Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$

1.4.5 Trivial Kan Fibrations

We now specialize the ideas of §1.4.4 to the category of simplicial sets.

Definition 1.4.5.1. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a map of simplicial sets. We say that $p$ is a trivial Kan fibration if, for each $n \geq 0$, every lifting problem

$\xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^ n \ar [d]^{i} \ar [r] & X_{\bullet } \ar [d]^{p} \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & Y_{\bullet } }$

admits a solution; here $i: \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ denotes the inclusion map.

Remark 1.4.5.2. Suppose we are given a pullback diagram of simplicial sets

$\xymatrix@R =40pt@C=40pt{ X'_{\bullet } \ar [d]^{p'} \ar [r] & X_{\bullet } \ar [d]^{p} \\ Y'_{\bullet } \ar [r] & Y_{\bullet }. }$

If $p$ is a trivial Kan fibration, then so is $p'$ (this follows from Proposition 1.4.4.5, applied to the opposite of the category $\operatorname{Set_{\Delta }}$).

Proposition 1.4.5.3. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a map of simplicial sets. The following conditions are equivalent:

$(1)$

The map $p$ is a trivial Kan fibration (in the sense of Definition 1.4.5.1).

$(2)$

The map $p$ has the right lifting property with respect to every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$. In other words, every lifting problem

$\xymatrix@C =40pt@R=40pt{ A_{\bullet } \ar [d]^{i} \ar [r] & X_{\bullet } \ar [d]^{p} \\ B_{\bullet } \ar@ {-->}[ur] \ar [r] & Y_{\bullet } }$

admits a solution, provided that $i$ is a monomorphism.

We will give the proof of Proposition 1.4.5.3 at the end of this section.

Corollary 1.4.5.4. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a trivial Kan fibration of simplicial sets. Then:

$(a)$

The map $p$ admits a section: that is, there is a map of simplicial sets $s: Y_{\bullet } \rightarrow X_{\bullet }$ such that the composition $p \circ s$ is the identity map $\operatorname{id}_{Y_{\bullet }}: Y_{\bullet } \rightarrow Y_{\bullet }$.

$(b)$

Let $s$ be any section of $p$. Then the composition $s \circ p: X_{\bullet } \rightarrow X_{\bullet }$ is fiberwise homotopic to the identity. That is, there exists a map of simplicial sets $h: \Delta ^1 \times X_{\bullet } \rightarrow X_{\bullet }$, compatible with the projection to $Y_{\bullet }$, such that $h|_{ \{ 0\} \times X_{\bullet } } = s \circ p$ and $h|_{ \{ 1\} \times X_{\bullet } } = \operatorname{id}_{X_{\bullet }}$.

Proof. To prove $(a)$, we observe that a section of $p$ can be described as a solution to the lifting problem

$\xymatrix@C =40pt@R=40pt{ \emptyset \ar [d] \ar [r] & X_{\bullet } \ar [d]^{p} \\ Y_{\bullet } \ar [r]^{\operatorname{id}} \ar@ {-->}[ur]^{s} & Y_{\bullet }, }$

which exists by virtue of Proposition 1.4.5.3. Given any section $s$, a fiberwise homotopy from $s \circ p$ to the identity can be identified with a solution to the lifting problem

$\xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^1 \times X_{\bullet } \ar [d] \ar [r]^-{(s \circ p, \operatorname{id})} & X_{\bullet } \ar [d]^{p} \\ \Delta ^1 \times X_{\bullet } \ar [r] \ar@ {-->}[ur]^{h} & Y_{\bullet }, }$

which again exists by virtue of Proposition 1.4.5.3. $\square$

Corollary 1.4.5.5. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a trivial Kan fibration of simplicial sets and let $i: A_{\bullet } \rightarrow B_{\bullet }$ be a monomorphism of simplicial sets. Then the canonical map

$\theta : \operatorname{Fun}( B_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Fun}( B_{\bullet }, Y_{\bullet }) \times _{ \operatorname{Fun}( A_{\bullet }, Y_{\bullet } ) } \operatorname{Fun}( A_{\bullet }, X_{\bullet } )$

is also a trivial Kan fibration.

