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3.1.2 Left and Right Fibrations

By definition, a morphism of simplicial sets $f: X_{} \rightarrow S_{}$ is a Kan fibration if and only if it has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ where $n > 0$. In particular, if $f$ is a Kan fibration, then it has the right lifting property with respect to both of the inclusion maps $\{ 0 \} \hookrightarrow \Delta ^1 \hookleftarrow \{ 1\} $. Concretely, this translates into the following pair of assertions:

[align=left]
(Left Path Lifting Property):

Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, let $x$ be a vertex of $X_{}$, and let $\overline{e}: f(x) \rightarrow \overline{y}$ be an edge of $S_{}$ beginning at the vertex $f(x)$. Then there exists an edge $e: x \rightarrow y$ in $X_{}$, beginning at the vertex $x$ and satisfying $f(e) = \overline{e}$.

(Right Path Lifting Property):

Let $f: X_{} \rightarrow S_{}$ be a Kan fibration of simplicial sets, let $y$ be a vertex of $X_{}$, and let $\overline{e}: \overline{x} \rightarrow f(y)$ be an edge of $S_{}$ ending at the vertex $f(y)$. Then there exists an edge $e: x \rightarrow y$ in $X_{}$, ending at the vertex $y$ and satisfying $f(e) = \overline{e}$.

We will see in a moment that every Kan fibration $f: X_{} \rightarrow S_{}$ satisfies stronger versions of both of these properties. We begin by observing that evaluation at the vertices of $\Delta ^1$ induces maps of simplicial sets

\[ \operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ) \]
\[ \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 1\} , X_{} ) \times _{ \operatorname{Fun}( \{ 1\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ). \]

The left path lifting property asserts that the first of these maps is surjective at the level of vertices, and the right path lifting property asserts that the second of these maps is surjective at the level of vertices. Our goal in this section is to show that $f$ is a Kan fibration if and only if both $\operatorname{ev}_{0}$ and $\operatorname{ev}_{1}$ are trivial Kan fibrations (Proposition 3.1.2.4). For later reference, it will be convenient to name these properties individually.

Definition 3.1.2.1. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. We will say that $f$ is a left fibration if evaluation at the vertex $\{ 0\} \subset \Delta ^1$ induces a trivial Kan fibration of simplicial sets

\[ \operatorname{ev}_0: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 0\} , X_{} ) \times _{ \operatorname{Fun}( \{ 0\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ). \]

We say that $f$ is a right fibration if evaluation at the vertex $\{ 1\} \in \Delta ^1$ induces a trivial Kan fibration

\[ \operatorname{ev}_1: \operatorname{Fun}( \Delta ^1, X_{}) \rightarrow \operatorname{Fun}( \{ 1\} , X_{} ) \times _{ \operatorname{Fun}( \{ 1\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ) \simeq X_{} \times _{ S_{} } \operatorname{Fun}( \Delta ^1, S_{} ). \]

Remark 3.1.2.2. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a left fibration if and only if the induced map $f^{\operatorname{op}}: X_{}^{\operatorname{op}} \rightarrow S_{}^{\operatorname{op}}$ is a right fibration.

Remark 3.1.2.3 (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 $f$.

We can now formulate our main result:

Proposition 3.1.2.4. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. Then $f$ is a Kan fibration if and only if it is both a left fibration and a right fibration.

Warning 3.1.2.5. In the statement of Proposition 3.1.2.4, 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.

Proposition 3.1.2.4 is an immediate consequence of the following dual pair of results:

Proposition 3.1.2.6. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is a left fibration.

$(2)$

For every pair of integers $0 \leq i < n$, every lifting problem

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

admits a solution (indicated by the dotted arrow).

Proposition 3.1.2.7. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. The following conditions are equivalent:

$(1)$

The morphism $f$ is a right fibration.

$(2)$

For every pair of integers $0 < i \leq n$, 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 (indicated by the dotted arrow).

We now give a proof of Proposition 3.1.2.7 (Proposition 3.1.2.6 then follows by passing to opposite simplicial sets). The proof will require some preliminaries.

Lemma 3.1.2.8. For every pair of integers $0 < i \leq n$, the horn inclusion $f_0: \Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ is a retract of the inclusion map $f: (\Delta ^1 \times \Lambda ^ n_ i) \coprod _{ \{ 1\} \times \Lambda ^{n}_ i} ( \{ 1\} \times \Delta ^ n) \hookrightarrow \Delta ^1 \times \Delta ^ n$.

Proof. Let $A_{}$ denote the simplicial subset of $\Delta ^1 \times \Delta ^ n$ given by the union of $\Delta ^1 \times \Lambda ^ n_ i$ with $\{ 1 \} \times \Delta ^ n$. To prove Lemma 3.1.2.8, it will suffice to show that there exists a commutative diagram of simplicial sets

\[ \xymatrix@R =50pt@C=50pt{ \{ 0\} \times \Lambda ^{n}_{i} \ar [r] \ar [d]^{f_0} & A_{} \ar [r] \ar [d]^{f} & \Lambda ^{n}_{i} \ar [d]^{f_0} \\ \{ 0\} \times \Delta ^ n \ar [r] & \Delta ^1 \times \Delta ^ n \ar [r]^-{r} & \Delta ^ n } \]

where the left horizontal maps are given by inclusion and the horizontal compositions are the identity maps. To achieve this, it suffices to choose $r$ to be given on vertices by the map of partially ordered sets

\[ r: [1] \times [n] \rightarrow [n] \quad \quad r(j,k) = \begin{cases} i & \text{ if $j=1$ and $k \leq i$ } \\ k & \text{ otherwise. } \end{cases} \]
$\square$

