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SUMMARY:It's all about monoids! Algebras\, coalgebras and Hopf algebras fr
 om a conceptual viewpoint. - Filip Bár
DTSTART:20120202T140000Z
DTEND:20120202T150000Z
UID:TALK36227@talks.cam.ac.uk
CONTACT:Guilherme Lima de Carvalho e Silva
DESCRIPTION:The aim of this talk is to illustrate one of MacLane's\nslogan
 s in his book 'Categories for the Working Mathematician'\, namely the cent
 rality of the concept of a monoid in mathematics.\n\nThe talk will consist
  of two parts. In the first part I will explain the basic definitions begi
 nning with internal monoids and internal groups in categories with finite 
 products and generalize them to the framework of (symmetric) monoidal cate
 gories\, i.e.\, to internal algebras and internal Hopf algebras. In course
  of the generalization we will understand the emergence of comonoids (coal
 gebras) and bimonoids (bialgebras). In the second part I will sketch a hie
 rarchy of algebraic structures with their respective tensor products apply
 ing the concepts and elementary observations from the first part to the ca
 tegory SETS. We will thereby encounter with familiar structures like group
 s\, rings and algebras (over commutative rings) as well as less\nfamiliar 
 structures like monoids\, rigs\, and algebras over commutative rigs. Final
 ly\, I will turn to certain adjunctions in this hierarchy and use them to 
 explain some of the standard (and universal) examples of Hopf algebras.
LOCATION:CMS\, MR15
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