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5.4.7 Small Kan Complexes

In the setting of Kan complexes, essential $\kappa $-smallness can be tested at the level of homotopy groups.

Proposition 5.4.7.1. Let $X$ be a Kan complex and let $\kappa $ be an uncountable regular cardinal. Then $X$ is essentially $\kappa $-small if and only if it satisfies the following pair of conditions:

$(1)$

The set $\pi _0(X)$ is $\kappa $-small.

$(2)$

For each vertex $x \in X$ and each integer $n > 0$, the homotopy group $\pi _{n}(X,x)$ is $\kappa $-small.

Proof. By virtue of Proposition 5.4.6.12, we may assume without loss of generality that the Kan complex $X$ is minimal. If $X$ is essentially $\kappa $-small, then it is $\kappa $-small (Corollary 5.4.6.9), so that conditions $(1)$ and $(2)$ follow immediately from the definitions. Conversely, suppose that $(1)$ and $(2)$ are satisfied; we wish to show that $X$ is $\kappa $-small. By virtue of Proposition 5.4.4.9, it will suffice to show that the collection of $n$-simplices of $X$ is $\kappa $-small, for each $n \geq 0$. Our proof proceeds by induction on $n$. Using our inductive hypothesis (together with Remark 5.4.3.4 and Proposition 5.4.3.5), we see that the set $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, X)$ is $\kappa $-small. Since $\kappa $ is regular, it will suffice to show that each fiber of the restriction map $\operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \Delta ^ n, X ) \rightarrow \operatorname{Hom}_{\operatorname{Set_{\Delta }}}( \operatorname{\partial \Delta }^ n, X)$ is $\kappa $-small.

Set $E = \operatorname{Fun}( \Delta ^{n}, X)$ and $B = \operatorname{Fun}( \operatorname{\partial \Delta }^{n}, X)$, so that the inclusion map $\operatorname{\partial \Delta }^{n} \hookrightarrow \Delta ^ n$ induces a Kan fibration $q: E \rightarrow B$ (Corollary 3.1.3.3) For each vertex $b \in B$, let $E_{b}$ denote the fiber $\{ b\} \times _{B} E$; we wish to show that the set of vertices of $E_{b}$ is $\kappa $-small. Since the Kan complex $X$ is minimal, each vertex of $E_{b}$ belongs to a different connected component. It will therefore suffice to show that the set $\pi _{0}(E_ b)$ is $\kappa $-small. If $n=0$, this follows from condition $(1)$. Let us therefore assume that $n > 0$, and identify $b$ with a morphism of simplicial sets $\operatorname{\partial \Delta }^{n} \rightarrow X$. If this morphism is not nullhomotopic, then the Kan complex $E_{b}$ is empty and there is nothing to prove. We may therefore assume that there is a homotopy from $b$ to a constant map $b': \operatorname{\partial \Delta }^{n} \rightarrow \{ x\} \hookrightarrow X$. In this case, Proposition 5.2.2.18 supplies a homotopy equivalence of $E_{b}$ with $E_{b'}$. We are therefore reduced to proving that the set $\pi _0( E_{b'} ) \simeq \pi _{n}(X,x)$ is $\kappa $-small, which follows from condition $(2)$. $\square$

Corollary 5.4.7.2. Let $\kappa $ be an uncountable regular cardinal and let $f: X \rightarrow Y$ be a Kan fibration between Kan complexes, where $Y$ is essentially $\kappa $-small. The following conditions are equivalent:

$(a)$

The Kan complex $X$ is essentially $\kappa $-small.

$(b)$

For each vertex $y \in Y$, the fiber $X_{y} = \{ y\} \times _{Y} X$ is essentially $\kappa $-small.

Proof. The implication $(a) \Rightarrow (b)$ follows from Corollary 5.4.5.16 (and does not require the regularity of $\kappa $). Assume that condition $(b)$ is satisfied; we will show that $X$ satisfies the criteria of Proposition 5.4.7.1:

$(1)$

Let $y$ be a vertex of $Y$ and let $[y]$ denote its image in $\pi _0(Y)$. Since $f$ is a Kan fibration, the tautological map $\pi _0( X_ y ) \rightarrow \{ [y] \} \times _{ \pi _0(Y) } \pi _0(X)$ is a surjection. Assumption $(b)$ guarantees that $\pi _0(X_ y)$ is $\kappa $-small, so that the fiber $\{ [y] \} \times _{ \pi _0(Y) } \pi _0(X)$ is also $\kappa $-small. Since $\pi _0(Y)$ is $\kappa $-small, the regularity of $\kappa $ guarantees that $\pi _0(X)$ is also $\kappa $-small.

$(2)$

Fix a vertex $x \in X$ having image $y = f(x)$, and let $n > 0$ be a positive integer. For each integer $n > 0$, Proposition 3.2.5.2 supplies an exact sequence of groups

\[ \pi _{n}( X_{y}, x ) \rightarrow \pi _{n}(X,x) \xrightarrow { \pi _{n}(f)} \pi _{n}(Y,y). \]

Consequently, every nonempty fiber of the group homomorphism $\pi _{n}(f)$ carries a transitive action of the $\kappa $-small group $\pi _{n}(X_ y, x)$, and is therefore $\kappa $-small. Since the group $\pi _{n}(Y,y)$ is $\kappa $-small, the regularity of $\kappa $ guarantees that $\pi _{n}(X,x)$ is $\kappa $-small.

$\square$

Exercise 5.4.7.3. Let $\kappa $ be an uncountable regular cardinal and let $f: X \rightarrow Y$ be Kan fibration between Kan complexes. Suppose that $X$ is essentially $\kappa $-small, that each fiber $X_{y} = \{ y\} \times _{Y} X$ is essentially $\kappa $-small, and that the morphism $\pi _0(f): \pi _0(X) \rightarrow \pi _0(Y)$ is surjective. Show that $Y$ is also essentially $\kappa $-small.