Proof. Fix an integer $n \geq 0$; we wish to show that every lifting problem

$\xymatrix@C =70pt@R=70pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \operatorname{Fun}( B_{\bullet }, X_{\bullet } ) \ar [d]^{\theta } \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B_{\bullet }, Y_{\bullet }) \times _{ \operatorname{Fun}( A_{\bullet }, Y_{\bullet } ) } \operatorname{Fun}( A_{\bullet }, X_{\bullet } ). }$

admits a solution. Unwinding the definitions, we see that this is equivalent to solving an associated lifting problem

$\xymatrix@C =70pt@R=70pt{ (\operatorname{\partial \Delta }^ n \times B_{\bullet }) \underset {\operatorname{\partial \Delta }^ n \times A_{\bullet }}{ \coprod } ( \Delta ^ n \times A_{\bullet } ) \ar [r] \ar [d]^{i} & X_{\bullet } \ar [d]^{p} \\ \Delta ^ n \times B_{\bullet } \ar@ {-->}[ur] \ar [r] & Y_{\bullet }. }$

This is possible by virtue of Proposition 1.4.5.3, since $p$ is a trivial Kan fibration and $i$ is a monomorphism. $\square$

Corollary 1.4.5.6. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a trivial Kan fibration of simplicial sets. Then, for every simplicial set $B_{\bullet }$, the induced map $\operatorname{Fun}( B_{\bullet }, X_{\bullet } ) \rightarrow \operatorname{Fun}( B_{\bullet }, Y_{\bullet } )$ is a trivial Kan fibration.

Proof. Apply Corollary 1.4.5.5 in the special case $A_{\bullet } = \emptyset$. $\square$

Definition 1.4.5.7. Let $X_{\bullet }$ be a simplicial set. We say that $X_{\bullet }$ is a contractible Kan complex if the projection map $X_{\bullet } \rightarrow \Delta ^0$ is a trivial Kan fibration (Definition 1.4.5.1). In other words, $X_{\bullet }$ is a contractible Kan complex if every map $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow X_{\bullet }$ can be extended to an $n$-simplex of $X_{\bullet }$.

Example 1.4.5.8. Let $X$ be a topological space. Then the singular simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a contractible Kan complex if and only if the space $X$ is weakly contractible: that is, if and only if every continuous map $\sigma _0: S^{n-1} \rightarrow X$ is nullhomotopic (here $S^{n-1} \simeq | \operatorname{\partial \Delta }^ n |$ denotes the sphere of dimension $n-1$, so that $\sigma _0$ is nullhomotopic if and only if extends to a continuous map defined on the disk $D^ n \simeq | \Delta ^ n |$). In particular, if the topological space $X$ is contractible, then the simplicial set $\operatorname{Sing}_{\bullet }(X)$ is a contractible Kan complex.

Applying Proposition 1.4.5.3 in the case $Y_{\bullet } = \Delta ^0$, we obtain the following:

Corollary 1.4.5.10. Let $X_{\bullet }$ be a simplicial set. The following conditions are equivalent:

$(1)$

The simplicial set $X_{\bullet }$ is a contractible Kan complex.

$(2)$

For every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ and every map of simplicial sets $f_0: A_{\bullet } \rightarrow X_{\bullet }$, there exists a map $f: B_{\bullet } \rightarrow X_{\bullet }$ such that $f_0 = f \circ i$.

Corollary 1.4.5.11. Let $X_{\bullet }$ be a contractible Kan complex. Then $X_{\bullet }$ is a Kan complex. In particular, $X_{\bullet }$ is an $\infty$-category.

We will deduce Proposition 1.4.5.3 from the following:

Proposition 1.4.5.12. Let $T$ be the collection of all monomorphisms in the category $\operatorname{Set_{\Delta }}$ of simplicial sets. Then:

$(a)$

The collection $T$ is weakly saturated, in the sense of Definition 1.4.4.15.

$(b)$

As a weakly saturated collection of morphisms, $T$ is generated by collection of inclusion maps $\{ \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n \} _{n \geq 0}$ (see Remark 1.4.4.17).

Proof. To prove $(a)$, we must establish the following:

• The collection $T$ is closed under pushouts. That is, if we are given a pushout diagram of simplicial sets

$\xymatrix { A_{\bullet } \ar [d]^{f} \ar [r] & A'_{\bullet } \ar [d]^{f'} \\ B_{\bullet } \ar [r] & B'_{\bullet } }$

where $f$ is a monomorphism, then $f'$ is also a monomorphism. This is clear, since we have a pushout diagram

$\xymatrix { A_{n} \ar [d] \ar [r] & A'_{n} \ar [d] \\ B_{n} \ar [r] & B'_{n} }$

in the category of sets for each $n \geq 0$ (where the left vertical map is injective, so the right vertical map is injective as well).

• The collection $T$ is closed under retracts. This is a special case of Example 1.4.4.8.

