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- Differentiable_stack abstract "In differential geometry, a differentiable stack is a stack over the category of differentiable manifolds (with the usual open covering topology). Every Lie groupoid Γ gives rise to a differentiable stack that is the category of Γ-torsors. In fact, every differentiable stack is of this form. Hence, roughly, "a differentiable stack is a Lie groupoid up to Morita equivalence."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.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 of differential p-forms over X is given by, for any x in X over a manifold U, letting be the space of p-forms on U. The sheaf is called the structure sheaf on X and is denoted by . comes with exterior derivative, obviously, and thus is a complex of sheaves of vector spaces over X: one thus has the notion of de Rham cohomology of X.An epimorphism between differentiable stacks is called a gerbe over X if is also an epimorphism. For example, if X is a stack, is a gerbe. A theorem of Giraud says that corresponds one-to-one to the set of gerbes over X that are locally isomorphic to and that come with trivializations of their bands.".
- Differentiable_stack wikiPageExternalLink 0605694.
- Differentiable_stack wikiPageExternalLink 0710.4340v2.pdf.
- Differentiable_stack wikiPageExternalLink differentiable+stack.
- Differentiable_stack wikiPageID "41375260".
- Differentiable_stack wikiPageRevisionID "591848605".
- Differentiable_stack subject Category:Differential_geometry.
- Differentiable_stack comment "In differential geometry, a differentiable stack is a stack over the category of differentiable manifolds (with the usual open covering topology). Every Lie groupoid Γ gives rise to a differentiable stack that is the category of Γ-torsors. In fact, every differentiable stack is of this form. Hence, roughly, "a differentiable stack is a Lie groupoid up to Morita equivalence."A differentiable space is a differentiable stack with trivial stabilizers.".
- Differentiable_stack label "Differentiable stack".
- Differentiable_stack sameAs m.0zrq83z.
- Differentiable_stack sameAs Q16960732.
- Differentiable_stack sameAs Q16960732.
- Differentiable_stack wasDerivedFrom Differentiable_stack?oldid=591848605.
- Differentiable_stack isPrimaryTopicOf Differentiable_stack.