nLab atlas

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 coordinate 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

1. an effective epimorphism;

2. 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, Sati & Schreiber 2020, p. 27).

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:

1. 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).

2. 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).

From here, the terminology generalizes to $\infty$-stacks in general $\infty$-toposes, see this Remark at groupoid objects in an (∞,1)-topos are effective.

Yet more generally, the notion generalizes to 2-topos theory and higher. Over the point this yields the notion of category with an atlas and flagged categories (depending on the truncation of the atlas) and relates to the notions of flagged higher categories (Ayala & Francis 2018).

Related concepts

• category with an atlas

• n-types cover

References

Review of the classical concept of atlases for geometric stacks:

• Eugene Lerman, Section 4.4 of: Orbifolds as stacks?, L’Enseign. Math. (2) 56 (2010), no. 3-4, 315–363 (arXiv:0806.4160)

Formalization in cohesive homotopy theory and cohesive/modal homotopy type theory:

• Igor Khavkine, Urs Schreiber, Synthetic geometry of differential equations (arXiv:1701.06238)

• Felix Wellen, Formalizing Cartan Geometry in Modal Homotopy Type Theory (arXiv:1806.05966)

• Hisham Sati, Urs Schreiber, p. 27 of: Proper Orbifold Cohomology (arXiv:2008.01101)