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4.3.7 Slices of Left and Right Fibrations

In this section, we collect some further applications of Lemma 4.3.6.10.

Proposition 4.3.7.1. Let $f: A \hookrightarrow A'$ and $g: B \hookrightarrow B'$ be monomorphisms of simplicial sets, and let

\[ \theta _{f,g}: (A \star B') \coprod _{ (A \star B) } (A' \star B) \hookrightarrow A' \star B' \]

be the pushout-join of Construction 4.3.6.3. If $f$ is anodyne, then $\theta _{f,g}$ is left anodyne. If $g$ is anodyne, then $\theta _{f,g}$ is right anodyne.

Proof. We will prove the first assertion; the proof of the second is similar. We proceed as in the proof of Proposition 4.3.6.4. Let us first regard the anodyne morphism $f$ as fixed, and let $T$ be the collection of all morphisms $g$ of simplicial sets for which $\theta _{f,g}$ is left anodyne. Then $T$ weakly saturated (in the sense of Definition 1.4.4.15). We wish to prove that $T$ contains every monomorphism of simplicial sets. By virtue of Proposition 1.4.5.12, we are reduced to proving that the morphism $\theta _{f,g}$ is left anodyne in the special case where $g$ is the boundary inclusion $\operatorname{\partial \Delta }^{q} \hookrightarrow \Delta ^{q}$ for some $q \geq 0$.

Let us now regard $g: \operatorname{\partial \Delta }^ q \hookrightarrow \Delta ^ q$ as fixed, and let $S$ denote the collection of all morphisms of simplicial sets for which $\theta _{f,g}$ is left anodyne. To complete the proof, we must show that $S$ contains every anodyne morphism of simplicial sets. As before, we note that $S$ is weakly saturated. It will therefore suffice to show that $S$ contains every horn inclusion $\Lambda ^{p}_{i} \hookrightarrow \Delta ^ p$ when $p > 0$ and $0 \leq i \leq p$. In other words, we are reduced to checking that the pushout-join

\[ \theta _{f,g}: (\Lambda ^{p}_{i} \star \Delta ^ q) \coprod _{ ( \Lambda ^{p}_{i} \star \operatorname{\partial \Delta }^ q) } ( \Delta ^ p \star \operatorname{\partial \Delta }^ q) \hookrightarrow \Delta ^{p} \star \Delta ^ q \]

is left anodyne. This is clear, since $\theta _{f,g}$ can be identified with the horn inclusion $\Lambda ^{p+1+q}_{i} \hookrightarrow \Delta ^{p+1+q}$ by virtue of Lemma 4.3.6.10. $\square$

Proposition 4.3.7.2. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets, let $K_0 \subseteq K$ be a simplicial subset, and set $f_0 = f|_{K_0}$. Then:

  • If $q$ is a left fibration, then the induced map

    \[ X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} \]

    is a Kan fibration.

  • If $q$ is a right fibration, then the induced map

    \[ X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/ } } S_{(q \circ f)/} \]

    is a Kan fibration.

Proof. We will prove the first assertion; the proof of the second is similar. Assume that $q$ is a left fibration; we wish to show that the map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)}$ is a Kan fibration. Equivalently, we wish to show that every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] & X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} } \]

admits a solution, provided that the left vertical map $A \rightarrow A'$ is anodyne. Unwinding the definitions, we see that this can be rephrased as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ (A \star K) \coprod _{ (A \star K_0) } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar [r] \ar@ {-->}[ur] & S. } \]

This problem admits a solution, since the vertical map on the left is left anodyne (Proposition 4.3.7.1) and $q$ is a left fibration. $\square$

Corollary 4.3.7.3. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets. Then:

  • If $q$ is a left fibration, then the induced map

    \[ X_{/f} \rightarrow X \times _{ S} S_{/ (q \circ f)} \]

    is a Kan fibration.

  • If $q$ is a right fibration, then the induced map

    \[ X_{f/} \rightarrow X \times _{ S} S_{(q \circ f)/} \]

    is a Kan fibration.

Proof. Apply Proposition 4.3.7.2 in the special case $K_0 = \emptyset $. $\square$

Corollary 4.3.7.4. Let $X$ be a Kan complex, let $f: K \rightarrow X$ be a morphism of simplicial sets, let $K_0 \subseteq K$ be a simplicial subset, and set $f_0 = f|_{K_0}$. Then the restriction maps

\[ X_{/f} \rightarrow X_{/f_0} \quad \quad X_{f/} \rightarrow X_{f_0/} \]

are Kan fibrations.

Proof. Apply Proposition 4.3.7.2 in the special case $S = \Delta ^{0}$. $\square$

Corollary 4.3.7.5. Let $X$ be a Kan complex and let let $f: K \rightarrow X$ be a morphism of simplicial sets. Then the projection maps

\[ X_{/f} \rightarrow X \quad \quad X_{f/} \rightarrow X \]

are Kan fibrations. In particular, the simplicial sets $X_{/f}$ and $X_{f/}$ are Kan complexes.

