Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

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 $f_{\gamma ,0}$ with the composition

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

    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 $i: A_{\bullet } \rightarrow B_{\bullet }$ 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 1.1.3.13 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 $\operatorname{\partial \Delta }^ k \hookrightarrow \Delta ^ k$. $\square$