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. 



Decomposition of simplicial G-set?
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.


