Kerodon

Proof. Without loss of generality, we may assume that $\lambda $ is regular (Remark 4.7.4.7), and therefore of uncountable cofinality. We will construct $\operatorname{\mathcal{C}}'$ as the union of a sequence

consisting of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$. Fix an integer $n \geq 0$, and assume that $\operatorname{\mathcal{C}}_{n}$ has been constructed. Let $\{ \sigma _{\alpha }: \Lambda ^{m}_{i} \rightarrow \operatorname{\mathcal{C}}_{n} \} _{\alpha \in A}$ be the collection of all inner horns in the simplicial set $\operatorname{\mathcal{C}}_{n}$ (so that $0 < i < m$). Since the simplicial set $\operatorname{\mathcal{C}}_{n}$ is $\lambda $-small, the set $A$ is also $\lambda $-small (Proposition 4.7.4.10). Since $\operatorname{\mathcal{C}}$ is an $\infty $-category, each $\sigma _{\alpha }$ can be extended to a simplex $\overline{\sigma }_{\alpha }: \Delta ^{m} \rightarrow \operatorname{\mathcal{C}}$. We then choose $\operatorname{\mathcal{C}}_{n+1}$ to be any $\lambda $-small simplicial subset of $\operatorname{\mathcal{C}}$ which contains $\operatorname{\mathcal{C}}_{n}$ and each of the simplices $\overline{\sigma }_{\alpha }$. It follows immediately from the construction that $\operatorname{\mathcal{C}}' = \bigcup _{n} \operatorname{\mathcal{C}}_{n}$ is a $\lambda $-small $\infty $-category containing $\operatorname{\mathcal{C}}_0$. $\square$

Proof. Let $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ be the collection of all $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$ which are $\infty $-categories. It follows from Proposition 9.1.7.1 that the index set $A$ is $\lambda $-directed (where we write $\alpha \leq \beta $ if $\operatorname{\mathcal{C}}_{\alpha }$ is contained in $\operatorname{\mathcal{C}}_{\beta }$), and that the canonical map $\varinjlim _{\alpha \in A} \operatorname{\mathcal{C}}_{\alpha } \rightarrow \operatorname{\mathcal{C}}$ is an isomorphism. The final assertion follows from Corollary 9.1.6.3. $\square$

Proof. Without loss of generality, we may assume that $\lambda $ is regular (Remark 4.7.4.7), and therefore of uncountable cofinality. As in the proof of Proposition 9.1.7.1, we will construct $\operatorname{\mathcal{C}}'$ as the union of an increasing sequence

of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$, where every inner horn in $\operatorname{\mathcal{C}}_{n}$ extends to a simplex of $\operatorname{\mathcal{C}}_{n+1}$. To guarantee that $\operatorname{\mathcal{C}}'$ is a filtered $\infty $-category, it will suffice to ensure that the following additional condition is satisfied (see Lemma 9.1.2.12):

• For $m,n \geq 0$, every morphism of simplicial sets $f: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}_{n}$ admits an extension $\overline{f}: (\operatorname{\partial \Delta }^{m})^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_{n+1}$.

This is possible, since the collection $\{ f: \operatorname{\partial \Delta }^{m} \rightarrow \operatorname{\mathcal{C}}_{n} \} _{m \geq 0}$ is $\lambda $-small (Proposition 4.7.4.10) and each of its members admits an extension $\overline{f}: ( \operatorname{\partial \Delta }^{m} )^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$. $\square$

Proof. For every ordinal $\alpha $, let $\mathrm{Ord}_{<\alpha }$ denote the set of ordinals smaller than $\alpha $. Then a subset $S_ i \subseteq S$ is $\kappa $-small if and only if it is the image of a function $F: \mathrm{Ord}_{< \alpha } \rightarrow S$, for some ordinal $\alpha < \kappa $. It will therefore suffice to show that the set of all such pairs $(\alpha ,F)$ is $\lambda $-small. The set $\mathrm{Ord}_{< \kappa }$ has cardinality $\kappa $, and is therefore $\lambda $-small by virtue of our assumption that $\kappa < \lambda $. Since $\lambda $ is regular, it will suffice to show that for every fixed ordinal $\alpha < \kappa $, the set of functions $F: \mathrm{Ord}_{< \alpha } \rightarrow S$ is $\lambda $-small. This follows from our assumption that $\lambda $ has exponential cofinality $\geq \kappa $. $\square$

Proof. The case $\kappa = \lambda $ is trivial (Example 9.1.7.6), and the case $\kappa < \lambda $ follows from Proposition 9.1.7.8. $\square$

