Understanding the definition of atlas of a stack over the category of manifolds

Question

I am reading https://arxiv.org/abs/0806.4160 to understand orbifolds as stacks.

Definition : Let $D\rightarrow Man$ be a stack over category of manifolds. An atlas for $D$ is a manifold $X$ and a map $p:X\rightarrow D$ such that for any map $f:M\rightarrow D$ from a manifold the fiber product $M\times_D X$ is a manifold and the map $pr_1:M\times_D X\rightarrow M$ is a surjective submersion.

I fail to understand what this means. I know what is a stack but I am not sure what it means to say a map from a manifold to a stack.

Any clarity would be welcome.

Before giving this definition, he says,

To keep the notation from getting out of control we drop the distinction between a manifold and the associated stack $\underline{M}$. We will also drop the distinction between stacks isomorphic to manifolds and manifolds.

This only made the definition complicated and not easier (for me) :D

Help me to understand the notion of atlas.

If it helps, I am trying to read about geometric stacks which are defined to be stacks over manifolds which possesses an atlas.

Answer 1

It is good to have a simple example in mind. An orbifold is a topological space for which each point $x$ has a neighborhood homeomorphic to the quotient of an open subset of $U_x$ of $\mathbb{R}^n$ by a finite group $G_x$. Here $X$ is $\coprod_xU_x$ and $p:X\rightarrow D$ is the restriction of the quotient map to $U_x$.

To be more concrete, if $D$ is the quotient of a finite group $G$ which acts on the manifold $X$ (with eventually fixed points) an atlas of $D$ is just $X$. To obtain an atlas of the orbifold you just have to blow-up the singularities.

You can also read the first section of my paper Differentiable Categories, gerbes and G-structures, Int. Journal of Contemp. Math. Sciences, Vol. 4, 2009, no. 29-32, 1547-1590, arXiv:0806.1357.

Answer 2

An atlas for a stack $\mathcal{D}\rightarrow Man$ is

• a smooth manifold $X$ and
• a map of stacks $p:\underline{X}\rightarrow \mathcal{D}$

such that, given

• a smooth manifold $M$ and
• a map of stacks $f:\underline{M}\rightarrow \mathcal{D}$

the fibered product stack $\underline{M}\times_{\mathcal{D}}\underline{X}\rightarrow Man$ is isomorphic to a stack coming from a manifold and something extra happens.

As we are saying $\underline{M}\times_{\mathcal{D}}\underline{X}\rightarrow Man$ is coming from a manifold, we better give it a name. We denote that manifold by $M\times_{\mathcal{D}}X$.

This has nothing to do with fibered product of manifolds $M$ and $X$, of course we can not talk of fibered product of manifolds with respect to a category $\mathcal{D}$, it is just a notation that the stack is $\underline{M}\times_\mathcal{D}\underline{X}$ is isomorphic to the stack $\underline{M\times_\mathcal{D}X}$ coming from the manifold $M\times_\mathcal{D}X$.

Now, let us come to something extra happens.

This fibered prodcut $\underline{M}\times_\mathcal{D}\underline{X}$ comes with projection map $pr_1:\underline{M}\times_\mathcal{D}\underline{X}\rightarrow \underline{M}$.

Corollary $4.16$ in that article says :

Let $M,M'$ be two manifolds. For any map $F:\underline{M}\rightarrow \underline{M'}$ of categories fibered in groupoids there is a unique map of manifolds $f:M\rightarrow M'$ defining $f$.

So, for $pr_1:\underline{M}\times_\mathcal{D}\underline{X}=\underline{M\times_DX}\rightarrow \underline{M}$, there is a unique map associated $f:M\times_D X\rightarrow M$ and by something extra happens we mean to say this map $f:M\times_D X\rightarrow M$ is a submersion.