Proof. Apply Corollary 4.3.7.4 in the special case $K_0 = \emptyset $ (or Corollary 4.3.7.3 in the special case $S = \Delta ^{0}$). $\square$

Proposition 4.3.7.6. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets, let $K_0 \subseteq K$ be a simplicial subset, and set $f_0 = f|_{K_0}$. Then:

  • If $q$ is a right fibration and the inclusion $K_0 \hookrightarrow K$ is anodyne, then the induced map

    \[ X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} \]

    is a trivial Kan fibration.

  • If $q$ is a left fibration and the inclusion $K_0 \hookrightarrow K$ is anodyne, then the induced map

    \[ X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/ } } S_{(q \circ f)/} \]

    is a trivial Kan fibration.

Proof. We will prove the first assertion; the proof of the second is similar. Assume that $q$ is a right fibration and that the inclusion $K_0 \hookrightarrow K$ is anodyne. We wish to show that the map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)}$ is a trivial Kan fibration. Equivalently, we wish to show that every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] & X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} } \]

admits a solution, provided that the left vertical map $A \rightarrow A'$ is a monomorphism. Unwinding the definitions, we see that this can be rephrased as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ (A \star K) \coprod _{ (A \star K_0) } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar [r] \ar@ {-->}[ur] & S. } \]

This problem admits a solution, since the vertical map on the left is right anodyne (Proposition 4.3.7.1) and $q$ is a right fibration. $\square$

Corollary 4.3.7.7. Let $X$ be a Kan complex, let $f: K \rightarrow X$ be a morphism of simplicial sets, and let $K_0 \subseteq K$ be a simplicial subset for which the inclusion $K_0 \hookrightarrow K$ is anodyne, and set $f_0 = f|_{K_0}$. Then the restriction maps

\[ X_{/f} \rightarrow X \quad \quad X_{f/} \rightarrow X \]

are trivial Kan fibrations.

Proof. Apply Proposition 4.3.7.6 in the special case $S = \Delta ^{0}$. $\square$

We now record some variants of the preceding results.

Lemma 4.3.7.8. Let $f: A \hookrightarrow B$ be a monomorphism of simplicial sets. Then the inclusion $f^{\triangleright }: A^{\triangleright } \hookrightarrow B^{\triangleright }$ is right anodyne, and the inclusion $A^{\triangleleft } \hookrightarrow B^{\triangleleft }$ is left anodyne.

Proof. We will prove the first assertion (the second follows by a similar argument). Let $T$ be the collection of all morphisms $f$ of simplicial sets for which $f^{\triangleright }$ is right anodyne. We wish to show that every monomorphism belongs to $T$. Since the collection $T$ is weakly saturated, it will suffice to show that every horn inclusion $f: \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ belongs to $T$ (Proposition 1.4.5.12). In this case, we can identify $f^{\triangleright }$ with the with the horn inclusion $\Lambda ^{n+1}_{n+1} \hookrightarrow \Delta ^{n}$ (see Example 4.3.3.24). $\square$

Lemma 4.3.7.8 immediately implies the following stronger assertion:

Proposition 4.3.7.9. Let $X$ and $Y$ be simplicial sets. If $X$ is weakly contractible and $Y'$ is a simplicial subset of $Y$, then the inclusion $X \star Y' \hookrightarrow X \star Y$ is left anodyne. If $Y$ is weakly contractible and $X'$ is a simplicial subset of $X$, then the inclusion $X' \star Y \hookrightarrow X \star Y$ is right anodyne.

Proof. We will prove the first assertion; the second follows by a similar argument. Fix a vertex $x \in X$, so that the inclusion morphism $\iota : X \star Y' \hookrightarrow X \star Y$ factors as a composition

\[ X \star Y' \xrightarrow {\iota '} (X \star Y') \coprod _{(\{ x\} \star Y')} (\{ x\} \star Y) \xrightarrow {\iota ''} X \star Y. \]

The morphism $\iota '$ is a pushout of the inclusion $Y'^{\triangleleft } \hookrightarrow Y^{\triangleleft }$, and is left anodyne by virtue of Lemma 4.3.7.8. It will therefore suffice to show that $\iota ''$ is left anodyne. This is a special case of Proposition 4.3.7.1, since the inclusion map $\{ x\} \hookrightarrow X$ is a weak homotopy equivalence (by virtue of our assumption that $X$ is weakly contractible) and therefore anodyne (by virtue of Corollary 3.3.7.5). $\square$

Example 4.3.7.10. Let $X$ and $Y$ be simplicial sets. If $X$ is weakly contractible, then Proposition 4.3.7.9 guarantees that the inclusion $\iota _{X}: X \hookrightarrow X \star Y$ is left anodyne. If $Y$ is weakly contractible, then Proposition 4.3.7.9 guarantees that the inclusion $\iota _{Y}: Y \hookrightarrow X \star Y$ is right anodyne.

Proposition 4.3.7.11. Let $q: X \rightarrow S$ and $f: K \rightarrow X$ be morphisms of simplicial sets. Then:

  • If $q$ is a right fibration and $K$ is weakly contractible, then the induced map $X_{/f} \rightarrow S_{/(q \circ f)}$ is a trivial Kan fibration.

