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5.4.4 Small Simplicial Sets

Definition 5.4.3.1 has a counterpart in the setting of simplicial sets.

Definition 5.4.4.1. Let $\kappa $ be an infinite cardinal. We say that a simplicial set $S$ is $\kappa $-small if the collection of nondegenerate simplices of $S$ is $\kappa $-small.

Remark 5.4.4.2. In the situation of Definition 5.4.4.1, the dimension of the simplices under consideration is not fixed. That is, a simplicial set $S_{\bullet }$ is $\kappa $-small if and only if the disjoint union $\coprod _{m \geq 0} S_{m}^{\mathrm{nd}}$ is a $\kappa $-small set, where $S_{m}^{\mathrm{nd}} \subseteq S_{m}$ denotes the set of nondegenerate $m$-simplices of $S_{\bullet }$.

Remark 5.4.4.3. Let $\kappa $ be an infinite cardinal. Then a simplicial set $S$ is $\kappa $-small if and only if the opposite simplicial set $S^{\operatorname{op}}$ is $\kappa $-small.

Remark 5.4.4.5 (Coproducts). Let $\kappa $ be an infinite cardinal and let $\{ S_{i} \} _{i \in I}$ be a collection of $\kappa $-small simplicial sets. Suppose that the cardinality of the index set $I$ is smaller than the cofinality $\mathrm{cf}(\kappa )$. Then the coproduct $\coprod _{i \in I} S_ i$ is also $\kappa $-small (see Corollary 5.4.3.9). In particular:

  • The collection of $\kappa $-small simplicial sets is closed under finite coproducts.

  • If $\kappa $ is regular, then the collection of $\kappa $-small simplicial sets is closed under $\kappa $-small coproducts.

Remark 5.4.4.6 (Colimits). Let $\kappa $ be an infinite cardinal and let $\{ S_{i} \} _{i \in \operatorname{\mathcal{I}}}$ be a diagram of simplicial sets indexed by a category $\operatorname{\mathcal{I}}$. Suppose that the set of objects $\mathrm{Ob}( \operatorname{\mathcal{I}})$ has cardinality smaller than the cofinality of $\kappa $. Then the colimit $\varinjlim _{i \in \operatorname{\mathcal{I}}} S_ i$ is also $\kappa $-small (since it can be realized as a quotient of the coproduct $\coprod S_ i$, which is $\kappa $-small by virtue of Remark 5.4.4.5).

Remark 5.4.4.7. Let $S$ be a simplicial set. Then there is a least infinite cardinal $\kappa $ for which $S$ is $\kappa $-small. If $S$ is finite, then $\kappa = \aleph _0$. If $S$ is not finite, then $\kappa = \lambda ^{+}$, where $\lambda $ is the cardinality of the set of all nondegenerate simplices of $S$. In particular, $\kappa $ is always a regular cardinal.

Remark 5.4.4.8. Let $\kappa $ be an infinite cardinal and let $T$ be a $\kappa $-small simplicial set. Then:

  • Every simplicial subset of $T$ is $\kappa $-small.

  • The simplicial set $T$ is $\lambda $-small for each $\lambda \geq \kappa $.

  • For every epimorphism of simplicial sets $T \twoheadrightarrow S$, the simplicial set $S$ is also $\kappa $-small.

See Remark 5.4.3.4.

Proposition 5.4.4.9. Let $\kappa $ be an infinite cardinal and $S_{\bullet }$ be a simplicial set. Assume that the cofinality of $\kappa $ is larger than $\aleph _0$ (this condition is satisfied, for example, if $\kappa $ is uncountable and regular). The following conditions are equivalent:

$(1)$

The simplicial set $S_{\bullet }$ is $\kappa $-small.

$(2)$

For every integer $n \geq 0$, the set $S_{n}$ is $\kappa $-small.

$(3)$

For every finite simplicial set $K$, the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}(K,S_{\bullet })$ is $\kappa $-small.

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 $X$. Using Proposition 1.1.3.4, 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 5.4.3.6).

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 3.5.1.7, 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

\[ \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, S_{\bullet } ) \hookrightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K', S_{\bullet }) \simeq \prod _{i \in I} S_{n_ i}. \]

Since the collection of $\kappa $-small sets is closed under finite products and passage to subsets (Proposition 5.4.3.5 and Remark 5.4.3.4), it follows that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( K, S_{\bullet } )$ is also $\kappa $-small. $\square$

Warning 5.4.4.10. The implications $(1) \Rightarrow (2) \Leftrightarrow (3)$ of Proposition 5.4.4.9 are valid for an arbitrary infinite cardinal $\kappa $. However, the implication $(2) \Rightarrow (1)$ is false if $\kappa $ has countable cofinality (for example, if $\kappa = \aleph _0$).

Corollary 5.4.4.11. Let $\kappa $ be an infinite cardinal. Then the collection of $\kappa $-small simplicial sets is closed under finite products.

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 $X_{i}$ is $\kappa $-small. Then $\kappa $ is regular (Remark 5.4.4.7). If $\kappa = \aleph _0$, then the desired result follows from Remark 3.5.1.6. We may therefore assume that $\kappa $ is uncountable. In this case, the desired result follows from the criterion of Proposition 5.4.4.9, since the collection of $\kappa $-small sets is closed under finite products (Proposition 5.4.3.5). $\square$

Corollary 5.4.4.12. Let $\kappa $ be an uncountable cardinal, let $S$ be a $\kappa $-small simplicial set, and let $K$ be a finite simplicial set. Then the simplicial set $\operatorname{Fun}(K, S)$ is $\kappa $-small.

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 5.4.4.7). By virtue of Proposition 5.4.4.9, 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 5.4.4.9, since the simplicial set $K \times L$ is finite (Remark 3.5.1.6). $\square$

Warning 5.4.4.13. The assertion of Corollary 5.4.4.12 is false in the case $\kappa = \aleph _0$. That is, if $K$ and $S$ are finite simplicial sets, then the simplicial set $\operatorname{Fun}(K, S)$ need not be finite.

We close by recording stronger forms of Corollaries 5.4.4.11 abnd 5.4.4.12.

Corollary 5.4.4.14. Let $\lambda $ be an infinite cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality (Definition 5.4.3.16). Then the collection of $\lambda $-small simplicial sets is closed under $\kappa $-small products.

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 5.4.4.11. We may therefore assume that $\kappa $ is uncountable. Then the cofinality $\mathrm{cf}(\lambda )$ is also uncountable (Remark 5.4.3.17). The desired result now follows from the criterion of Proposition 5.4.4.9, since the collection of $\lambda $-small sets is closed under $\kappa $-small products. $\square$

Corollary 5.4.4.15. Let $\lambda $ be an infinite cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ be its exponential cofinality. If $S$ is a $\lambda $-small simplicial set and $K$ be a $\kappa $-small simplicial set. Then $\operatorname{Fun}(K,S)$ is $\lambda $-small.

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 5.4.4.12 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 5.4.4.14. $\square$