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SUMMARY:PB-groupoids vs VB-groupoids
DTSTART:20261013T090000Z
DTEND:20261013T100100Z
DTSTAMP:20261012T202800Z
UID:indico-event-17494@indico.math.cnrs.fr
DESCRIPTION:Speakers: Francesco Cattafi (Julius-Maximilians-Universität W
 ürzburg)\n\nIt is well known that the collection of linear frames of a sm
 ooth $n$-manifold $M$ defines a principal $GL(n\,\\mathbb{R})$-bundle over
  $M$ (called the frame bundle)\; more generally\, this construction makes 
 sense for any vector bundle over $M$. Conversely\, any principal bundle to
 gether with a representation induces an associated vector bundle\; these p
 rocesses establish therefore a correspondence between vector bundles on on
 e side\, and principal bundles with representations on the other side.\nIn
  differential geometry there are several natural instances where diagrams 
 of Lie groupoids and vector bundles\, together with suitable compatibiliti
 es\, appear. They are known as vector bundle groupoids (VB-groupoids) and 
 their theory has been fairly developed in the past decades\, with applicat
 ions e.g. to Poisson geometry\, representations up to homotopy\, deformati
 on theory\, and non-commutative geometry. On the other hand\, little is kn
 own about the principal bundle counterpart of these objects.\nIn this talk
 \, I will recall all the notions mentioned above\, and then introduce a sp
 ecial class of frames of VB-groupoids which interact nicely with the group
 oid structure. I will then use them to associate to any given VB-groupoid 
 a diagram of Lie groupoids and principal bundles\, together with the actio
 n of a (strict) Lie 2-groupoid $GL(l\,k)$\; this will lead to the general 
 notion of a principal bundle groupoid (PB-groupoid). Moreover\, I will ske
 tch how to generalise the standard correspondence between vector bundles a
 nd principal bundles to a correspondence between VB-groupoids and PB-group
 oids. I will conclude discussing a few examples\, current works in progres
 s\, and future applications.\nThis is joint work with Alfonso Garmendia.\n
 \nhttps://indico.math.cnrs.fr/event/17494/
LOCATION:125
URL:https://indico.math.cnrs.fr/event/17494/
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