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### 4.1.1 Definitions

We now introduce the main objects of study in this chapter.

Definition 4.1.1.1. 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 4.1.1.2. Any isomorphism of simplicial sets is both a left fibration and a right fibration.

Example 4.1.1.4. 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 4.1.1.5. In the statement of Example 4.1.1.4, 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).