Kerodon

Proof. We first show that $(1)$ implies $(2)$. Assume that $S_{\bullet }$ is $\kappa $-small and let $n \geq 0$ be an integer. For each integer $m \geq 0$, let $S_{m}^{\mathrm{nd}}$ denote the set of nondegenerate $m$-simplices of $S_{\bullet }$. Using Proposition 1.1.3.8, we can identify $S_ n$ with the coproduct ${\coprod }_{ \alpha : [n] \twoheadrightarrow [m] } S_{m}^{\mathrm{nd}}$, where $\alpha $ ranges over all surjective maps of linearly ordered sets $[n] \twoheadrightarrow [m]$. Our assumption that $S_{\bullet }$ is $\kappa $-small guarantees that each of the sets $S_{m}^{\mathrm{nd}}$ is $\kappa $-small, so that $S_{n}$ is also $\kappa $-small (Corollary 4.7.3.6).

We now show that $(2)$ implies $(1)$. Assume that, for each $n \geq 0$, the set $S_{n}$ is $\kappa $-small. Since $\kappa $ has cofinality $> \aleph _0$ it follows that the coproduct ${\coprod }_{n \geq 0} S_{n}$ is also $\kappa $-small. In particular, the coproduct ${\coprod }_{n \geq 0} S_{n}^{\mathrm{nd}}$ is $\kappa $-small: that is, the simplicial set $S_{\bullet }$ is $\kappa $-small.

The implication $(3) \Rightarrow (2)$ is immediate from the definition. We will complete the proof by showing that $(2) \Rightarrow (3)$. Assume that, for each $n \geq 0$, the set $S_{n}$ is $\kappa $-small, and let $K$ be a finite simplicial set. By virtue of Proposition 3.6.1.7, there exists an epimorphism $f: K' \twoheadrightarrow K$, where $K' = {\coprod }_{i \in I} \Delta ^{n_ i}$ is a disjoint union of finitely many standard simplices. Then precomposition with $f$ induces a monomorphism

Since the collection of $\kappa $-small sets is closed under finite products and passage to subsets (Proposition 4.7.3.5 and Remark 4.7.3.4), it follows that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, S_{\bullet } )$ is also $\kappa $-small. $\square$

Proof. Let $\{ S_ i \} _{i \in I}$ be a collection of $\kappa $-small simplicial sets indexed by a finite set $I$; we wish to show that the product $S = {\prod }_{i \in I} S_ i$ is $\kappa $-small. Without loss of generality, we may assume that $\kappa $ is the least infinite cardinal for which each of the simplicial sets $S_{i}$ is $\kappa $-small. Then $\kappa $ is regular (Remark 4.7.4.7). If $\kappa = \aleph _0$, then the desired result follows from Remark 3.6.1.6. We may therefore assume that $\kappa $ is uncountable. In this case, the desired result follows from the criterion of Proposition 4.7.4.10, since the collection of $\kappa $-small sets is closed under finite products (Proposition 4.7.3.5). $\square$

Proof. The case $\kappa = \aleph _0$ follows from Remark 4.3.3.21. We may therefore assume that $\kappa $ is uncountable. In this case, it will suffice to show that for every $n \geq 0$, the collection of $n$-simplices of $K \star L$ is $\kappa $-small (Proposition 4.7.4.10). This follows from Remark 4.3.3.17, since the collection of $\kappa $-small sets is closed under finite products and coproducts (Proposition 4.7.3.5 and Corollary 4.7.3.6). $\square$

Proof. Without loss of generality, we may assume that $\kappa $ is the least uncountable cardinal for which $S$ is $\kappa $-small. In particular, $\kappa $ is regular (Remark 4.7.4.7). By virtue of Proposition 4.7.4.10, it will suffice to show that for every finite simplicial set $L$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( L, \operatorname{Fun}(K, S) ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K \times L, S)$ is $\kappa $-small. This is a special case of Proposition 4.7.4.10, since the simplicial set $K \times L$ is finite (Remark 3.6.1.6). $\square$

Proof. By virtue of Remark 4.7.4.7, we may assume that $\kappa $ is a regular cardinal, In particular, $\kappa $ has cofinality larger than $\aleph _0$. It will therefore suffice to prove that $\operatorname{Ex}^{n}(S)$ is $\kappa $-small, for each integer $n \geq 0$. By virtue of Proposition 4.7.4.10, it will suffice to show that for every finite simplicial set $K$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, \operatorname{Ex}^{n}(S) ) \simeq \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{Sd}^{n}(K), S )$ is $\kappa $-small. This follows from Proposition 4.7.4.10, since $S$ is $\kappa $-small and $\operatorname{Sd}^{n}(K)$ is finite (see Remark 3.3.3.6). $\square$

Proof. Let $\{ S_ i \} _{i \in I}$ be a collection of $\lambda $-small simplicial sets indexed by a $\kappa $-small set $I$; we wish to show that the product $S = {\prod }_{i \in I} S_ i$ is $\lambda $-small. If $\kappa = \aleph _0$, this follows from Corollary 4.7.4.12. We may therefore assume that $\kappa $ is uncountable. Then the cofinality $\mathrm{cf}(\lambda )$ is also uncountable (Remark 4.7.3.17). The desired result now follows from the criterion of Proposition 4.7.4.10, since the collection of $\lambda $-small sets is closed under $\kappa $-small products. $\square$

Proof. Since $K$ is $\kappa $-small, we can choose an epimorphism of simplicial sets ${\coprod }_{i \in I} \Delta ^{n_{i}} \twoheadrightarrow K$, where $I$ is a $\kappa $-small set. It follows that $\operatorname{Fun}(K,S)$ can be identified with a simplicial subset of the product ${\prod }_{ i \in I} \operatorname{Fun}( \Delta ^{n_ i}, S)$. Corollary 4.7.4.14 guarantees that each factor $\operatorname{Fun}( \Delta ^{n_ i}, S)$ is $\lambda $-small, so that the product ${\prod }_{ i \in I} \operatorname{Fun}( \Delta ^{n_ i}, S)$ is $\lambda $-small by virtue of Corollary 4.7.4.18. $\square$