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4.2 Left and Right Fibrations

Let $q: X \rightarrow S$ be a morphism of simplicial sets. Recall that $q$ is a Kan fibration if and only if it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $n > 0$ and $0 \leq i \leq n$ (Definition 3.1.1.1). In particular, if $q$ 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 $q: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, let $x$ be a vertex of $X_{}$, and let $\overline{e}: q(x) \rightarrow \overline{y}$ be an edge of $S_{}$ originating at the vertex $q(x)$. Then there exists an edge $e: x \rightarrow y$ in $X_{}$ which originates at the vertex $x$ and satisfies $q(e) = \overline{e}$.

(Right Path Lifting Property):

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

In §4.2.1, we introduce stronger versions of these lifting properties. We say that a morphism of simplicial sets $q: X \rightarrow S$ 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 $0 < i \leq n$ (Definition 4.2.1.1). Setting $n = 1$, we see that every left fibration satisfies the left path lifting property, and that every right fibration satisfies the right path lifting property. Moreover, this assertion has a partial converse. 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 §4.2.6, 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 4.2.6.1). 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 §4.2.5 (Propositions 4.2.5.1 and 4.2.5.4). Our proofs will make use of some basic facts about left anodyne and right anodyne morphisms of simplicial sets, which we establish in §4.2.4.

The notions of left and right fibration have antecedents in classical category theory. In §4.2.2, we show that the induced map of simplicial sets $\operatorname{N}_{\bullet }(U): \operatorname{N}_{\bullet }(\operatorname{\mathcal{E}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a right fibration if and only if $U$ is a fibration in groupoids (see Definition 4.2.2.1). We will be particularly interested in the special case where $U$ is a fibration in groupoids for which each fiber $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a discrete category. In §4.2.3, we show that this is equivalent to the condition that the induced map of simplicial sets $\operatorname{N}_{\bullet }(U)$ is a right covering of simplicial sets (Proposition 4.2.3.16): that is, it satisfies a unique lifting property for horn inclusions $\Lambda ^{n}_{i} \hookrightarrow \Delta ^{n}$ with $0 < i \leq n$ (Definition 4.2.3.8).

Structure

  • Subsection 4.2.1: Left and Right Fibrations of Simplicial Sets
  • Subsection 4.2.2: Fibrations in Groupoids
  • Subsection 4.2.3: Left and Right Covering Maps
  • Subsection 4.2.4: Left Anodyne and Right Anodyne Morphisms
  • Subsection 4.2.5: Exponentiation for Left and Right Fibrations
  • Subsection 4.2.6: The Homotopy Extension Lifting Property