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4.2.6 The Homotopy Extension Lifting Property

We now show that left and right fibrations can be characterized by homotopy lifting properties.

Proposition 4.2.6.1. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then:

  • The morphism $f$ is a left fibration if and only if the evaluation map

    \[ \operatorname{ev}_{0}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 0\} , X) \times _{ \operatorname{Fun}( \{ 0\} , S)} \operatorname{Fun}( \Delta ^1, S) \]

    is a trivial Kan fibration.

  • The morphism $f$ is a right fibration if and only if the evaluation map

    \[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S)} \operatorname{Fun}( \Delta ^1, S) \]

    is a trivial Kan fibration.

Proof. We prove the second assertion; the first follows by passing to opposite simplicial sets. If $f$ is a right fibration, then the evaluation map $\operatorname{ev}_1$ is a trivial Kan fibration by virtue of Proposition 4.2.5.4 (since the inclusion $\{ 1\} \hookrightarrow \Delta ^1$ is right anodyne). Conversely, suppose that $\operatorname{ev}_1$ is a trivial Kan fibration. Then every lifting problem

\[ \xymatrix@C =100pt{ ( \Delta ^1 \times \Lambda ^{n}_{i} ) \coprod _{ \{ 1\} \times \Lambda ^ n_ i} (\{ 1\} \times \Delta ^ n ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^1 \times \Delta ^ n \ar [r] \ar@ {-->}[ur] & S_{} } \]

admits a solution. In other words, $f$ has the right lifting property with respect to the inclusion map

\[ u: ( \Delta ^1 \times \Lambda ^{n}_{i} ) \coprod _{ \{ 1\} \times \Lambda ^ n_ i} (\{ 1\} \times \Delta ^ n ) \hookrightarrow \Delta ^1 \times \Delta ^ n. \]

If $0 < i \leq n$, then the horn inclusion $u_0: \Lambda ^ n_ i \hookrightarrow \Delta ^ n$ is a retract of $u$ (Lemma 3.1.2.9). It follows that $f$ also has the right lifting property with respect to $u_0$ (Proposition 1.4.4.9): that is, 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. $\square$

Corollary 4.2.6.2. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a Kan fibration if and only if both of the evaluation maps

\[ \operatorname{ev}_{0}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 0\} , X) \times _{ \operatorname{Fun}( \{ 0\} , S)} \operatorname{Fun}( \Delta ^1, S) \]
\[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S)} \operatorname{Fun}( \Delta ^1, S) \]

are trivial Kan fibrations.

Remark 4.2.6.3 (The Homotopy Extension Lifting Property). Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Unwinding the definitions, we see that the following conditions are equivalent:

  • The morphism $f$ is a left fibration.

  • For every monomorphism of simplicial sets $i: A_{} \hookrightarrow B_{}$, every lifting problem

    \[ \xymatrix@C =100pt{ A_{} \ar [d]^{i} \ar [r] & \operatorname{Fun}( \Delta ^1, X_{} ) \ar [d]^{\operatorname{ev}_0} \\ B_{} \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) } \]

    admits a solution (indicated by the dotted arrow in the diagram).

  • For every monomorphism of simplicial sets $i: A_{} \hookrightarrow B_{}$, every lifting problem

    \[ \xymatrix@C =100pt{ ( \Delta ^1 \times A_{} ) \coprod _{ \{ 0\} \times A_{} } (\{ 0\} \times B_{} ) \ar [r] \ar [d] & X_{} \ar [d]^{f} \\ \Delta ^1 \times B_{} \ar [r]^-{\overline{h}} \ar@ {-->}[ur]^-{h} & S_{} } \]

    admits a solution (indicated by the dotted arrow in the diagram).

  • Let $u: B_{} \rightarrow X_{}$ be a map of simplicial sets and let $\overline{h}: \Delta ^1 \times B_{} \rightarrow S_{}$ be a map satisfying $\overline{h}|_{ \{ 0\} \times B_{} } = f \circ u$: that is, $\overline{h}$ is a homotopy from $f \circ u$ to another map $\overline{v} = \overline{h}|_{ \{ 1\} \times B_{} }$. Then we can choose a map of simplicial sets $h: \Delta ^1 \times B_{} \rightarrow X_{}$ satisfying $f \circ h = \overline{h}$ and $h|_{ \{ 0\} \times B_{} } = u$: in other words, $\overline{h}$ can be lifted to a homotopy $h$ from $u$ to another map $v = h|_{ \{ 1\} \times B_{} }$. Moreover, given any simplicial subset $A_{} \subseteq B_{}$ and any map $h_0: \Delta ^1 \times A_{} \rightarrow X_{}$ satisfying $f \circ h_0 = \overline{h}|_{ \Delta ^1 \times A_{}}$ and $h_0|_{ \{ 0\} \times A_{} } = u|_{ A_{}}$, we can arrange that $h$ is an extension of $h_0$.

In the special case where $B_{} = \Delta ^0$ and $A_{} = \emptyset $, each of these assertions reduces to the left path lifting property of $f$.

Exercise 4.2.6.4. Let $f: X \rightarrow S$ be a morphism of simplicial sets. Show that:

  • The morphism $f$ is a left covering map if and only if the evaluation map

    \[ \operatorname{ev}_{0}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 0\} , X) \times _{ \operatorname{Fun}( \{ 0\} , S)} \operatorname{Fun}( \Delta ^1, S) \]

    is an isomorphism of simplicial sets

  • The morphism $f$ is a right covering map if and only if the evaluation map

    \[ \operatorname{ev}_{1}: \operatorname{Fun}( \Delta ^1, X) \rightarrow \operatorname{Fun}( \{ 1\} , X) \times _{ \operatorname{Fun}( \{ 1\} , S)} \operatorname{Fun}( \Delta ^1, S) \]

    is an isomorphism of simplicial sets.

  • The morphism $f$ is a covering map if and only if both $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ are isomorphisms.