• The collection $T$ is closed under transfinite composition. Suppose we are given an ordinal $\alpha$ and a functor $S: [\alpha ] \rightarrow \operatorname{Set_{\Delta }}$, given by a collection of simplicial sets $\{ S(\beta )_{\bullet } \} _{ \beta \leq \alpha }$ and transition maps $f_{\gamma ,\beta }: S(\beta )_{\bullet } \rightarrow S(\gamma )_{\bullet }$. Assume that the maps $f_{\beta +1,\beta }$ are monomorphisms for $\beta < \alpha$ and that, for every nonzero limit ordinal $\lambda \leq \alpha$, the induced map $\varinjlim _{\beta < \lambda } S(\beta )_{\bullet } \rightarrow S(\lambda )_{\bullet }$ is an isomorphism. We must show that the map $f_{\alpha ,0}: S(0)_{\bullet } \rightarrow S(\alpha )_{\bullet }$ is a monomorphism of simplicial sets. In fact, we claim that for each $\gamma \leq \alpha$, the map $f_{\gamma ,0}: S(0)_{\bullet } \rightarrow S(\gamma )_{\bullet }$ is a monomorphism. The proof proceeds by transfinite induction on $\gamma$. In the case $\gamma = 0$, the map $f_{\gamma ,0} = \operatorname{id}_{ S(0)_{\bullet } }$ is an isomorphism. If $\gamma$ is a nonzero limit ordinal, then the desired result follows from our inductive hypothesis, since the collection of monomorphisms in $\operatorname{Set_{\Delta }}$ is closed under filtered colimits. If $\gamma = \beta + 1$ is a successor ordinal, then we can identify $f_{0, \gamma }$ with the composition

$S(0)_{\bullet } \xrightarrow { f_{\beta ,0} } S(\beta )_{\bullet } \xrightarrow { f_{\gamma ,\beta } } S(\gamma )_{\bullet },$

where $f_{\beta , \gamma }$ is a monomorphism by assumption and $f_{0, \beta }$ is a monomorphism by virtue of our inductive hypothesis.

We now prove $(b)$. Let $T'$ be a collection of morphisms in $\operatorname{Set_{\Delta }}$ which is weakly saturated and contains each of the inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$; we wish to show that every monomorphism $i: A_{\bullet } \rightarrow B_{\bullet }$ belongs to $T'$. For each $k \geq -1$, let $B(k)_{\bullet } \subseteq B_{\bullet }$ denote the simplicial subset given by the union of the skeleton $\operatorname{sk}_ k( B_{\bullet } )$ (Construction 1.1.3.5) with the image of $i$. Then the inclusion $i$ can be written as a transfinite composition

$A_{\bullet } \simeq B(-1)_{\bullet } \hookrightarrow B(0)_{\bullet } \hookrightarrow B(1)_{\bullet } \hookrightarrow B(2)_{\bullet } \hookrightarrow \cdots$

Since $T'$ is closed under transfinite composition, it will suffice to show that each of the inclusion maps $B(k-1)_{\bullet } \hookrightarrow B(k)_{\bullet }$ belongs to $T'$. Applying Proposition 1.1.3.11 to both $A_{\bullet }$ and $B_{\bullet }$, we obtain a pushout diagram

$\xymatrix { \underset { \sigma \in Q }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { \sigma \in Q }{\coprod } \Delta ^{k} \ar [d] \\ B(k-1)_{\bullet } \ar [r] & B(k)_{\bullet } }$

where $Q$ denotes the collection of all nondegenerate $k$-simplices of $B_{\bullet }$ which do not belong to the image of $i$. Since $T'$ is closed under pushouts, we are reduced to showing that the inclusion map

$j: \coprod _{\sigma \in Q} \operatorname{\partial \Delta }^ k \hookrightarrow \coprod _{ \sigma \in Q} \Delta ^ k$

belongs to $T'$. Choosing a well-ordering of $Q$, we see that $j$ can be written as a transfinite composition of morphisms

$j_{\sigma }: (\coprod _{ \tau \leq \sigma } \operatorname{\partial \Delta }^ k ) \amalg ( \coprod _{\tau > \sigma } \Delta ^ k ) \hookrightarrow (\coprod _{ \tau < \sigma } \operatorname{\partial \Delta }^ k ) \amalg ( \coprod _{\tau \geq \sigma } \Delta ^ k ),$

each of which is a pushout of the inclusion $\operatorname{\partial \Delta }^ k \hookrightarrow \Delta ^ k$. $\square$

Proof of Proposition 1.4.5.3. Let $p: X_{\bullet } \rightarrow Y_{\bullet }$ be a trivial Kan fibration of simplicial sets and let $T$ be the collection of all morphisms in $\operatorname{Set_{\Delta }}$ which have the left lifting property with respect to $p$. Then $T$ contains each of the inclusions $\operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n$ (by virtue of our assumption that $p$ is a trivial Kan fibration) and is weakly saturated (Proposition 1.4.4.16). It follows from Proposition 1.4.5.12 that every monomorphism of simplicial sets $i: A_{\bullet } \hookrightarrow B_{\bullet }$ belongs to $T$ (and therefore has the left lifting property with respect to $p$). $\square$