  • If $q$ is a left fibration and $K$ is weakly contractible, then the induced map $X_{f/} \rightarrow S_{(q \circ f)/}$ is a trivial Kan fibration.

Proof. We will prove the first assertion; the second follows by a similar argument. To show that the morphism $X_{/f} \rightarrow S_{/(q \circ f)}$ is a trivial Kan fibration, we must prove that every lifting problem

\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\partial \Delta }^{n} \ar [r] \ar [d] & X_{/q} \ar [d] \\ \Delta ^{n} \ar [r] \ar@ {-->}[ur] & S_{/q \circ f} } \]

admits a solution. Unwinding the definitions, we can rephrase this as a lifting problem

\[ \xymatrix@R =50pt@C=50pt{ ( \operatorname{\partial \Delta }^{n} ) \star K \ar [r] \ar [d] & X \ar [d]^{q} \\ (\Delta ^{n} ) \star K \ar@ {-->}[ur] \ar [r] & S. } \]

This lifting problem admits a solution, since $q$ is assumed to be a right fibration and the left vertical map is right anodyne (Proposition 4.3.7.9). $\square$

Corollary 4.3.7.12. Let $q: X \rightarrow S$ be a morphism of simplicial sets, and let $x \in X$ be a vertex having image $s = q(x)$ in $S$. Then:

  • If $q$ is a right fibration, then the induced map $X_{/x} \rightarrow S_{/s}$ is a trivial Kan fibration.

  • If $q$ is a left fibration, then the induced map $X_{x/} \rightarrow S_{s/}$ is a trivial Kan fibration.

Corollary 4.3.7.13. Let $X$ be a Kan complex containing a vertex $x$. Then the simplicial sets $X_{/x}$ and $X_{x/}$ are contractible Kan complexes.

Proof. Apply Corollary 4.3.7.12 in the special case $S = \Delta ^{0}$. $\square$

Proposition 4.3.7.14. Let $f: K \rightarrow X$ and $q: X \rightarrow S$ be morphisms of simplicial sets, let $K_0 \subseteq K$ be a simplicial subset, and set $f_0 = f|_{K_0}$. If $q$ is a trivial Kan fibration, then the induced maps

\[ X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} \quad \quad X_{f/} \rightarrow X_{f_0/} \times _{ S_{(q \circ f_0)/ } } S_{(q \circ f)/} \]

are also trivial Kan fibrations.

Proof. To show that the map $X_{/f} \rightarrow X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)}$ is a trivial Kan fibration, we must show that every lifting problem every lifting problem

\[ \xymatrix@C =50pt@R=50pt{ A \ar [r] \ar [d] & X_{/f} \ar [d] \\ A' \ar [r] \ar@ {-->}[ur] & X_{/f_0} \times _{ S_{ / (q \circ f_0) } } S_{/ (q \circ f)} } \]

admits a solution, provided that the left vertical map $A \rightarrow A'$ is a monomorphism. Unwinding the definitions, we see that this can be rephrased as a lifting problem

\[ \xymatrix@C =50pt@R=50pt{ (A \star K) \coprod _{ (A \star K_0) } (A' \star K_0) \ar [r] \ar [d] & X \ar [d]^{q} \\ A' \star K \ar [r] \ar@ {-->}[ur] & S. } \]

This problem admits a solution, since the vertical map on the left is a monomorphism (Proposition 4.3.7.1) and $q$ is a trivial Kan fibration. $\square$

Corollary 4.3.7.15. Let $q: X \rightarrow S$ be a trivial Kan fibrations of simplicial sets, and let $f: K \rightarrow X$ be any morphism of simplicial sets. Then the induced maps

\[ X_{/f} \rightarrow X \times _{ S} S_{/ (q \circ f)} \quad \quad X_{f/} \rightarrow X \times _{ S} S_{(q \circ f)/} \]

are trivial Kan fibrations.

Proof. Apply Proposition 4.3.7.14 in the special case $K_0 = \emptyset $. $\square$

Corollary 4.3.7.16. Let $X$ be a contractible Kan complex, let $f: K \rightarrow X$ be a morphism of simplicial sets, let $K_0$ be a simplicial subset of $K$, and set $f_0 = f|_{K_0}$. Then the restriction maps

\[ X_{/f} \rightarrow X_{/f_0} \quad \quad X_{f/} \rightarrow X_{f_0/} \]

are trivial Kan fibrations.

Proof. Apply Proposition 4.3.7.14 in the special case $S = \Delta ^{0}$. $\square$

Corollary 4.3.7.17. Let $X$ be a contractible Kan complex and let $f: K \rightarrow X$ be a morphism of simplicial sets. Then the projection maps

\[ X_{/f} \rightarrow X \quad \quad X_{f/} \rightarrow X \]

are trivial Kan fibrations. In particular, $X_{/f}$ and $X_{f/}$ are also contractible Kan complexes.

Proof. Apply Corollary 4.3.7.15 in the special case $S = \Delta ^{0}$ (or Corollary 4.3.7.16 in the special case $K_0 = \emptyset $). $\square$