Proof. Assume that $\kappa \triangleleft \lambda $; we will show that $\lambda $ has exponential cofinality $\geq \kappa $ (the converse follows from Proposition 9.1.7.8). If $\lambda $ is strongly inaccessible, this is immediate. Otherwise, we may assume that $\lambda = \lambda _0^{+}$ is a successor cardinal (Remark 4.7.3.23). Let $\{ S_ i \} _{i \in I}$ be a $\kappa $-small collection of $\lambda $-small sets; we wish to show that the product $\prod _{i \in I} S_ i$ is $\lambda $-small. Fix a set $S$ of cardinality $\lambda _0$. Each of the sets $S_ i$ has cardinality $\leq \lambda _0$, and is therefore isomorphic to a subset of $S$. We may therefore assume without loss of generality that each $S_ i$ is equal to $S$, so that $\prod _{i \in I} S_ i$ can be identified with the set $\operatorname{Fun}(I, S)$ of functions from $I$ to $S$. Set $T = I \times S$. Note that every function $f: I \rightarrow S$ is determined by its graph $\Gamma _{f} = \{ (i,s) \in I \times S: f(i) = s \} $, which is a $\kappa $-small subset of $T$. We will complete the proof by showing that the collection of $\kappa $-small subsets of $T$ is $\lambda $-small.

Our assumption that $\kappa \triangleleft \lambda $ guarantees that there is a $\lambda $-small collection $\{ T_ j \} _{j \in J}$, where each $T_ j$ is a $\kappa $-small subset of $T$ and every $\kappa $-small subset of $T_ j$ is contained in some $T_ j$. It will therefore suffice to show that, for each $j \in J$, the collection of subsets of $T_{j}$ is $\lambda $-small. This is clear: since $T_ j$ has cardinality $< \kappa $, the generalized continuum hypothesis guarantees that the collection of subsets of $T_ j$ has cardinality $\leq \kappa < \lambda $. $\square$

Proof. Let $S$ be a $\mu $-small set. Using our assumption that $\lambda \trianglelefteq \mu $, we can choose a $\mu $-small collection $\{ S_ i \} _{i \in I}$ of $\lambda $-small subsets of $S$ such that every $\lambda $-small subset of $S$ is contained in some $S_ i$. For each $i \in I$, our assumption that $\kappa \trianglelefteq \lambda $ guarantees that we can choose a $\lambda $-small collection $\{ S_{i,j} \} _{j \in J_{i} }$ of $\kappa $-small subsets of $S_{i}$ such that every $\kappa $-small subset of $S_{i}$ is contained in some $S_{i,j}$. Then $\{ S_{i,j} \} _{i \in I, j \in J_ i}$ is a $\mu $-small collection of $\kappa $-small subsets of $S$, and every $\kappa $-small subset of $S$ is contained in some $S_{i,j}$. $\square$

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). $\square$

Proof. If $\kappa = \aleph _0$, we can take $\{ X(i)_{\bullet } \} $ to be the collection of all finite simplicial subsets of $X_{\bullet }$. We may therefore assume without loss of generality that $\kappa $ is uncountable. Let $S = \coprod _{n \geq 0} X_{n}$ denote the collection of all simplices of $X$. The assumption $\kappa \trianglelefteq \lambda $ guarantees that there is a $\lambda $-small collection $\{ S(i) \} _{i \in I}$ of $\kappa $-small subsets of $S$ such that every $\kappa $-small subset of $S$ is contained in some $S(i)$. Enlarging the subsets $S(i)$ if necessary, we may assume that they are closed under the face and degeneracy operators for the simplicial set $X_{\bullet }$. It follows that each $S(i)$ can be identified with the collection of simplices for some $\kappa $-small simplicial subset $X(i)_{\bullet } \subseteq X$. By construction, the collection of simplicial subsets $\{ X(i)_{\bullet } \} _{i \in I}$ has the desired property. $\square$

Proof. Using Lemma 9.1.7.18, we can choose a $\mu $-small collection $\{ \operatorname{\mathcal{C}}_{\alpha } \} _{\alpha \in A}$ of $\lambda $-small simplicial subsets of $\operatorname{\mathcal{C}}$ such that every $\lambda $-small simplicial subset of $\operatorname{\mathcal{C}}$ is contained in some $\operatorname{\mathcal{C}}_{\alpha }$. Enlarging the simplicial subsets $\operatorname{\mathcal{C}}_{\alpha }$ if necessary, we may assume that they are $\kappa $-filtered $\infty $-categories (Proposition 9.1.7.14). The index set $A$ is then $\lambda $-directed by inclusion (Remark 9.1.7.19), and $\operatorname{\mathcal{C}}= \bigcup _{\alpha } \operatorname{\mathcal{C}}_{\alpha }$ is the colimit of the diagram $\{ \operatorname{\mathcal{C}}_{\alpha } \} $ in the category of simplicial sets. The final assertion follows from Corollary 9.1.6.3. $\square$

Proof. Combine Lemma 9.1.7.18 with Proposition 9.1.7.1 (or Proposition 9.1.7.3, in the case where $\operatorname{\mathcal{C}}$ is filtered). $\square$