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Doctoral Thesis
DOI
https://doi.org/10.11606/T.45.2020.tde-27042020-232832
Document
Author
Full name
Fernando Studzinski Carvalho
E-mail
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2019
Supervisor
Committee
Gonzalez, Cristian Andres Ortiz (President)
Brahic, Olivier
Hoyo, Matias Luis del
Jardim, Marcos Benevenuto
Struchiner, Ivan
Title in English
On the cohomology of representations up to homotopy of Lie groupoids
Keywords in English
Cohomology
Lie groupoid
Representation up to homotopy
Simplicial manifold
Abstract in English
We study the concept of representations up to homotopy of Lie groupoids. Our main result is the proof that the cohomology of a Lie groupoid with coefficients in a representation up to homotopy is a Morita invariant of the groupoid. This can be interpreted as a way to provide cohomological invariants for orbifolds and more generally for differentiable stacks, which are spaces with singularities whose isomorphism classes are in one-to-one correspondence with Morita equivalence classes of Lie groupoids. To prove this result, we rely on the theory of simplicial objects in smooth categories e.g. simplicial manifolds, sim- plicial vector bundles, and equivalences between them which are defined in terms of maps called hypercovers. We also prove results on the invariance of the simplicial cohomology of these spaces under hypercovers.
Title in Portuguese
Sobre a cohomologia de representações a menos de homotopia de grupoides de Lie
Keywords in Portuguese
Cohomology
Lie groupoid
Representation up to homotopy
Simplicial manifold
Abstract in Portuguese
Estudamos o conceito de representações a menos de homotopia de grupoides de Lie e a cohomologia naturalmente associada a tais representações. Nosso principal resultado é a prova de que a cohomologia de um grupoide de Lie com valores em uma representação a menos de homotopia é um invariante de Morita, o que pode ser interpretado como uma forma de introduzir invariantes cohomologicos para orbifolds e mais geralmente para stacks diferenciáveis, que são espaços com singularidades cujas classes de isomorfismo estão em correspondência biunvoca com classes de equivalência de Morita de grupoides de Lie. Para provar tal resultado, utilizamos a teoria de objetos simpliciais em categorias suaves e.g. variedades simpliciais, fibrados vetoriais simpliciais e equivalências entre eles, definidas a partir de mapas chamados hypercovers. Demonstramos também a invariância da cohomologia simplicial destes objetos sob hypercovers.
 
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ThesisFerStudzinski.pdf (874.15 Kbytes)
Publishing Date
2020-04-28
 
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