# The stack classifying non-oriented triangles is equivalent to a quotient stack $[\widetilde{T}/S_3]$

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 $$X$$.

• 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. May 12 at 15:03
• Are you familiar with covering theory and, specifically, regular coverings? Do you realize that these are principal fiber bundles? May 12 at 19:06
• @Shaun Sorry, I have changed my question and add more details. It seems better. Thanks for your reminding. May 13 at 3:38