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SUMMARY:UCLouvain-ULB-VUB Category Theory Seminar
DTSTART:20260323T151500Z
DTEND:20260323T171500Z
DTSTAMP:20260531T023900Z
UID:indico-event-5750@agenda.irmp.ucl.ac.be
DESCRIPTION:Speakers: Tim Van der Linden (Université catholique de Louvai
 n)\, Andrea Montoli (University of Milan)\n\n4.15 p.m. Andrea Montoli (Uni
 versity of Milan) : On the categorical behaviour of V-groups\nAbstract :We
  will consider compatible group structures on a V-category\, where V is a 
 quantale\, and we will explore some categorical properties of such groups.
  Examples of such structures are preordered groups\, metric and ultrametri
 c groups\, probabilistic (ultra)metric groups. In particular\, we show tha
 t\, when V is a cartesian quantale\, symmetric V-groups satisfy very stron
 g categorical-algebraic properties\, typical of the category of groups\, w
 hile the whole category of V-groups satisfies similar properties relativel
 y to a suitable class of split epimorphisms\, similarly to what happens fo
 r the category of monoids.\nJoint work with Maria Manuel Clementino.\n5.25
  p.m. Tim Van der Linden (UCLouvain and VUB): The bilinear product: an int
 rinsic approach to tensor products\nAbstract :In the context of a Janelidz
 e-Márki-Tholen semi-abelian category\, we establish an intrinsic definiti
 on of a bilinear product\, a tensor-like operation on objects of a categor
 y\, constructed in terms of limits and colimits. Given two objects in the 
 category\, their bilinear product is the abelian object obtained as the co
 smash product in the category of two-nilpotent objects of the reflections 
 of these objects. In many concrete cases\, this operation\, applied to a p
 air of abelian objects\, captures a classical tensor product. We explain t
 his by means of a recognition theorem\, which states that any symmetric\, 
 bi-cocontinuous bifunctor on an abelian variety of algebras can be recover
 ed as the bilinear product within a suitable semi-abelian variety\, namely
  of algebras over a certain two-nilpotent operad. In other words\, the ext
 ra structure carried by such a bifunctor on the abelian variety (for insta
 nce\, a tensor product\, known in the literature) is encoded by means of a
  surrounding semi-abelian variety whose abelian core is the original varie
 ty.\nWe illustrate the construction with several examples\, develop its ba
 sic properties\, and compare it to the semi-abelian analogue of the Brown-
 Loday non-abelian tensor product. As an application\, we present a categor
 ical version of Ganea's six-term exact homology sequence.\nThis is joint w
 ork with Bo Shan Deval and Manfred Hartl\, based on the preprint arXiv:251
 2.03951.\n\nhttps://agenda.irmp.ucl.ac.be/event/5750/
LOCATION:B/2nd floor-B.203 - Seminar room (Marc de Hemptinne (chemin du Cy
 clotron\, 2\, Louvain-la-Neuve))
URL:https://agenda.irmp.ucl.ac.be/event/5750/
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