Lemma 3.1.2.9. Let $n$ be a nonnegative integer. Then there exists a chain of simplicial subsets

\[ X(0) \subset X(1) \subset \cdots \subseteq X(n) \subset X(n+1) = \Delta ^1 \times \Delta ^ n \]

with the following properties:

$(a)$

The simplicial $X(0)$ is given by the union of $\Delta ^1 \times \operatorname{\partial \Delta }^ n$ with $\{ 1\} \times \Delta ^ n$ (and can therefore be described abstractly as the pushout $(\Delta ^1 \times \operatorname{\partial \Delta }^ n) \coprod _{ \{ 1\} \times \operatorname{\partial \Delta }^ n} ( \{ 1 \} \times \Delta ^ n )$).

$(b)$

For $0 \leq i \leq n$, the inclusion map $X(i) \hookrightarrow X(i+1)$ fits into a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \Lambda ^{n+1}_{i+1} \ar [r] \ar [d] & X(i) \ar [d] \\ \Delta ^{n+1} \ar [r] & X(i+1). } \]

Proof. For $0 \leq i \leq n$, let $\sigma _ i: \Delta ^{n+1} \rightarrow \Delta ^1 \times \Delta ^ n$ denote the map of simplicial sets given on vertices by the formula $\sigma _ i(j) = \begin{cases} (0,j) & \text{ if } j \leq i \\ (1,j-1) & \text{ if } j > i. \end{cases}$ We define simplicial subsets $X(i) \subseteq \Delta ^1 \times \Delta ^ n$ inductively by the formulae

\[ X(0) = (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \cup (\{ 1\} \times \Delta ^ n) \quad \quad X(i+1) = X(i) \cup \operatorname{im}( \sigma _{i} ), \]

where $\operatorname{im}( \sigma _{i} )$ denotes the image of the morphism $\sigma _{i}$. Note that $\Delta ^1 \times \Delta ^ n$ is the union of the simplicial subsets $\{ \operatorname{im}(\sigma _ i) \} _{0 \leq i \leq n}$, and is therefore equal to $X(n+1)$. This definition satisfies condition $(a)$ by construction. To verify $(b)$, it will suffice to show that for $0 \leq i \leq n$, the inverse image $A_{} = \sigma _{i}^{-1} X(i)$ is equal to $\Lambda ^{n+1}_{i+1}$ (as a simplicial subset of $\Delta ^{n+1}$). Regarding $\sigma _{i}$ as an $(n+1)$-simplex of $\Delta ^1 \times \Delta ^ n$, we are reduced to showing that the faces $d_{j}( \sigma _{i} )$ belong to $X(i)$ if and only if $j \neq i+1$. One direction is clear: the face $d_{j}( \sigma _{i} )$ is contained in $\Delta ^1 \times \operatorname{\partial \Delta }^ n$ for $j \notin \{ i, i+1 \} $, the face $d_{i}(\sigma _ i) = d_{i}( \sigma _{i-1} )$ is contained in $\operatorname{im}( \sigma _{i-1} ) \subseteq X(i)$ for $i > 0$, and $d_0( \sigma _0)$ is contained in $\{ 1\} \times \Delta ^ n$. To complete the proof, it suffices to show that the face $d_{i+1}( \sigma _ i )$ is not contained in $X(i)$, which follows by inspection. $\square$

Proof of Proposition 3.1.2.7. Let $f: X_{} \rightarrow S_{}$ be a morphism of simplicial sets. If $f$ is a right 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 (this follows from the dual of Remark 3.1.2.3). 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 3.1.2.8). 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.

Conversely, suppose that $f$ has the right lifting property with respect to every horn inclusion $\Lambda ^{n}_{i} \hookrightarrow \Delta ^ n$ for $0 < i \leq n$. Combining Proposition 1.4.4.14, Proposition 1.4.4.5, and Lemma 3.1.2.9, we conclude that $f$ also has the right lifting property with respect to the inclusion map

\[ (\Delta ^1 \times \operatorname{\partial \Delta }^ n) \coprod _{ \{ 1\} \times \operatorname{\partial \Delta }^ n} ( \{ 1 \} \times \Delta ^ n ) \hookrightarrow \Delta ^1 \times \Delta ^ n \]

for each $n \geq 0$. More concretely, this asserts that every

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

admits a solution, or equivalently that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \operatorname{Fun}( \Delta ^1, X_{} ) \ar [d]^{\operatorname{ev}_1} \\ \Delta ^ n \ar [r] \ar@ {-->}[ur] & \operatorname{Fun}( \{ 1\} , X_{} ) \times _{ \operatorname{Fun}( \{ 1\} , S_{} ) } \operatorname{Fun}( \Delta ^1, S_{} ). } \]

admits a solution. It follows that 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 of simplicial sets, so that $f: X_{} \rightarrow S_{}$ is a right fibration. $\square$