nLab
differentiable stack

Context

Higher geometry

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

\infty-Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

A differentiable stack is a geometric stack on the site SmoothMfd of smooth manifolds: a stack which is represented by a Lie groupoid.

This means that a differentiable stack is in a way nothing else but a Lie groupoid: but the point is that equivalence of stacks captures the notion of Morita equivalence of their presenting (Lie) groupoids.

This means that looking at a Lie groupoid in terms of the stack it presents provides a direct way to recognizing the right notion of equivalence of these objects. The notion of Morita equivalence, on the other hand, proceeds via the reasoning of homotopy theory.

Notice that the stack presented by a (Lie) groupoid GG is really the stack which sends every test manifold to the category of GG-principal bundles over that manifold. Such a GG-principal bundle, also known as a torsor, is analogous to a module for an algebra. This explains the terminology of “Morita morphisms”, which originates in algebra:

Just as two algebras are Morita equivalent if their categories of modules are equivalent, two groupoids are Morita equivalent if their stacks of torsors are equivalent.

Properties

Characterization by Lie groupoids

Sending a Lie groupoid to the smooth stack it represents constitutes an equivalence of (infinity,1)-categories between Lie groupoids with Morita morphisms/bibundles between them and differentiable stacks.

(e.g. Blohmann 07, Carchedi 11). For review in a broader context see also Nuiten 13, around prop. 2.2.34

Other descriptions

A differentible stack is in particular an object in the cohesive (∞,1)-topos Smooth∞Grpd that is

It is however more special than that. The general 1-truncated concrete smooth ∞-groupoids are internal groupoids in diffeological spaces.

Mapping stacks

For XX, YY two differentiable stacks, there is the mapping stack [X,Y][X,Y], which a priory is only a smooth stack. It should be true that a sufficient condition for this itself to be a Frechet-differentiable stack is that XX has an atlas by a compact smooth manifold.

(See also manifold structure of mapping spaces.)

References

See also the references at geometric stack and topological stack.

Revised on March 17, 2015 14:37:19 by Urs Schreiber (80.92.246.195)