \\ f_{42} \circ f_{21} \circ f_{10} \ar@ {=>}[rr]^(.4){ \mu _{210} } \ar@ {=>}[dr]^{ \mu _{421} } & & f_{42} \circ f_{20} \ar@ {=>}[dr]^{\mu _{420}}_-{\sim } & \\ & f_{41} \circ f_{10} \ar@ {=>}[rr]^{ \mu _{410} } & & f_{40} } \]
in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X_0, X_4)$ and applying the two-out-of-six property to the chain of $2$-morphisms
\[ f_{43} \circ f_{32} \circ f_{21} \circ f_{10} \xRightarrow {\mu _{210} } f_{43} \circ f_{32} \circ f_{20} \xRightarrow { \mu _{320} } f_{43} \circ f_{30} \xRightarrow { \mu _{430} } f_{40}. \]
Proof of Proposition 5.4.6.11.
Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $\sigma : \Delta ^4 \rightarrow \operatorname{\mathcal{C}}$ be a $4$-simplex. For every triple of integers $0 \leq i < j < k \leq 4$, let $\sigma _{kji}$ denote the $2$-simplex of $\operatorname{\mathcal{C}}$ given by the restriction of $\sigma $ to $\operatorname{N}_{\bullet }( \{ i < j < k \} )$. Assume that the $2$-simplices $\sigma _{310}$, $\sigma _{420}$, $\sigma _{321}$, and $\sigma _{432}$ are thin. We wish to show that $\sigma _{430}$ is also thin.
Set $X = \sigma (0) \in \operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{E}}$ denote the fiber product $\operatorname{\mathcal{C}}_{X/} \times _{\operatorname{\mathcal{C}}} \operatorname{Pith}(\operatorname{\mathcal{C}})$ and let $\pi : \operatorname{\mathcal{E}}\rightarrow \operatorname{Pith}(\operatorname{\mathcal{C}})$ be the projection map, so that $\pi $ is a cocartesian fibration of $\infty $-categories (Proposition 5.4.5.14). For $1 \leq i \leq 4$, let $\operatorname{\mathcal{E}}_{i}$ denote the $\infty $-category $\{ \sigma (i) \} \times _{\operatorname{Pith}(\operatorname{\mathcal{C}}) } \operatorname{\mathcal{E}}$, so that the edge $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < i \} ) }$ of $\operatorname{\mathcal{C}}$ can be identified with an object $Y_{i} \in \operatorname{\mathcal{E}}_{i}$. For $1 \leq i < j \leq 4$, let us identify the $2$-simplex $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < i < j \} )}$ with a morphism $f_{j,i}: Y_ i \rightarrow Y_ j$ in $\operatorname{\mathcal{E}}$. By virtue of Proposition 5.4.5.14), it will suffice to show that the morphism $f_{4,3}: Y_{3} \rightarrow Y_{4}$ is $\pi $-cocartesian.
For $2 \leq i \leq 4$, let $F_{i}: \operatorname{\mathcal{E}}_{i-1} \rightarrow \operatorname{\mathcal{E}}_{i}$ be given by covariant transport along the edge $\sigma |_{ \operatorname{N}_{\bullet }( \{ i-1 < i \} )}$ of $\operatorname{Pith}(\operatorname{\mathcal{C}})$ (see Definition 5.2.2.4) so that we have a sequence of functors
\[ \operatorname{\mathcal{E}}_{1} \xrightarrow {F_2} \operatorname{\mathcal{E}}_{2} \xrightarrow { F_3} \operatorname{\mathcal{E}}_{3} \xrightarrow { F_4} \operatorname{\mathcal{E}}_{4}. \]
Let $H_ i: \Delta ^{1} \times \operatorname{\mathcal{E}}_{i-1} \rightarrow \operatorname{\mathcal{E}}$ be a functor which witnesses that $F_{i}$ is given by covariant transport along $\sigma |_{ \operatorname{N}_{\bullet }( \{ i-1 < i \} )}$, so that $h_{i} = H_ i|_{ \Delta ^1 \times \{ Y_{i-1} \} }$ is a $\pi $-cocartesian morphism of $\operatorname{\mathcal{E}}$. It follows that the morphism $f_{i,i-1}$ can be written as a composition
\[ Y_{i-1} \xrightarrow { h_ i } F_ i(Y_{i-1}) \xrightarrow { g_ i} Y_{i}, \]
where $g_{i}$ is a morphism in the $\infty $-category $\operatorname{\mathcal{E}}_{i}$. To complete the proof, it will suffice to show that the morphism $g_{4}: F_4(Y_3) \rightarrow Y_4$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{E}}_{4}$ (see Remark 5.1.3.8).
