Kerodon

Proof. We use a variant of the proof of Proposition 9.1.7.3. We first claim that for every $\lambda $-small simplicial subset $X \subseteq \operatorname{\mathcal{C}}$, we can find a larger $\lambda $-small simplicial subset $X^{+} \subseteq \operatorname{\mathcal{C}}$ satisfying the following conditions:

$(1)$

For $0 < i < n$, every morphism $\Lambda ^{n}_{i} \rightarrow X$ can be extended to an $n$-simplex of $X^{+}$.

$(2)$

For $n \geq 0$, every morphism $\operatorname{\partial \Delta }^{n} \rightarrow X$ can be extended to a morphism $( \operatorname{\partial \Delta }^{n})^{\triangleright } \rightarrow X^{+}$.

$(3)$

For every object $C \in \operatorname{\mathcal{C}}$ which is contained in $X$ and every $\kappa $-small collection of morphisms $\{ f_ i: C \rightarrow D_ i \} _{i \in I}$ which are also contained $X$, there exists a morphism $h: C \rightarrow E$ which is contained in $X^{+}$ and a collection of $2$-simplices $\{ \sigma _ i \} _{i \in I}$ of $X^{+}$ with boundary indicated in the diagram

\[ \xymatrix@R =50pt@C=50pt{ & D_ i \ar [dr] & \\ C \ar [ur]^{f_ i} \ar [rr]^{h} & & E. } \]

Arranging $(1)$ and $(2)$ is straightforward (see the proofs of Proposition 9.1.7.1 and 9.1.7.3, respectively). To guarantee that condition $(3)$ is satisfied, we need to work a bit harder. Let us first regard the object $C$ as fixed, and let $\{ f_ s: C \rightarrow D_ s \} _{s \in S}$ be the collection of all morphisms with source $C$ which are contained in $X$. Since the simplicial set $X$ is $\lambda $-small, the set $S$ is $\lambda $-small. Invoking our assumption that $\kappa \triangleleft \lambda $, we can choose a $\lambda $-small collection $\{ S_ j \} _{j \in J}$, where each $S_ j$ is a $\kappa $-small subset of $S$ and every $\kappa $-small subset of $S$ is contained in some $S_ j$. It follows that, to verify condition $(3)$ for the object $C$, we may assume that $I = S_ j$ for some $j \in J$. If we regard $j$ as fixed, then our assumption that $\operatorname{\mathcal{C}}$ is $\kappa $-filtered guarantees that there exists a morphism $h_ j: C \rightarrow E_ j$ and a collection of $2$-simplices $\{ \sigma _{i,j} \} _{i \in S_ j}$ with boundary indicated in the diagram

We can therefore guarantee that condition $(3)$ is satisfied by demanding that $X^{+}$ contains each of the morphisms $h_{j}$ and each of the $2$-simplices $\sigma _{i,j}$, where $C$ ranges over all vertices of $X$, $j$ ranges over all elements of the set $J$ (which depends on $C$), and $i$ ranges over all elements of the set $S_{j}$.

We now construct a transfinite sequence $\{ \operatorname{\mathcal{C}}_{\beta } \} _{\beta < \kappa }$ of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$ as follows:

• If $\beta = \alpha +1$ is a successor ordinal, we take $\operatorname{\mathcal{C}}_{\beta } = \operatorname{\mathcal{C}}_{\alpha }^{+}$.

• If $\beta < \kappa $ is a nonzero limit ordinal, we take $\operatorname{\mathcal{C}}_{\beta } = \bigcup _{ \alpha < \beta } \operatorname{\mathcal{C}}_{\alpha }$.

Set $\operatorname{\mathcal{C}}' = \bigcup _{\beta < \kappa } \operatorname{\mathcal{C}}_{\beta }$. Since $\kappa < \lambda $, our assumption that $\lambda $ is regular guarantees that $\operatorname{\mathcal{C}}'$ is also $\lambda $-small. Moreover, our construction guarantees that the simplicial set $\operatorname{\mathcal{C}}'^{+} = \operatorname{\mathcal{C}}'$ satisfies condition $(1)$, $(2)$, and $(3)$ above. Condition $(1)$ guarantees that $\operatorname{\mathcal{C}}'$ is an $\infty $-category, condition $(2)$ guarantees that $\operatorname{\mathcal{C}}'$ is filtered (Lemma 9.1.2.12), and condition $(3)$ guarantees that it is also $\kappa $-filtered (Proposition 9.1.5.8). $\square$