$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

### 3.1.1 Kan Fibrations

Recall that a simplicial set $X_{}$ is said to be a *Kan complex* if it has the extension property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $n > 0$ (Definition 1.1.9.1). For many purposes, it is useful to consider a relative version of this notion, which applies to a morphism between simplicial sets.

Definition 3.1.1.1. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. We say that $f$ is a *Kan fibration* if, for each $n > 0$ and each $0 \leq i \leq n$, every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n}_{i} \ar [r]^-{\sigma _0} \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^{n} \ar@ {-->}[ur]^-{\sigma } \ar [r]^-{ \overline{\sigma } } & S_{} } \]

admits 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 $f \circ \sigma = \overline{\sigma }$.

Example 3.1.1.2. Let $X_{}$ be a simplicial set. Then the projection map $X_{} \rightarrow \Delta ^0$ is a Kan fibration if and only if $X_{}$ is a Kan complex.

Example 3.1.1.3. Any isomorphism of simplicial sets is a Kan fibration.

Example 3.1.1.4. Let $q: X \rightarrow S$ be a morphism of simplicial sets which induces an isomorphism from $X$ to a summand of $S$ (Definition 1.1.6.1). Then $q$ is a Kan fibration.