Question

Let $X_{\bullet}$ be a simplicial topological space. There is a truncation functor $tr^n \colon Fun(\Delta^{op}, Top) \to Fun(\Delta_n^{op},Top)$ (where $\Delta_n$ is the full subcategory of $\Delta$ that has the objects $[0], \dots, [n]$) which is just given by pullback with the inclusion. The functor $tr^n$ has a left adjoint called the skeleton $sk^n$ and a right adjoint called the coskeleton $cosk^n$. If I am not mistaken, we can use the following inductive definition of $cosk^n$:

$cosk^n(X)_m = X_m$ if $m \leq n$ and

$cosk^n(X)_m = \{ (x_0, \dots, x_m) \ | \ x_i \in cosk^n(X)_{m-1} \ {\rm with }\ d_i(x_j) = d_{j-1}(x_i)\ \text{if}\ i < j \}$ for $m > n$.

Now suppose I have a simplicial principal bundle $P_{\bullet} \to X_{\bullet}$ for a simplicial topological group $G_{\bullet}$. By this I mean that $P_{\bullet}$ carries a simplicial action of the simplicial group $G_{\bullet}$, such that $P_m \to X_m$ is a principal $G_m$-bundle at each level $m$.

If I apply $cosk^n$ to all spaces, I get a simplicial space $cosk^n(P)_{\bullet}$ with a projection map $cosk^n(P)_{\bullet} \to cosk^n(X)_{\bullet}$ and a simplicial action by the group $cosk^n(G)_{\bullet}$, since $cosk^n$ is well-behaved with respect to products.

Are there sensible conditions on $X$, $P$ and $G$ that allow to conclude that $cosk^n(P)_{\bullet} \to cosk^n(X)_{\bullet}$ is a $cosk^n(G)_{\bullet}$-principal bundle? In particular, when is this map locally trivial?

Answer 1

If you interpret locally trivial as follows, then the answer is always, though it will actually be a $Cosk^{n-1}G_{\bullet}$ bundle.

The interpretation of locally trivial which I want to suggest is that $P_{\bullet}\rightarrow X_{\bullet}$ is locally trivial if there exists a hypercover $c:U_{\bullet}\rightarrow X_{\bullet}$, such that $c^*P_{\bullet}\rightarrow U_{\bullet}$ fits into a pullback diagram (which I'll describe since I can't figure out how to draw it in MathJax).

The diagram expresses $f^*P_{\bullet}$ as the pullback of the universal $G_{\bullet}$ bundle $WG_{\bullet}\rightarrow \bar{W}G_{\bullet}$ along a cocycle $U_{\bullet}\rightarrow\bar{W}G_{\bullet}$.

With this definition of locally trivial, we can see that $Cosk^n$ preserves local triviality as follows: namely, $Cosk^n$ preserves limits (it's a right adjoint), so applying it to the pullback square, we get another pullback square. Now, a calculation in the definitions of $W$, $\bar{W}$ and the combinatorics of simplices shows that $Cosk^n\bar{W}=\bar{W}Cosk^{n-1}$. Working backward through the definitions, we see $Cosk^nP_{\bullet}$ is a $Cosk^{n-1}G_{\bullet}$ bundle over $Cosk^nX_{\bullet}$.