$\DeclareMathOperator\Aut{Aut}\DeclareMathOperator\Gal{Gal}$Recall that $n$-simplices of the classifying space (simplicial set) $BG$ of a group are orbits of the diagonal action by shifts
on $n+1$-tuples of its elements. But what if we take elements from some other set the group acts on ?


That is, the same definition can be made for an arbitrary action of a group on a set,
e.g. the Galois action on $\Bbb{\bar Q}$, or a symmetric group acting by permutations. Note that for the trivial action
of a group on a singleton I get the trivial simplicial object, and thus this is not the Borel construction.
Where can I read about this ? What is the right terminology ?


In notation: let $G$ be a group acting on a set $X$, i.e. we have $\tau:G\rightarrow \Aut X$.
Recall that $B(G)\_n = G^{n+1}/G$, and I define
$$B'(G \xrightarrow{\tau} \Aut X):=X^{n+1}/G$$
Note that for $X=G$ and $\tau:G\rightarrow \Aut G$ the action by shifts,
$B(G)=B'(\tau)$, and for the trivial action of a group on a singleton,
$B'(G\rightarrow \Aut(\{\mathrm{pt}\}))$ is trivial as well.


What what can be said about $B'(\tau)$ and where can I read about it ?
In particular, for $G=\Gal( \Bbb{\bar Q/ Q})$ action on $\mathbb{\bar Q}$, or perhaps
$\Gal(K/\Bbb{Q})$ action on $K$ for a number field ? Has this been considered in number theory ?
Is this construction trivial for some reason ?


For the permutation representation of an infinite symmetric group one gets a simplicial set representing equivalence relations;
for the finite symmetric group $S\_n$ it classifies equivalence relations with at most $n$ equivalence classes.


I am also interested in the following "ordered" modification of the construction for the group $G=\Aut(\Bbb{Q}^\leq)$
of automorphisms of a dense linear order $\Bbb{ Q}$: an $n$-simplex is an orbit of the action on *non-decreasing* $n+1$-tuples.
This simplicial set has the property that in each dimension there is a unique non-degenerate simplex, and, moreover,
all its faces are non-degenerate, and is weakly contractible, according to [answers here](https://mathoverflow.net/questions/439687/the-simplicial-set-with-a-unique-non-degenerate-simplex-in-each-dimension).If you truncate the simplicial set, for odd dimensions you get a sphere up to homotopy, and for even
dimensions something weakly contractible, e.g. for $n=2$ the dunce hat.



Defining the classifying space of a group acting on a set
As in Tom Goodwillie's comment, if you take the construction that you discuss for a $G$-set $X$, the equivariant homotopy type of the space obtained before you quotient out by the action of $G$ is called a classifying space for a family of subgroups of $G$; the family being the subgroups that fix some point of $X$.  

There is a survey by Wolfgang Lueck "Survey on Classifying Spaces for Families of Subgroups", available on arXiv at <https://arxiv.org/abs/math/0312378> or as a chapter of a volume published by Birkhauser <https://link.springer.com/chapter/10.1007/3-7643-7447-0_7>. The survey considers topological groups, but as he says in the introduction, if you just want to consider discrete groups you should just ignore Sections 2 and 3.  

The cases when stabilizers in $X$ contain exactly the finite subgroups and exactly the virtually cyclic subgroups have attracted a lot of interest, partly due to their appearance in "assembly conjectures" as discussed in Section 7 of Lueck's survey.


