Diffeological Coarse Moduli Spaces of Stacks over Manifolds

Watts, Jordan; Wolbert, Seth

doi:10.48550/arxiv.1406.1392

Cited by 2 publications

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“…A different strategy, which also proves useful, but that we shall not discuss in this text is that of equipping an orbispace with the structure of a diffeology. We refer to [67,69,71] for details on this approach.…”

Section: Introductionmentioning

confidence: 99%

Orbispaces as differentiable stratified spaces

Crainic

Mestre

2017

Lett Math Phys

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We present some features of the smooth structure and of the canonical stratification on the orbit space of a proper Lie groupoid. One of the main features is that of Morita invariance of these structures—it allows us to talk about the canonical structure of differentiable stratified space on the orbispace (an object analogous to a separated stack in algebraic geometry) presented by the proper Lie groupoid. The canonical smooth structure on an orbispace is studied mainly via Spallek’s framework of differentiable spaces, and two alternative frameworks are then presented. For the canonical stratification on an orbispace, we extend the similar theory coming from proper Lie group actions. We make no claim to originality. The goal of these notes is simply to give a complementary exposition to those available and to clarify some subtle points where the literature can sometimes be confusing, even in the classical case of proper Lie group actions.

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Section: Introductionmentioning

confidence: 99%

Orbispaces as differentiable stratified spaces

Crainic

Mestre

2017

Lett Math Phys

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“…The category Mfd also embeds into the category of differentiable stacks Stacks Mfd with the Yoneda embedding Y. Moreover, the category Stacks Mfd connects with Diff using the Grothendieck construction functor and the coarse moduli space functor Coarse, which are adjoint to each other; see [53] for more details.…”

Section: Appendix Bmentioning

confidence: 99%

“…The category Diff is indeed equivalent to the category of concrete sheaves on a concrete site; see [3]. Moreover, Watts and Wolbert [53] have shown that Diff is closely related to the category of differentiable stacks with adjoint functors between them. As Baez and Hoffnung have mentioned in [3,Introduction], we can use the larger category Diff for abstract constructions and the smaller one Mfd for theorems that rely on good control over local structure.…”

Section: Introductionmentioning

confidence: 99%

Simplicial cochain algebras for diffeological spaces

Kuribayashi

2019

Preprint

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We introduce a de Rham complex endowed with an integration map into the singular cochain complex which gives the de Rham theorem for every diffeological space. The theorem allows us to conclude that the Chen complex for a simply-connected manifold is quasi-isomorphic to the new de Rham complex of the free loop space of the manifold with an appropriate diffeology. This result is generalized from a diffeological point of view. In consequence, the de Rham complex behaves as a relevant codomain of Chen's iterated integrals. Furthermore, the precess of the generalization yields the Leray-Serre spectral sequence and the Eilenberg-Moore spectral sequence in the diffeological setting. The spectral sequences enable us to obtain computational examples of the new de Rham cohomology algebras for diffeological spaces containing the irrational torus and its related loop space.

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Diffeological Coarse Moduli Spaces of Stacks over Manifolds

Cited by 2 publications

References 0 publications

Orbispaces as differentiable stratified spaces

Orbispaces as differentiable stratified spaces

Simplicial cochain algebras for diffeological spaces

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