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FUNCTORIAL ASPECTS OF THE RECONSTRUCTION OF LIE GROUPOIDS FROM THEIR BISECTIONS

Part of: Lie groups Pseudogroups, differentiable groupoids and general structures on manifolds General theory of categories and functors Categories and theories

Published online by Cambridge University Press:  14 March 2016

ALEXANDER SCHMEDING  and
CHRISTOPH WOCKEL
ALEXANDER SCHMEDING
Affiliation:
NTNU Trondheim, Alfred Getz’ vei 1, 7034 Trondheim, Norway email alexander.schmeding@math.ntnu.no
CHRISTOPH WOCKEL*
Affiliation:
Georg-August-Universität Göttingen, Bunsenstrasse 3–5, D-37073 Göttingen, Germany email christoph@wockel.eu
*
✉christoph@wockel.eu
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Abstract

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To a Lie groupoid over a compact base $M$, the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from its Lie group of bisections. In the present article we consider functorial aspects of these construction principles. The first observation is that this procedure is functorial (for morphisms fixing $M$). Moreover, it gives rise to an adjunction between the category of Lie groupoids over $M$ and the category of Lie groups acting on $M$. In the last section we then show how to promote this adjunction to almost an equivalence of categories.

Keywords

Lie groupoidinfinite-dimensional Lie groupmapping spacebisection functoradjoint functorscomonad

MSC classification

Primary: 18A40: Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
Secondary: 22E65: Infinite-dimensional Lie groups and their Lie algebras 58H05: Pseudogroups and differentiable groupoids 18C15: Triples (= standard construction, monad or triad), algebras for a triple, homology and derived functors for triples
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 101 , Issue 2 , October 2016 , pp. 253 - 276
Copyright
© 2016 Australian Mathematical Publishing Association Inc. 

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