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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.

Remark 3.1.1.4. The collection of Kan fibrations is closed under retracts. That is, given a diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X_{} \ar [r] \ar [d]^{f} & X'_{} \ar [d]^{f'} \ar [r] & X_{} \ar [d]^{f} \\ S_{} \ar [r] & S'_{} \ar [r] & S_{} } \]

where both horizontal compositions are the identity, if $f'$ is a Kan fibration, then so is $f$.

Remark 3.1.1.5. The collection of Kan fibrations is closed under pullback. That is, given a pullback diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ X'_{} \ar [d]^{f'} \ar [r] & X_{} \ar [d]^{f} \\ S'_{} \ar [r] & S_{} } \]

where $f$ is a Kan fibration, $f'$ is also a Kan fibration.

Remark 3.1.1.6. The collection of Kan fibrations is closed under filtered colimits. That is, if $\{ f_{\alpha }: X_{\alpha } \rightarrow S_{\alpha } \} $ is any filtered diagram in the arrow category $\operatorname{Fun}( [1], \operatorname{Set_{\Delta }})$ having colimit $f: X \rightarrow S$, and each $f_{\alpha }$ is a Kan fibration of simplicial sets, then $f$ is also a Kan fibration of simplicial sets.

Remark 3.1.1.7. Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets. Then, for every vertex $s \in S_0$, the fiber $\{ s\} \times _{ S_{} } X_{}$ is a Kan complex (this follows from Remark 3.1.1.5 and Example 3.1.1.2).

Remark 3.1.1.8. Let $f: X_{} \rightarrow Y_{}$ and $g: Y_{} \rightarrow Z_{}$ be Kan fibrations. Then the composite map $(g \circ f): X_{} \rightarrow Z_{}$ is a Kan fibration.