Criterion for a map of stacks to be an atlas

Definition : A stack \(\mathcal{D}\rightarrow \text{Man}\) is differentiable if there exists a manifold \(X\) with an atlas \(p:\underline{X}\rightarrow \mathcal{D}\) i.e., \(p\) is representable surjective submersion.

We see a criterion for a map \(p:\underline{X}\rightarrow \mathcal{D}\) to be an atlas. By \(p:\underline{X}\rightarrow \mathcal{D}\) to be representable surjective submersion, we mean given a map of stacks \(\underline{Y}\rightarrow \mathcal{D}\) the fibered product \(\underline{X}\times_{\mathcal{D}}\underline{Y}\) is representable by a manifold and that the map of manifolds \(\underline{X}\times_{\mathcal{D}}\underline{Y}\rightarrow \underline{Y}\) is a surjective submersion. As \(\underline{X}\times_{\mathcal{D}}\underline{Y}\) is representable by a manifold for any map of stacks \(\underline{Y}\rightarrow \mathcal{D}\), in particular, taking \(\underline{Y}\rightarrow \mathcal{D}\) to be the same map \(\underline{X}\rightarrow \mathcal{D}\) we see that, in particular \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold.

Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold.

As \(\underline{X}\times_{\mathcal{D}}\underline{Y}\rightarrow \underline{Y}\) is a submersion for any map of stacks \(\underline{Y}\rightarrow \mathcal{D}\), in particular, taking \(\underline{Y}\rightarrow \mathcal{D}\) to be the same map \(\underline{X}\rightarrow \mathcal{D}\) we see that, projecion map \(\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) is a submersion. It is not relevant which projection is it as both maps are same. So, both projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions.

Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for stack \(\mathcal{D}\) then projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions.

As any representable surjective submersion is an epimorphism we have following remark.

Remark : If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism.

Combining all these remarks we have following remark.

If \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\) then, \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism.

It turns out that converse of above remark is true.

Proposition : Let \(p:\underline{X}\rightarrow \mathcal{D}\) is a morphism of stacks such that \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and that  \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. Then, Then, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable surjective submersion i.e., an atlas for \(\mathcal{D}\).

Before we give proof of this, we recall a result.

Lemma : Let \(\mathcal{D}\rightarrow\mathcal{C}\) be a morphism of stacks. Suppose \(U\) be a manifold and \(\underline{U}\rightarrow \mathcal{C}\) is an epimorphism of stacks such that fiber product \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\) is represented by a manifold and the map of manifolds \(\mathcal{D}\times_{\mathcal{C}}\underline{U}\rightarrow U\) is a submersion. Then, \(\mathcal{D}\rightarrow \mathcal{C}\) is a representable submersion.

To prove \(\underline{X}\rightarrow \mathcal{D}\) is a representable submersion, consider an epimorphism of stacks, namely \(\underline{X}\rightarrow \mathcal{D}\) (it is given to be an epimorphism, condition \(2\) above). See that the fibre product \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold (it is given in condition \(1\) above) and that the projection map \(\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow X\) is a submersion (it is in condition \(1\) above). Thus, by above lemma, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable submersion. Note that, a representable submersion that is an epimorphism is a representable surjective submersion.  Thus, \(p:\underline{X}\rightarrow \mathcal{D}\) is an atlas for \(\mathcal{D}\). So, we have the following result.

Proposition : Let \(p:\underline{X}\rightarrow \mathcal{D}\) is a morphism of stacks such that \(\underline{X}\times_{\mathcal{D}}\underline{X}\) is representable by a manifold and projection maps \(pr_1:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) and \(pr_2:\underline{X}\times_{\mathcal{D}}\underline{X}\rightarrow \underline{X}\) are submersions and that  \(p:\underline{X}\rightarrow \mathcal{D}\) is an epimorphism. Then, Then, \(p:\underline{X}\rightarrow \mathcal{D}\) is a representable surjective submersion i.e., an atlas for \(\mathcal{D}\).