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4 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 $0 \leq i \leq n$. 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 this chapter, we will introduce parametrized versions of these path lifting properties. We will 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 $0 < i \leq n$ (see Definition 4.1.0.1). In §4.1, we study the notions of left and right fibration and establish their basic formal properties.

In §4.2, we consider some prototypical examples of left and right fibrations which arise frequently in practice. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category. To each object $X \in \operatorname{\mathcal{C}}$, one can associate a simplicial set $\operatorname{\mathcal{C}}_{/X}$, whose $n$-simplices are given by maps $\sigma : \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$ which carry the final vertex of $\Delta ^{n+1}$ to the object $X \in \operatorname{\mathcal{C}}$. In particular, vertices of $\operatorname{\mathcal{C}}_{/X}$ can be identified with morphisms $f: Y \rightarrow X$ in $\operatorname{\mathcal{C}}$ having target $X$, and edges of $\operatorname{\mathcal{C}}_{/X}$ can be identified with commutative diagrams

\[ \xymatrix { Z \ar [rr] \ar [dr] & & Y \ar [dl] \\ & X. & } \]

in the $\infty $-category $\operatorname{\mathcal{C}}$ (see Notation 4.2.5.6 for a precise definition). The simplicial set $\operatorname{\mathcal{C}}_{/X}$ is itself an $\infty $-category, which we we will refer to as the slice $\infty $-category of $\operatorname{\mathcal{C}}$ over the object $X$. Moreover, the evident forgetful functor $\operatorname{\mathcal{C}}_{/X} \rightarrow \operatorname{\mathcal{C}}$ (given on objects by the construction $(f: Y \rightarrow X) \mapsto Y$) is a right fibration (Proposition 4.2.6.1). A dual version of this construction produces another $\infty $-category $\operatorname{\mathcal{C}}_{X/}$ whose objects are morphisms $f: X \rightarrow Y$ in the $\infty $-category $\operatorname{\mathcal{C}}$, which we refer to as the coslice $\infty $-category of $\operatorname{\mathcal{C}}$ under the object $X$. The slice and coslice constructions (and generalizations thereof) provide a rich supply of right and left fibrations between simplicial sets, and will play an essential role in the theory developed in this book.

Let $q: X \rightarrow S$ be a morphism of simplicial sets. For each vertex $s \in S$, let $X_{s}$ denote the fiber product $\{ s\} \times _{S} X$, which we refer to as the fiber of $q$ over $s$. In §4.3, we study the properties of these fibers in the special case where $q$ is a left or right fibration. In particular, we establish the following:

  • If $q$ is either a left fibration or a right fibration, then each fiber $X_{s}$ is a Kan complex (Proposition 4.3.1.1).

  • If $q$ is a left fibration, then the construction $s \mapsto X_{s}$ can be promoted to a functor from the homotopy category $\mathrm{h} \mathit{S}$ of the simplicial set $S$ to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes (Construction 4.3.2.7).

  • If $q$ is a right fibration, then the construction $s \mapsto X_{s}$ can be promoted to a functor from the opposite category $\mathrm{h} \mathit{S}^{\operatorname{op}}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ (Construction 4.3.2.12).

We will return to this idea in Chapter , where we show that the datum of a left fibration $q: X \rightarrow S$ is essentially equivalent to the datum of a map from $S$ to the $\infty $-category $\operatorname{\mathcal{S}}$ of Kan complexes (see Theorem ).

Structure

  • Section 4.1: Left and Right Fibrations of Simplicial Sets
    • Subsection 4.1.1: Left Anodyne and Right Anodyne Morphisms
    • Subsection 4.1.2: Exponentiation for Left and Right Fibrations
    • Subsection 4.1.3: The Homotopy Extension Lifting Property
    • Subsection 4.1.4: Example: Fibrations in Groupoids
  • Section 4.2: The Slice and Join Constructions
    • Subsection 4.2.1: Slices of Categories
    • Subsection 4.2.2: Joins of Categories
    • Subsection 4.2.3: Joins of Simplicial Sets
    • Subsection 4.2.4: Joins of Topological Spaces
    • Subsection 4.2.5: Slices of Simplicial Sets
    • Subsection 4.2.6: Slices of $\infty $-Categories
    • Subsection 4.2.7: Slices of Left and Right Fibrations
  • Section 4.3: Fibers of Left and Right Fibrations
    • Subsection 4.3.1: Fibrations over a Point
    • Subsection 4.3.2: Covariant Transport
    • Subsection 4.3.3: A Characterization of Kan Fibrations
    • Subsection 4.3.4: Relative Homotopy Equivalences
    • Subsection 4.3.5: Covariant and Contravariant Equivalences