Note that we have a chain of $1$-morphisms
\[ (F_4 \circ F_3 \circ F_2)(Y_1) \xrightarrow { (F_4 \circ F_3)(g_2) } (F_4 \circ F_3)(Y_2) \xrightarrow { F_4( g_3) } F_4(Y_3) \xrightarrow { g_4 } Y_4 \]
in the $\infty $-category $\operatorname{\mathcal{E}}_4$. Since the collection of isomorphisms in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}_{4}}$ satisfies the two-out-of-six property, it will suffice to prove the following:
- $(a)$
The composition
\[ (F_4 \circ F_3 \circ F_2)(Y_1) \xrightarrow { [(F_4 \circ F_3)(g_2)] } (F_4 \circ F_3)(Y_2) \xrightarrow { [F_4( g_3)] } F_4(Y_3) \]
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}_{4}}$.
- $(b)$
The composition
\[ (F_4 \circ F_3)(Y_2) \xrightarrow { [F_4( g_3)] } F_4(Y_3) \xrightarrow { [g_4] } Y_4 \]
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}_{4}}$.
We will deduce $(a)$ from the following slightly stronger assertion:
- $(a')$
The composition
\[ (F_3 \circ F_2)(Y_1) \xrightarrow { [F_3(g_2)] } F_3(Y_2) \xrightarrow {[g_3]} Y_3 \]
is an isomorphism in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}_3}$.
To prove $(a')$, we first note that the $2$-simplex $\sigma _{321}$ is thin, and can therefore be regarded as a $2$-simplex of $\operatorname{Pith}(\operatorname{\mathcal{C}})$. Let $\operatorname{\mathcal{E}}'$ denote the fiber product $\operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} ) \times _{\operatorname{Pith}(\operatorname{\mathcal{C}})} \operatorname{\mathcal{E}}$, and let $\pi ': \operatorname{\mathcal{E}}' \rightarrow \operatorname{N}_{\bullet }( \{ 1 < 2 < 3 \} )$ be the projection map. In the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}'}$, we have a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ Y_1 \ar [r]^-{ [h_2] } \ar [dr]_{ [f_{2,1}] } & F_2(Y_1) \ar [r] \ar [d]^-{[g_2]} & (F_3 \circ F_2)(Y_1) \ar [d]^-{ [ F_3(g_2)] } \\ & Y_2 \ar [r]^-{ [h_3] } \ar [dr]_{ [f_{3,2}] } & F_3(Y_2) \ar [d]^-{ [ g_3] } \\ & & Y_3, } \]
where the upper horizontal composition is the homotopy class of a $\pi '$-cocartesian morphism (Corollary 5.1.2.4). It follows that the vertical composition on the right is an isomorphism if and only if the diagonal composition is also the homotopy class of a $\pi '$-cocartesian morphism (Remark 5.1.3.8). We now observe that the $3$-simplex $\sigma |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 2 < 3 \} )}$ witnesses the identity $[ f_{3,2} ] \circ [ f_{2,1}] = [ f_{3,1} ]$ in the homotopy category $\mathrm{h} \mathit{\operatorname{\mathcal{E}}'}$. It will therefore suffice to show that $f_{3,1}$ is a $\pi '$-cocartesian morphism of the $\infty $-category $\operatorname{\mathcal{E}}'$, which follows from Proposition 5.4.5.14 and our assumption that $\sigma _{310}$ is thin. This completes the proof of $(a)$. The proof of $(b)$ follows by the same argument, using the thinness of the $2$-simplices $\sigma _{432}$ and $\sigma _{420}$ in place of $\sigma _{321}$ and $\sigma _{310}$.
$\square$
We now prove a partial converse to Proposition 5.4.6.11, which can be regarded as an $(\infty ,2)$-categorical analogue of Proposition 5.4.6.6.
Proposition 5.4.6.14. Let $\operatorname{\mathcal{C}}$ be an $(\infty ,2)$-category and let $T$ be a collection of $2$-simplices of $\operatorname{\mathcal{C}}$. Assume that:
- $(1)$
Every degenerate $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$.
