Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary 3.5.9.14 (Transitivity). Let $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ be morphisms of Kan complexes and let $n$ be an integer. Then:

$(a)$: If the morphisms $f$ and $g$ are $n$-truncated, then the composition $(g \circ f): X \rightarrow Z$ is $n$-truncated.
$(b)$: If $(g \circ f)$ is $n$-truncated and $g$ is $(n+1)$-truncated, then $f$ is $n$-truncated.
$(c)$: If $(g \circ f)$ is $n$-truncated, $f$ is $(n-1)$-truncated, and $\pi _0(f)$ is surjective, then $g$ is $n$-truncated.

Proof. Using Proposition 3.1.7.1, we can reduce to the case where $Z$ is a Kan complex and the morphisms $f$ and $g$ are Kan fibrations. Using the criterion of Proposition 3.5.9.8, we can further reduce to the case $Z = \Delta ^0$. In this case, Corollary 3.5.9.14 is a restatement of Proposition 3.5.9.13. $\square$