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Proposition 5.3.6.11 (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.

Proof of Proposition 5.3.6.11. 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 5.3.5.13). 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 5.3.5.13), 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 5.2.2.1) 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 5.3.5.13 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 $\sigma _{310}$. $\square$