# Decomposition of simplicial G-sets as a colimits of its simplicial G-subsets

Every simplicial set is the colimit of its finite simplicial subsets. Suppose $G$ be a finite discrete set.

Is every simplicial $G$-set a colimit of its finite simplicial $G$-subsets? I'm particularly interested in the case where $G={\mathbb{Z}}/2$.

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"Every simplicial set is the colimit of its finite simplicial subsets. " Really? I doubt that this is true. Or what notion of "finite" are you working with here? –  Martin Brandenburg Oct 25 '12 at 15:15
The obvious way of making the statement true is to replace "finite" by "finitely presented" (in the sense of Gabriel and Ulmer). But I think this turns out to be the same thing as a simplicial set with finitely many non-degenerate simplices. –  Zhen Lin Oct 25 '12 at 16:29

I assume you meant $G$ should be a finite discrete group? Also, I think the answer is yes. Every simplex of a $G$-sset is contained in a finite sub-sset, and then you you can take the closure under the $G$-action (which will still be finite, since $G$ is). This closure will be a sub-sset because the face and degeneracy maps are $G$-equivariant.