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Let $G$ be a simplicial group. Let $X$ be a simplicial $G$-set,i.e. for each level, $X_n$ is a $G_n$-set.

Can $X$ be written as a union of finite (or finite type) simplicial $G$-subsets? Here "finit simplicial set" means that elements of high levels are all degenerate and "finite type simplicial set" means each level $X_n$ has finite elements.

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  • $\begingroup$ Do you mean $X_n$ has finitely many orbits? Otherwise any infinite group (considered as a constant simplicial group) acting on itself would be a counterexample. $\endgroup$
    – Justin Noel
    Commented Dec 10, 2011 at 8:26
  • $\begingroup$ The condition "elements of high levels are all degenerate" also seems a bit too strong. If $G$ has non-degenerate simplices in all degrees, then so will any non-trivial simplicial $G$-set... $\endgroup$
    – André Henriques
    Commented Dec 10, 2011 at 8:48
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    $\begingroup$ You may consider the set $G_n/X_n$ of orbits, and these constitute a simplicial set $G/X$. If, as Justin asks, you mean finitely many orbits, then the answer is yes: any simplicial set is a union of simplicial subsets that are both finite and of finite type in your terminology; in particular this is true for the simplicial set $G/X$. (By the way, your "finite simplicial set" means that $|X|$ is a finite-dimensional cell complex, your "finite type simplicial set" means finitely many cells in each dimension, and the two together mean that $|X|$ is a finite cell complex, i.e. a compact space.) $\endgroup$
    – Tom Goodwillie
    Commented Dec 10, 2011 at 12:34
  • $\begingroup$ You definition of simplicial $G$-set is missing the condition that the face and degeneracy maps in $X$ are compatible with the maps in $G$. $\endgroup$
    – S. Carnahan ♦
    Commented Dec 11, 2011 at 4:18

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Let $H$ be your favorite infinite abelian group, and let $G$ be the simplicial abelian group $BH$. In other words, $G$ is the nerve of the category with one object whose morphism monoid is $H$. Specifically, $G_i \cong H^i$, with structure homomorphisms given by projections and identity insertions.

Let $X$ be a copy of $G$, with the action by left multiplication. The action of $G_i$ on $X_i$ is transitive for all $i$, and $X_i$ is infinite for $i > 0$, so $X$ is not a union of finite type $G$-simplicial sets. Similarly, $X$ is not a union of "finite" (or finite dimensional) $G$-simplicial sets, because any $G$-simplicial subset containing the base point (i.e., any nonempty subobject) contains an element of $X_i$ for all $i$ by degeneracy maps, and all of $X$ by transitivity of the $G$-action.

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