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SUMMARY:PB-groupoids vs VB-groupoids
DTSTART:20261009T120000Z
DTEND:20261009T130000Z
DTSTAMP:20260930T200100Z
UID:indico-event-17354@indico.math.cnrs.fr
CONTACT:kellendonk@math.univ-lyon1.fr\;athomas@math.univ-lyon1.fr
DESCRIPTION:Speakers: Francesco Cattafi\n\nIt is well known that the colle
 ction of linear frames of a smooth $n$-manifold $M$ defines a principal $G
 L(n\,\\mathbb{R})$-bundle over $M$ (called the frame bundle)\; more genera
 lly\, this construction makes sense for any vector bundle over $M$. Conver
 sely\, any principal bundle together with a representation induces an asso
 ciated vector bundle\; these processes establish therefore a correspondenc
 e between vector bundles on one side\, and principal bundles with represen
 tations on the other side.\nIn differential geometry there are several nat
 ural instances where diagrams of Lie groupoids and vector bundles\, togeth
 er with suitable compatibilities\, appear. They are known as vector bundle
  groupoids (VB-groupoids) and their theory has been fairly developed in th
 e past decades\, with applications e.g. to Poisson geometry\, representati
 ons up to homotopy\, deformation theory\, and non-commutative geometry. On
  the other hand\, little is known about the principal bundle counterpart o
 f these objects.\nIn this talk\, I will recall all the notions mentioned a
 bove\, and then introduce a special class of frames of VB-groupoids which 
 interact nicely with the groupoid structure. I will then use them to assoc
 iate to any given VB-groupoid a diagram of Lie groupoids and principal bun
 dles\, together with the action of a (strict) Lie 2-groupoid $GL(l\,k)$\; 
 this will lead to the general notion of a principal bundle groupoid (PB-gr
 oupoid). Moreover\, I will sketch how to generalise the standard correspon
 dence between vector bundles and principal bundles to a correspondence bet
 ween VB-groupoids and PB-groupoids. I will conclude discussing a few examp
 les\, current works in progress\, and future applications.\nThis is joint 
 work with Alfonso Garmendia.\n\nhttps://indico.math.cnrs.fr/event/17354/
LOCATION:Salle Fokko du Cloux (Bat. Braconnier)
URL:https://indico.math.cnrs.fr/event/17354/
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