- $(2)$
For every pair of morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, there exists a thin $2$-simplex $\sigma $ of $\operatorname{\mathcal{C}}$ which belongs to $T$ and satisfies $d^{2}_2(\sigma ) = f$ and $d^{2}_0(\sigma ) = g$, as indicated in the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr] & & Z. } \]
- $(3)$
The collection $T$ has the inner exchange property (Definition 5.4.5.7).
- $(4)$
The collection $T$ has the four-out-of-five property (Definition 5.4.6.8).
Then every thin $2$-simplex of $\operatorname{\mathcal{C}}$ belongs to $T$.
Proof.
Let $\sigma $ be a thin $2$-simplex of $\operatorname{\mathcal{C}}$, whose $1$-skeleton we represent by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^-{h} & & Z. } \]
Applying assumption $(2)$, we can choose a thin $2$-simplex $\sigma '$ of $\operatorname{\mathcal{C}}$ which belongs to $T$ whose restriction to the $1$-skeleton of $\Delta ^2$ is represented by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & Y \ar [dr]^{g} & \\ X \ar [ur]^{f} \ar [rr]^-{h'} & & Z. } \]
The edge $g$ determines a morphism of simplicial sets $\Delta ^1 \rightarrow \operatorname{\mathcal{C}}$. Let $\operatorname{\mathcal{E}}$ denote the fiber product $\Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_{X/}$. Since the projection map $\operatorname{\mathcal{C}}_{X/} \rightarrow \operatorname{\mathcal{C}}$ is an interior fibration (Proposition 5.4.3.1), it follows from Remark 5.4.2.4 and Example 5.4.2.2 that the projection map $\pi : \operatorname{\mathcal{E}}\rightarrow \Delta ^1$ is an inner fibration; in particular, $\operatorname{\mathcal{E}}$ is an $\infty $-category. Moreover, we can identify the edges $f$, $h$, and $h'$ of $\operatorname{\mathcal{C}}$ with objects $\widetilde{Y}$, $\widetilde{Z}$, and $\widetilde{Z}'$ of $\operatorname{\mathcal{E}}$, and the $2$-simplices $\sigma $ and $\sigma '$ with morphisms $\widetilde{h}: \widetilde{Y} \rightarrow \widetilde{Z}$ and $\widetilde{h}': \widetilde{Y} \rightarrow \widetilde{Z}'$. Since $\sigma $ and $\sigma '$ are both thin, the morphisms $\widetilde{h}$ and $\widetilde{h}'$ are both $\pi $-cocartesian (Theorem 5.4.4.1). It follows that $\widetilde{h}$ and $\widetilde{h}'$ are isomorphic when viewed as objects of the $\infty $-category $\operatorname{\mathcal{E}}_{ \widetilde{Y} / }$ (see Remark 5.1.3.8). We can therefore choose a $2$-simplex $\rho $ of $\operatorname{\mathcal{E}}_{ \widetilde{Y} / }$ whose $1$-skeleton is given by the diagram
\[ \xymatrix@R =50pt@C=50pt{ & \widetilde{h}' \ar [dr] & \\ \widetilde{h} \ar [rr]^-{ \operatorname{id}_{ \widetilde{h} }} \ar [ur] & & \widetilde{h}, } \]
which we can identify with a $4$-simplex $\tau : \Delta ^4 \rightarrow \operatorname{\mathcal{C}}$. For $0 \leq i < j < k \leq 4$, let $\tau _{kji}$ denote the $2$-simplex of $\operatorname{\mathcal{C}}$ given by $\tau |_{ \operatorname{N}_{\bullet }( \{ i < j < k \} )}$. By construction, the $2$-simplex $\tau _{310}$ is equal to $\sigma '$, and therefore belongs to $T$. Moreover, the $2$-simplices $\tau _{420}$, $\tau _{321}$, $\tau _{431}$, and $\tau _{432}$ are right-degenerate, and therefore belong to $T$ by virtue of assumption $(1)$. Since $T$ has the four-out-of-five-property, it follows that $\tau _{430}$ belongs to $T$. Applying the inner exchange property to the $3$-simplex $\tau |_{ \operatorname{N}_{\bullet }( \{ 0 < 1 < 3 < 4 \} )}$, we deduce that the $2$-simplex $\sigma = \tau _{410}$ also belongs to $T$, as desired.
$\square$