Kerodon

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

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

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

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

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

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

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

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 $\sigma _0 = \sigma |_{ \Lambda ^{n}_{i} }$. $\square$

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

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

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

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

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 $p$). $\square$