### 1.5.5 Trivial Kan Fibrations

We now specialize the ideas of ยง1.5.4 to the category of simplicial sets.

Definition 1.5.5.1. Let $q: X \rightarrow Y$ be a morphism of simplicial sets. We say that $q$ is a *trivial Kan fibration* if, for each $n \geq 0$, every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ \operatorname{\partial \Delta }^ n \ar [d]^{i} \ar [r] & X \ar [d]^{q} \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & Y } \]

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

Proposition 1.5.5.4. Let $p: X \rightarrow Y$ 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 1.5.5.1).

- $(2)$
The map $p$ is weakly right orthogonal to every monomorphism of simplicial sets $i: A \hookrightarrow B$. In other words, every lifting problem

\[ \xymatrix@C =40pt@R=40pt{ A \ar [d]^{i} \ar [r] & X \ar [d]^{p} \\ B \ar@ {-->}[ur] \ar [r] & Y } \]

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

We will give the proof of Proposition 1.5.5.4 at the end of this section.

Corollary 1.5.5.5. Let $p: X \rightarrow Y$ 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 $s: Y \rightarrow X$ such that the composition $p \circ s$ is the identity map $\operatorname{id}_{Y}: Y \rightarrow Y$.

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

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

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

which exists by virtue of Proposition 1.5.5.4. Given any section $s$, a fiberwise homotopy from $s \circ p$ to the identity can be identified with a solution to the lifting problem

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

which again exists by virtue of Proposition 1.5.5.4.
$\square$

Corollary 1.5.5.6. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets and let $i: A \rightarrow B$ be a monomorphism of simplicial sets. Then the canonical map

\[ \theta : \operatorname{Fun}( B, X ) \rightarrow \operatorname{Fun}( B, Y) \times _{ \operatorname{Fun}( A, Y ) } \operatorname{Fun}( A, X ) \]

is also a trivial Kan fibration.

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

\[ \xymatrix@C =70pt@R=70pt{ \operatorname{\partial \Delta }^ n \ar [r] \ar [d] & \operatorname{Fun}( B, X ) \ar [d]^{\theta } \\ \Delta ^ n \ar@ {-->}[ur] \ar [r] & \operatorname{Fun}( B, Y) \times _{ \operatorname{Fun}( A, Y ) } \operatorname{Fun}( A, X ) } \]

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

\[ \xymatrix@C =70pt@R=70pt{ (\operatorname{\partial \Delta }^ n \times B) \underset {\operatorname{\partial \Delta }^ n \times A}{ \coprod } ( \Delta ^ n \times A ) \ar [r] \ar [d]^{i} & X \ar [d]^{p} \\ \Delta ^ n \times B \ar@ {-->}[ur] \ar [r] & Y. } \]

This is possible by virtue of Proposition 1.5.5.4, since $p$ is a trivial Kan fibration and $i$ is a monomorphism.
$\square$

Corollary 1.5.5.7. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then, for every simplicial set $B$, the induced map $\operatorname{Fun}( B, X ) \rightarrow \operatorname{Fun}( B, Y )$ is a trivial Kan fibration.

**Proof.**
Apply Corollary 1.5.5.6 in the special case $A = \emptyset $.
$\square$

Definition 1.5.5.8. Let $X$ be a simplicial set. We say that $X$ is a *contractible Kan complex* if the projection map $X \rightarrow \Delta ^0$ is a trivial Kan fibration (Definition 1.5.5.1). In other words, $X$ is a contractible Kan complex if every map $\sigma _0: \operatorname{\partial \Delta }^ n \rightarrow X$ can be extended to an $n$-simplex of $X$.

Example 1.5.5.9. 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 it 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.

Proposition 1.5.5.11. Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets. Then:

- $(1)$
If $X$ is a Kan complex, then $Y$ is a Kan complex.

- $(2)$
If $X$ is a contractible Kan complex, then $Y$ is a contractible Kan complex.

- $(3)$
If $X$ is an $\infty $-category, then $Y$ is an $\infty $-category.

**Proof.**
We will prove $(1)$; the proofs of $(2)$ and $(3)$ are similar. Suppose we are given a pair of integers $0 \leq i \leq n$ with $n > 0$; we wish to show that every morphism of simplicial sets $\sigma _0: \Lambda ^{n}_{i} \rightarrow Y$ can be extended to an $n$-simplex of $Y$. Since $p$ is a trivial Kan fibration, we can write $\sigma _0 = p \circ \tau _0$ for some morphism $\tau _0: \Lambda ^{n}_{i} \rightarrow X$ (Proposition 1.5.5.4). If $X$ is a Kan complex, we can extend $\tau _0$ to an $n$-simplex $\tau $ of $X$. Then $\sigma = p \circ \tau $ is an $n$-simplex of $Y$ satisfying $\sigma _0 = \sigma |_{ \Lambda ^{n}_{i} }$.
$\square$

Applying Proposition 1.5.5.4 in the case $Y = \Delta ^0$, we obtain the following:

Corollary 1.5.5.12. Let $X$ be a simplicial set. The following conditions are equivalent:

- $(1)$
The simplicial set $X$ is a contractible Kan complex.

