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4.3 Fibers of Left and Right Fibrations

In §4.1, we defined left and right fibration between simplicial sets (Definition 4.1.0.1), and established some of their basic formal properties. Our goal in this section is to analyze the structure of left and right fibrations in more detail. We begin in §4.3.1 by showing that a simplicial set $X$ is a Kan complex if and only if the projection map $q: X \rightarrow \Delta ^0$ is a left (or right) fibration (Proposition 4.3.1.1). The proof has two main steps. First, we argue that if $q: X \rightarrow \Delta ^0$ is a left (or right) fibration, then $X$ is an $\infty $-category and every morphism in $X$ is an isomorphism. We then argue that these properties automatically guarantee that $X$ is a Kan complex. The proof of the latter fact is based on a characterization of isomorphisms in terms of horn filling conditions (Theorem 4.3.1.3), which is of some independent interest.

Let $q: X \rightarrow S$ be an arbitrary morphism of simplicial sets. It follows from Proposition 4.3.1.1 that, if $q$ is a left or a right fibration, then each fiber $X_{s} = \{ s\} \times _{S} X$ is a Kan complex (Corollary 4.3.1.2). In §4.3.2, we study the relationship between these fibers as the vertex $s$ varies. When $q$ is a left fibration, we associate to each edge $e: s \rightarrow s'$ of $S$ a morphism of Kan complexes $e_{!}: X_ s \rightarrow X_{s'}$, which is well-defined up to homotopy and which we refer to as the covariant transport morphism (Construction 4.3.2.5). By means of this construction, we can promote the construction $s \mapsto X_{s}$ to a functor from the homotopy category $\mathrm{h} \mathit{S}$ to the homotopy category $\mathrm{h} \mathit{\operatorname{Kan}}$ of Kan complexes, which we will refer to as the covariant transport functor (Construction 4.3.2.7). Dually, if $q$ is a right fibration, we can associate to each edge $e: s \rightarrow s'$ a morphism of Kan complexes $e^{\ast }: X_{s'} \rightarrow X_{s}$ which we refer to as the contravariant transport morphism (Notation 4.3.2.11), which allows us to promote the construction $s \mapsto X_{s}$ to a functor $\mathrm{h} \mathit{S}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ which we refer to as the contravariant transport functor (Construction 4.3.2.12).

Let $q: X \rightarrow S$ be a left fibration between simplicial sets, and let $e_{!}: X_{s} \rightarrow X_{s'}$ be the covariant transport morphism associated to an edge $e: s \rightarrow s'$ of the simplicial set $S$. If $q$ is a Kan fibration, then the morphism $e_{!}$ is a homotopy equivalence; more precisely, it is homotopy inverse to the contravariant transport morphism $e^{\ast }: X_{s'} \rightarrow X_{s}$ (Proposition 4.3.3.1). In §4.3.3, we will establish a converse to this: if $q: X \rightarrow S$ is a left fibration for which every covariant transport morphism $e_{!}$ is a homotopy equivalence, then $q$ is a Kan fibration (Theorem 4.3.3.2). Note that in some special cases, the latter hypothesis is automatically satisfied: for example, if $S$ is a Kan complex (see Lemma 4.3.3.4) or if the fibers of $q$ are contractible Kan complexes (see Lemma 4.3.3.7).

In §4.3.4, we study the relationship between different left fibrations over the same simplicial set $S$. Suppose we are given a commutative diagram

\[ \xymatrix { X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S, & } \]

where $q_{X}$ and $q_{Y}$ are left fibrations. For each vertex $s \in S$, the morphism $f$ restricts to a map of Kan complexes $f_{s}: X_{s} \rightarrow Y_{s}$. Our main result asserts that, if each $f_{s}$ is a homotopy equivalence of Kan complexes, then $f$ is a homotopy equivalence relative to $S$ (Theorem 4.3.4.7): that is, there exists a morphism of simplicial sets $g: Y \rightarrow X$ compatible with the projection to $S$, such that $g \circ f$ and $f \circ g$ are homotopic to the identity morphisms $\operatorname{id}_{X}$ and $\operatorname{id}_{Y}$ via homotopies which are compatible with the projection to $S$ (see Definition 4.3.4.5). Beware that this implication need not hold if $q_{X}$ and $q_{Y}$ are not assumed to be left fibrations. To remedy the situation, it is convenient to introduce a weaker notion of relative homotopy equivalence which we refer to as covariant equivalence (Definition 4.3.5.1), which we introduce and study in §4.3.5.

Remark 4.3.0.1. Let $\mathrm{h} \mathit{\operatorname{Kan}}$ denote the homotopy category of Kan complexes (Construction 3.1.4.10). Then $\mathrm{h} \mathit{\operatorname{Kan}}$ can be identified with the homotopy category of the $\infty $-category $\operatorname{\mathcal{S}}$ of Construction 3.1.4.12. If $q: X \rightarrow S$ is a left fibration of simplicial sets, then Construction 4.3.2.7 determines a covariant transport functor

\[ T: \mathrm{h} \mathit{S} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}. \]

In Chapter , we show that (up to equivalence) $T$ can be promoted to a map of simplicial sets $S \rightarrow \operatorname{\mathcal{S}}$. In other words, the formation of covariant transport morphisms $(e: s \rightarrow s') \mapsto (e_{!}: X_{s} \rightarrow X_{s'})$ is compatible with composition not only up to homotopy, but up to coherent homotopy.

Structure

  • 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