Proposition 7.6.3.25 (Transitivity). Let $U: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories and let $\sigma : \Delta ^2 \times \Delta ^1 \rightarrow \operatorname{\mathcal{C}}$ be diagram, which we depict informally as
7.61
\begin{equation} \begin{gathered}\label{equation:transitivity-pullback-square} \xymatrix@R =50pt@C=50pt{ X' \ar [r] \ar [d] & Y' \ar [r] \ar [d] & Z' \ar [d] \\ X \ar [r] & Y \ar [r] & Z. } \end{gathered} \end{equation}
Then:
- $(1)$
Assume that the right square of (7.61) is a $U$-pullback. Then the left square is a $U$-pullback if and only if the outer rectangle is a $U$-pullback.
- $(2)$
Assume that the left square of (7.61) is a $U$-pushout. Then the left square is a $U$-pushout if and only if the outer rectangle is a $U$-pushout.
Proof.
We will prove $(1)$; the proof of $(2)$ is similar. Let $A$ denote the partially ordered set $([2] \times [1] ) \setminus \{ (0,0) \} $. Note that the inclusion maps
\[ A \setminus \{ (2,0), (2,1) \} \hookrightarrow A \quad \quad A \setminus \{ (1,0), (1,1) \} \hookrightarrow A \setminus \{ (1,0) \} \]
admit right adjoints, and therefore induce left cofinal morphisms
\[ \operatorname{N}_{\bullet }( A \setminus \{ (2,0), (2,1) \} ) \hookrightarrow \operatorname{N}_{\bullet }(A) \quad \quad \operatorname{N}_{\bullet }( A \setminus \{ (1,0), (1,1) \} ) \hookrightarrow \operatorname{N}_{\bullet }(A \setminus \{ (1,0) \} ) \]
(Corollary 7.2.3.7). Applying Corollary 7.2.2.2, we obtain the following:
The left square of (7.61) is a $U$-pullback diagram if and only if $\sigma $ is a $U$-limit diagram.
The outer rectangle of (7.61) is a $U$-pullback diagram if and only if the restriction $\sigma |_{ \operatorname{N}_{\bullet }( ([2] \times [1] ) \setminus \{ (1,0) \} )}$ is a $U$-limit diagram.
If the right square of (7.61) is a $U$-pullback diagram, then $\sigma |_{\operatorname{N}_{\bullet }(A)}$ is $U$-right Kan extended from $\sigma |_{ \operatorname{N}_{\bullet }( A \setminus \{ (1,0) \} )}$, so the desired equivalence follows from Proposition 7.3.8.1.
$\square$