## Coskeleta of simplicial principal bundles

### Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

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?

-

## 1 Answer

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 $Cosk^nX_{\bullet}$.

-
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 42, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9196189641952515, "perplexity_flag": "head"}