- $(2)$
For every monomorphism of simplicial sets $i: A \hookrightarrow B$ and every map of simplicial sets $f_0: A \rightarrow X$, there exists a map $f: B \rightarrow X$ such that $f_0 = f \circ i$.

Corollary 1.5.5.13. Let $X$ be a contractible Kan complex. Then $X$ is a Kan complex. In particular, $X$ is an $\infty $-category.

We will deduce Proposition 1.5.5.4 from the following:

Proposition 1.5.5.14. 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 1.5.4.12.

- $(b)$
As a weakly saturated collection of morphisms, $T$ is generated by the collection of inclusion maps $\{ \operatorname{\partial \Delta }^ n \hookrightarrow \Delta ^ n \} _{n \geq 0}$ (see Remark 1.5.4.14).

**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

\[ \xymatrix@R =50pt@C=50pt{ A \ar [d]^{f} \ar [r] & A' \ar [d]^{f'} \\ B \ar [r] & B' } \]

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

\[ \xymatrix@R =50pt@C=50pt{ A_{n} \ar [d] \ar [r] & A'_{n} \ar [d] \\ B_{n} \ar [r] & B'_{n} } \]

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 Exercise 1.5.4.8.

The collection $T$ is closed under transfinite composition. Suppose we are given an ordinal $\alpha $ and a functor $S: \mathrm{Ord}_{\leq \alpha } \rightarrow \operatorname{Set_{\Delta }}$, given by a collection of simplicial sets $\{ S(\beta ) \} _{ \beta \leq \alpha }$ and transition maps $f_{\gamma ,\beta }: S(\beta ) \rightarrow S(\gamma )$. 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 $\varinjlim _{\beta < \lambda } S(\beta ) \rightarrow S(\lambda )$ is an isomorphism. We must show that the map $f_{\alpha ,0}: S(0) \rightarrow S(\alpha )$ is a monomorphism of simplicial sets. In fact, we claim that for each $\gamma \leq \alpha $, the map $f_{\gamma ,0}: S(0) \rightarrow S(\gamma )$ is a monomorphism. The proof proceeds by transfinite induction on $\gamma $. In the case $\gamma = 0$, the map $f_{\gamma ,0} = \operatorname{id}_{ S(0) }$ 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 $f_{\gamma ,0}$ with the composition

\[ S(0) \xrightarrow { f_{\beta ,0} } S(\beta ) \xrightarrow { f_{\gamma ,\beta } } S(\gamma ), \]

where $f_{\gamma ,\beta }$ is a monomorphism by assumption and $f_{\beta ,0}$ 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 $i: A \rightarrow B$ belongs to $T'$. For each $k \geq -1$, let $B(k) \subseteq B$ denote the simplicial subset given by the union of the skeleton $\operatorname{sk}_ k( B )$ (Construction 1.1.4.1) with the image of $i$. Then the inclusion $i$ can be written as a transfinite composition

\[ A \simeq B(-1) \hookrightarrow B(0) \hookrightarrow B(1) \hookrightarrow B(2) \hookrightarrow \cdots \]

Since $T'$ is closed under transfinite composition, it will suffice to show that each of the inclusion maps $B(k-1) \hookrightarrow B(k)$ belongs to $T'$. Applying Proposition 1.1.4.12 to both $A$ and $B$, we obtain a pushout diagram

\[ \xymatrix@R =50pt@C=50pt{ \underset { \sigma \in Q }{\coprod } \operatorname{\partial \Delta }^{k} \ar [r] \ar [d] & \underset { \sigma \in Q }{\coprod } \Delta ^{k} \ar [d] \\ B(k-1) \ar [r] & B(k) } \]

where $Q$ denotes the collection of all nondegenerate $k$-simplices of $B$ 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 $T'$. By virtue of Theorem 4.7.1.34, the set $Q$ admits a well-ordering. Then $j$ can be written as a transfinite composition of morphisms

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

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

**Proof of Proposition 1.5.5.4.**
Let $p: X \rightarrow Y$ be a trivial Kan fibration of simplicial sets and let $S$ be the collection of all morphisms in $\operatorname{Set_{\Delta }}$ which are weakly left orthogonal to $p$. Then $S$ 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 1.5.4.13). It follows from Proposition 1.5.5.14 that every monomorphism of simplicial sets $i: A \hookrightarrow B$ belongs to $S$ (and is therefore weakly left orthogonal to $p$).
$\square$