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I come across problems when I try to compute the stack classifying non-oriented triangles. My reference is the book Algebriac Stacks ( K. Behrend, B. Conrad, D. Edidin, B. Fantechi, W. Fulton, L. Göttsche & A. Kresch). Sorry I can't find the online book project link.

A family of triangles over $S$ is a continuous proper map $X\rightarrow S$ making $X$ form a fiber bundle over $S$ with a distance function $d:X\times_SX\rightarrow \mathbb{R}_{\geq 0}$ making fibers isometric to some triangles. Then we can define a category $\mathfrak{T}$ over $\mathbf{Top}$ whose objects are families of triangle and morphisms are pairs $(X'\rightarrow X,S'\rightarrow S)$ making such a diagram commutative and the first map induces isomortries on fibers.

If we consider the stack classifying oriented triangles, then this stack is represented by a fiine moduli space $\widetilde{T}$ which is defined by inequations $a+b>c>0,a+c>b>0,b+c>a>0$.

For a given triangle, exchange its end ponts we will obtain an isometric one. Therefore we let the symmetric group $S_3$ act on $\widetilde{T}$ and obtain the coarse moduli space $\widetilde{T}/S_3$.

In the book above, it says the category $\mathfrak{T}$ is equivalent to the quotient stack $[\widetilde{T}/S_3]$. Actually in its definition, for any family of triangles $X\rightarrow S$ it associates the base space $S$ with a principal $S_3$-bundle $\widetilde{S}\rightarrow S$ where $\widetilde{S}:=(s,\text{ordering of the edges of}\ X_s)$. This will define a functor $\mathfrak{T}\rightarrow [\widetilde{T}/S_3]$.

My difficulty lies in proving this functor is essentially surjective. If it's true, it will mean given any principal $S_3$-bundle $E\rightarrow S$ there will exist some isomorphism $\widetilde{S}\rightarrow E$ and this means all principal $S_3$-bundles are trivial. But this corollary seems not to be correct.

Note that the quotient stack $[X/G]$ for a space $X$ and a topological group $G$ is the stack of $G$-torsors (principal bundles) with an equivariant map from the torsor to $X$.

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    $\begingroup$ Welcome to Mathematics SE. Take a tour. You'll find that simple "Here's the statement of my question, solve it for me" posts will be poorly received. What is better is for you to add context (with an edit): What you understand about the problem, what you've tried so far, etc.; something both to show you are part of the learning experience and to help us guide you to the appropriate help. You can consult this link for further guidance. $\endgroup$
    – Shaun
    May 12 at 15:03
  • $\begingroup$ Are you familiar with covering theory and, specifically, regular coverings? Do you realize that these are principal fiber bundles? $\endgroup$ May 12 at 19:06
  • $\begingroup$ @Shaun Sorry, I have changed my question and add more details. It seems better. Thanks for your reminding. $\endgroup$ May 13 at 3:38

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