5.1 Fibers of Left and Right Fibrations
Let $q: X \rightarrow S$ be a left fibration of simplicial sets (Definition 4.2.1.1). For each vertex $s \in S$, let $X_{s} = \{ s\} \times _{S} X$ denote the fiber of $q$ over $s$. Our assumption that $q$ is a left fibration guarantees that each fiber $X_{s}$ is a Kan complex (Corollary 4.4.2.2). Our goal in this section is to study the relationship between these Kan complexes as $s$ varies. In §5.1.1, we associate to each edge $e: s \rightarrow s'$ of $S$ a morphism of Kan complexes $e_{!}: X_ s \rightarrow X_{s'}$, which is welldefined up to homotopy and which we refer to as the covariant transport morphism (Construction 5.1.1.5). 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 5.1.1.7). 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 5.1.1.11), 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 5.1.1.12).
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 5.1.2.1). In §5.1.2, 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 5.1.2.2). Note that in some special cases, the latter hypothesis is automatically satisfied: for example, if $S$ is a Kan complex (see Lemma 5.1.2.4) or if the fibers of $q$ are contractible Kan complexes (see Lemma 5.1.2.7).
In §5.1.3, we study the relationship between different left fibrations over the same simplicial set $S$. Suppose we are given a commutative diagram
\[ \xymatrix@R =50pt@C=50pt{ X \ar [rr]^{f} \ar [dr]_{q_ X} & & Y \ar [dl]^{q_ Y} \\ & S, & } \]
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 5.1.3.3): 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 5.1.3.1). 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 5.1.4.1), which we introduce and study in §5.1.4.
Structure

Subsection 5.1.1: Covariant Transport

Subsection 5.1.2: A Characterization of Kan Fibrations

Subsection 5.1.3: Relative Homotopy Equivalences

Subsection 5.1.4: Covariant and Contravariant Equivalences