Kerodon

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4.1 Left and Right Fibrations of Simplicial Sets

Let $f: X \rightarrow S$ be a morphism of simplicial sets. Recall that $f$ is a Kan fibration if and only if it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i}$ for $n > 0$ and $0 \leq i \leq n$. In particular, if $f$ is a Kan fibration, then it has the right lifting property with respect to both of the inclusion maps $\{ 0 \} \hookrightarrow \Delta ^1 \hookleftarrow \{ 1\} $. Concretely, this translates into the following pair of assertions:

(Left Path Lifting Property):

Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, let $x$ be a vertex of $X_{}$, and let $\overline{e}: f(x) \rightarrow \overline{y}$ be an edge of $S_{}$ beginning at the vertex $f(x)$. Then there exists an edge $e: x \rightarrow y$ in $X_{}$, beginning at the vertex $x$ and satisfying $f(e) = \overline{e}$.

(Right Path Lifting Property):

Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, let $y$ be a vertex of $X_{}$, and let $\overline{e}: \overline{x} \rightarrow f(y)$ be an edge of $S_{}$ ending at the vertex $f(y)$. Then there exists an edge $e: x \rightarrow y$ in $X_{}$, ending at the vertex $y$ and satisfying $f(e) = \overline{e}$.

In this section, we study parametrized versions of these lifting properties. We begin in §4.1.1 by introducing the notions left fibration and right fibration between simplicial sets (Definition 4.1.1.1). By definition, a morphism of simplicial sets $f: X \rightarrow S$ is a left fibration if it has the right lifting property with respect to the horn inclusions $\Lambda ^ n_{i} \hookrightarrow \Delta ^ n$ for $0 \leq i < n$, and a right fibration if it has the right lifting property with respect the horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i \leq n$. Specializing to the case $n=1$, we see that every left fibration has the left path lifting property, and that every right fibration has the right path lifting property. The primary goal of this section is to establish a partial converse to this observation. Note that evaluation at the vertices of $\Delta ^1$ induces morphisms of simplicial sets

\[ \operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ) \]
\[ \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 1\} , X_{} ) \times _{ \operatorname{Fun}( \{ 1\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ). \]

In §4.1.4, we show that $f$ is a left fibration if and only if the evaluation map $\operatorname{ev}_0$ is a trivial Kan fibration, and that $f$ is a right fibration if and only if $\operatorname{ev}_{1}$ is a trivial Kan fibration (Proposition 4.1.4.1). The “only if” direction of this assertion is a special case of general stability properties of left and right fibrations under exponentiation, which we prove in §4.1.3 (Propositions 4.1.3.1 and 4.1.3.4). Our proofs will make use of some basic facts about left anodyne and right anodyne morphisms of simplicial sets, which we establish in §4.1.2.

The notions of left and right fibration introduced in this section have a counterpart in classical category theory. In §4.1.5, we recall the notion a fibration in groupoids (Definition 4.1.5.1), and show that a functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is a fibration in groupoids if and only if the induced map of nerves $\operatorname{N}_{\bullet }(F): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a right fibration of simplicial sets (Proposition 4.1.5.11). Similarly, a functor of ordinary categories $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is an opfibration in groupoids if only if the map $\operatorname{N}_{\bullet }(F)$ is a left fibration of simplicial sets.

Structure

  • Subsection 4.1.1: Definitions
  • Subsection 4.1.2: Left Anodyne and Right Anodyne Morphisms
  • Subsection 4.1.3: Exponentiation for Left and Right Fibrations
  • Subsection 4.1.4: The Homotopy Extension Lifting Property
  • Subsection 4.1.5: Fibrations in Groupoids