Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Theorem 5.1.2.2. Let $q: X \rightarrow S$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $q$ is a Kan fibration.

$(2)$

The morphism $q$ is a left fibration and, for every edge $e: s \rightarrow s'$ of the simplicial set $S$, the covariant transport morphism $e_{!}: X_{s} \rightarrow X_{s'}$ is a homotopy equivalence.

$(3)$

The morphism $q$ is a right fibration and, for every edge $e: s \rightarrow s'$ of the simplicial set $S$, the contravariant transport morphism $e^{\ast }: X_{s'} \rightarrow X_{s}$ is a homotopy equivalence.

Proof of Theorem 5.1.2.2. We will show that $(1) \Leftrightarrow (2)$; the proof of the equivalence $(1) \Leftrightarrow (3)$ is similar. The implication $(1) \Rightarrow (2)$ is immediate from Proposition 5.1.2.1. For the converse, assume that $q: X \rightarrow S$ is a left fibration of simplicial sets and that, for every edge $e: s \rightarrow s'$ of $S$, the covariant transport morphism $e_{!}: X_{s} \rightarrow X_{s'}$ is a homotopy equivalence. We wish to show that $q$ is a Kan fibration. By virtue of Example 4.2.1.5, it will suffice to show that $q$ is a right fibration. By virtue of Proposition 4.2.4.1, this is equivalent to the assertion that the induced map

\[ \theta : \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S) } \operatorname{Fun}( \Delta ^1, S) \]

is a trivial Kan fibration. Note that our assumption that $q$ is a left fibration guarantees that $\theta $ is also a left fibration (Proposition 4.2.3.1). It will therefore suffice to show that the fibers of $\theta $ are contractible Kan complexes (Lemma 5.1.2.7).

Fix an edge $e: s \rightarrow s'$ of the simplicial set $S$ and set $Y = \operatorname{Fun}( \Delta ^1, X) \times _{ \operatorname{Fun}( \Delta ^1, S) } \{ e\} $. Then evaluation at the vertex $1 \in \Delta ^1$ induces a morphism $\theta _{e}: Y \rightarrow X_{s'}$. Note that $\theta _{e}$ is a pullback of $\theta $, and is therefore also a left fibration. Since $X_{s'}$ is a Kan complex (Corollary 4.4.2.2), Lemma 5.1.2.4 guarantees that $\theta _{e}$ is Kan fibration (so $Y$ is also a Kan complex). Evaluation at the vertex $0 \in \Delta ^1$ induces another morphism of simplicial sets $u: Y \rightarrow X_{s}$. Since $q$ is a left fibration, the morphism $u$ is a trivial Kan fibration. By construction, the covariant transport morphism $e_{!}: X_{s} \rightarrow X_{s'}$ can be realized as a composition

\[ X_{s} \xrightarrow {v} Y \xrightarrow { \theta _{e} } X_{s'}, \]

where $v$ is a section of $u$ (and therefore a homotopy equivalence). Consequently, our assumption that $e_{!}$ is a homotopy equivalence of Kan complexes guarantees that $\theta _{e}$ is a homotopy equivalence of Kan complexes (Remark 3.1.5.16). Applying Corollary 3.2.6.9, we deduce that the fibers of $\theta _{e}$ are contractible Kan complexes. Since every fiber of $\theta $ can also be viewed as a fiber of $\theta _{e}$ for some edge $e$ of the simplicial set $S$, it follows that the fibers of $\theta $ are also contractible Kan complexes, as desired. $\square$