Let $X$ be a simplicial $G$-set, where $G$ is a simplicial group. What is the homotopy type of the simplicial action groupoid $X//G$?
1 Answer
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Take the nerve of $X//G \in Gpd(sSet)$ to get a bisimplicial set (http://ncatlab.org/nlab/show/bisimplicial+set), then take the diagonal, or the Artin-Mazur codiagonal (http://ncatlab.org/nlab/show/codiagonal). Either of the resulting simplicial sets represent the homotopy type. Pick the one which is better for what you need it for.
answered Nov 16, 2011 at 5:03
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2$\begingroup$ From this you also see that it is just the Borel construction (associated if you prefer to geometric realisations). $\endgroup$ Commented Nov 16, 2011 at 5:05
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1$\begingroup$ Torsten, the Borel construction know is for a discrete group $G$. Does the Borel construction also have a analogue for a simplicial group $G$? $\endgroup$– user2529Commented Nov 16, 2011 at 8:54
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1$\begingroup$ It works for a topological group too! You just need a contractible free $G$-simplicial set, and this is supplied by $WG$. I'm not sure of the exact relation that Torsten mentions, off the top of my head... $\endgroup$– David Roberts ♦Commented Nov 16, 2011 at 10:25
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$\begingroup$ David, your geometric realization ought to work for an simplicial object in ${\bf{Cat}}$ too, right? $\endgroup$– user2529Commented Nov 27, 2011 at 2:49
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