nLab atlas (Rev #6, changes)

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Context

Higher geometry

Manifolds and cobordisms

Idea

In basic topology and differential geometry, by an atlas of/for a topological-, differentiable- or smooth manifold $X$ one means a collection of coordinate charts $U_i \subset X$ which form an open cover of $X$ .

If one considers here the disjoint union $\mathcal{U} \coloneqq \underset{i}{\sqcup} U_i$ of all the choordinate charts, then the separate chart embeddings $U_i \subset X$ give rise to a single map (continuous/differentiable function)

\mathcal{U} \longrightarrow X

and now the condition for an atlas is that this is a surjective étale map/local diffeomorphism.

If, next, one regards this morphism, under the Yoneda embedding, inside the topos of formal smooth sets, then these conditions on an atlas say that this morphism is

an effective epimorphism;
a formally étale morphism.

In this abstract form the concept of an atlas generalizes to any cohesive higher geometry (KS 17, Def. 3.3, Wellen 18, Def 4.13).

Next, for a geometric stack $\mathcal{X}$ , an atlas is a smooth manifold $\mathcal{U}$ (for differentiable stacks) or scheme $\mathcal{U}$ (for algebraic stacks) or similar, equipped with a morphism

\mathcal{U} \longrightarrow \mathcal{X}

that is an effective epimorphism and formally étale morphism in the corresponding higher topos (for instance in that of formal smooth infinity-groupoids).

Here the terminology has a bifurcation:

In the general context of geometric stacks one typically drops the second condition and calls any effective epimorphism from a smooth manifold or scheme to a differentiable stack or algebraic stack, respectively, an atlas . (e.g.Leman 10, 4.4).
If in addition the condition is imposed that such an effective epimorphism exists which is also formally étale, then the geometric stack is called an orbifold or Deligne-Mumford stack (often with various further conditions imposed).