Differentiable stack

A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.

Differentiable stacks are particularly useful to handle spaces with singularities (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory.

Definition

Categorical definition (via 2-functors)

Recall that a prestack (of groupoids) on a category ${\mathcal {C}}$, also known as a 2-presheaf, is a 2-functor $X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp}$, where $\mathrm {Grp}$ is the 2-category of (set-theoretical) groupoids, their morphisms, and the natural transformations between them. A stack is a prestack satisfying further glueing properties (analogously to the glueing properties satisfied by a sheaf). In order to state such properties precisely, one needs to define (pre)stacks on a site, i.e. a category equipped with a Grothendiek topology.

Any object $M\in \mathrm {Obj} ({\mathcal {C}})$ defines a stack ${\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)$, which associated to another object $N\in \mathrm {Obj} ({\mathcal {C}})$ the groupoid $\mathrm {Hom} _{\mathcal {C}}(N,M)$ of morphisms from $N$ to $M$. A stack $X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp}$ is called geometric if there is an object $M\in \mathrm {Obj} ({\mathcal {C}})$ and a special kind of morphism of stacks ${\underline {M}}\to X$, often call atlas, presentation or cover of the stack $X$.

A differentiable stack is a stack on the category ${\mathcal {C}}=\mathrm {Mfd}$ of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor $X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp}$, which is also geometric, i.e. admits an atlas ${\underline {M}}\to X$ as described above.

Note that, replacing the category of differentiable manifolds with that of affine schemes, one obtains the standard notion of algebraic stack. Similarly, replacing $\mathrm {Mfd}$ with the category of topological spaces, one obtains the definition of topological stack.

Geometric definition (via Morita equivalences)

Recall that a Lie groupoid consists of two differentiable manifolds $G$ and $M$, together with two surjective submersions $s,t:G\to M$, as well as a partial multiplication map $m:G\times _{M}G\to G$, a unit map $u:M\to G$, and an inverse map $i:G\to G$, satisfying group-like compatibilities.

Two Lie groupoids $G\rightrightarrows M$ and $H\rightrightarrows N$ are Morita equivalent if there is a principal bi-bundle $P$ between them, i.e. a principal right $H$-bundle $P\to M$, a principal left $G$-bundle $P\to N$, such that the two actions on $P$ commutes. Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.

A differentiable stack is a Morita equivalence class of a Lie groupoid.

Equivalence between the two definitions

Every Lie groupoid $G\rightrightarrows M$ gives rise to the differentiable stack $BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp}$, which sends any manifold $N$ to the category of $G$-torsors on $N$ (i.e. $G$-principal bundles). Any other Lie groupoid in the Morita class of $G\rightrightarrows M$ induces an isomorphic stack.

Conversely, any differentiable stack $X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp}$ is of the form $BG$, i.e. it can be represented by a Lie groupoid. More precisely, if ${\underline {M}}\to X$ is an atlas of the stack $X$, then one defines the Lie groupoid $G_{X}:=M\times _{X}M\rightrightarrows M$ and checks that $BG_{X}$ is isomorphic to $X$.

A theorem by Dorette Pronk states an equivalence of bicategories between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.

Differential space

A differentiable space is a differentiable stack with trivial stabilizers. For example, if a Lie group acts freely but not necessarily properly on a manifold, then the quotient by it is in general not a manifold but a differentiable space.

With Grothendieck topology

A differentiable stack $X$ may be equipped with Grothendieck topology in a certain way (see the reference). This gives the notion of a sheaf over $X$. For example, the sheaf $\Omega _{X}^{p}$ of differential $p$-forms over $X$ is given by, for any $x$ in $X$ over a manifold $U$, letting $\Omega _{X}^{p}(x)$ be the space of $p$-forms on $U$. The sheaf $\Omega _{X}^{0}$ is called the structure sheaf on $X$ and is denoted by ${\mathcal {O}}_{X}$. $\Omega _{X}^{*}$ comes with exterior derivative and thus is a complex of sheaves of vector spaces over $X$: one thus has the notion of de Rham cohomology of $X$.

Gerbes

An epimorphism between differentiable stacks $G\to X$ is called a gerbe over $X$ if $G\to G\times _{X}G$ is also an epimorphism. For example, if $X$ is a stack, $BS^{1}\times X\to X$ is a gerbe. A theorem of Giraud says that $H^{2}(X,S^{1})$ corresponds one-to-one to the set of gerbes over $X$ that are locally isomorphic to $BS^{1}\times X\to X$ and that come with trivializations of their bands.

References

• Kai Behrend, Ping Xu, Differentiable Stacks and Gerbes, 2008
• Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008
• Grégory Ginot, Introduction to Differentiable Stacks (and gerbes, moduli spaces …), 2013
• Jochen Heinloth: Some notes on differentiable stacks, Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32.

External links

• http://ncatlab.org/nlab/show/differentiable+stack