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I have ’stumbled on’ something that is quite fun (I doubt it is new). I was working on the higher generation by subgroups stuff, with a plan to go from there to Haefliger’s complexes of groups and then to orbihedra and orbifolds to close up a gap in the nLab linkages and put this also in the Menagerie. I therefore needed to look at group actions on simplicial complexes as that is the starting point of Haefliger’s theory. The notion of regular G-complex is defined in several (probably inequivalent) ways in the sources, so I wanted a categorical one to be clear about things and to try to make sense of the mess. Here is what seems to be the case.
Simplicial complexes give simplicial sets either by choosing a total order on the vertices or by taking multiple copies of each simplex one for each possible order, and then forming degeneracies. Suppose that $G$ acts on $K$ and we denote by $K^{simp}$ the corresponding simplicial set. We can form $(K/G)^{simp}$ and $(K^{simp})/G$ and they are not in general isomorphic. There is a simplicial map from $(K^{simp})/G$ to $(K/G)^{simp}$ and these would seem to be ’onto’. It it 1-1 (I think) if and only if the action is ’regular’.
(The action is regular if given elements $g_0, \ldots, g_n\in G$ and a simplex, $\sigma = \{v_0,\ldots, v_n\}$ of $K$ such that $\tau = \{v_0 g_0,\ldots, v_n g_n\}$ is also a simplex of $K$, then there is a single element, $g\in G$ such that $\sigma\cdot g = \tau$.)
Any thoughts on the categorical interpretation of ’regular’. Simplicial complexes are strange categorically so it may be not obvious. (The usual treatments of regularity use the geometric realisation not the simplicialisation (or whetever this functor should be called.))
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