4.1 Left and Right Fibrations of Simplicial Sets
Let us begin by introducing the main objects of study in this chapter.
Definition 4.1.0.1. Let $f: X \rightarrow S$ be a morphism of simplicial sets. We will say that $f$ is a left fibration if, for every pair of integers $0 \leq i < n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {>}[ur]^{\sigma } \ar [r]^{ \overline{\sigma } } & S_{} } \]
has a solution (as indicated by the dotted arrow). That is, for every map of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow X_{}$ and every $n$simplex $\overline{\sigma }: \Delta ^ n \rightarrow S_{}$ extending $f \circ \sigma _0$, we can extend $\sigma _0$ to an $n$simplex $\sigma : \Delta ^{n} \rightarrow X_{}$ satisfying $f \circ \sigma = \overline{\sigma }$.
We say that $f$ is a right fibration if, for every pair of integers $0 \leq i < n$, every lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {>}[ur]^{\sigma } \ar [r]^{ \overline{\sigma } } & S_{} } \]
has a solution.
Example 4.1.0.2. Any isomorphism of simplicial sets is both a left fibration and a right fibration.
Example 4.1.0.5. A morphism of simplicial sets $f: X \rightarrow S$ is a Kan fibration if and only if it is both a left fibration and a right fibration.
Warning 4.1.0.6. In the statement of Example 4.1.0.5, both hypotheses are necessary: a left fibration of simplicial sets need not be a right fibration and vice versa. For example, the inclusion map $\{ 1 \} \hookrightarrow \Delta ^1$ is a left fibration, but not a right fibration (and therefore not a Kan fibration).
Let $q: X \rightarrow S$ be a morphism of simplicial sets. 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 §4.1.3, 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 4.1.3.1). 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 §4.1.2 (Propositions 4.1.2.1 and 4.1.2.4). Our proofs will make use of some basic facts about left anodyne and right anodyne morphisms of simplicial sets, which we establish in §4.1.1.
The notions of left and right fibration introduced in this section have a counterpart in classical category theory. In §4.1.4, we recall the notion a fibration in groupoids (Definition 4.1.4.1), 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 4.1.4.12). 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: 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