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.

EDIT(Credits to მამუკაჯიბლაძე) : I read it another time and understand little more but still some notion is not clear.

Given any manifold $M$ there is a stack associated to it. It is $B(M\rightrightarrows M)\rightarrow Man$ which usually denoted by $\underline{M}$. So, by a map $p:X\rightarrow D$ they mean a map of stacks $p:\underline{X}\rightarrow D$. Similarly, for a manifold $M$, by a map $f:M\rightarrow D$ they mean a map of stacks $f:\underline{M}\rightarrow D$.

Given these two maps of stacks $f:\underline{M}\rightarrow D$ and $p:\underline{X}\rightarrow D$ there is a notion of $2$-fibered product, denoted by $\underline{M}\times_D\underline{X}$. Now, what they are asking is this stack $\underline{M}\times_D \underline{X}\rightarrow Man$ is coming from a manifold, i.e., isomorphic to some stack of the form $\underline{N}\rightarrow Man$, they are denoting $M\times_D X$ for $N$. Till here it is clear. I do not understand what does it mean to say $M\times_D X\rightarrow M$ is a submersion.. Is it that this map $M\times_D X\rightarrow M$ is some god given map coming from morphism of stacks $\underline{M}\times_D\underline{X}\rightarrow \underline{M}$?

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.

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.

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.

EDIT(Credits to მამუკაჯიბლაძე) : I read it another time and understand little more but still some notion is not clear.

Given any manifold $M$ there is a stack associated to it. It is $B(M\rightrightarrows M)\rightarrow Man$ which usually denoted by $\underline{M}$. So, by a map $p:X\rightarrow D$ they mean a map of stacks $p:\underline{X}\rightarrow D$. Similarly, for a manifold $M$, by a map $f:M\rightarrow D$ they mean a map of stacks $f:\underline{M}\rightarrow D$.

Given these two maps of stacks $f:\underline{M}\rightarrow D$ and $p:\underline{X}\rightarrow D$ there is a notion of $2$-fibered product, denoted by $\underline{M}\times_D\underline{X}$. Now, what they are asking is this stack $\underline{M}\times_D \underline{X}\rightarrow Man$ is coming from a manifold, i.e., isomorphic to some stack of the form $\underline{N}\rightarrow Man$, they are denoting $M\times_D X$ for $N$. Till here it is clear. I do not understand what does it mean to say $M\times_D X\rightarrow M$ is a submersion.. Is it that this map $M\times_D X\rightarrow M$ is some god given map coming from morphism of stacks $\underline{M}\times_D\underline{X}\rightarrow \underline{M}$?