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Fundamental group functors and higher Hopf formulae
 
 
Description:  In 1988, R. Brown and G. J. Ellis gave Hopf formulae for the integral homology of a group using topological methods. G. Janelidze first recognized that these descriptions are deeply connected with the categorical Galois-theoretic notion of covering morphism. Using this perspective, T. Everaert, M. Gran and T. Van der Linden proved that the same formulae can describe homology objects in a semi-abelian context by taking a Birkhoff reflector as coefficient functor. It turns out that these homology objects coincide with the so called fundamental groups arising in categorical Galois theory. In my thesis, I have developed this approach to homology and I have showed that one can easily work with both a bigger class of reflectors (those which preserve pullbacks of split epimorphisms along regular epimorphisms) and a bigger class of categories (the homological categories in which every regular epimorphism is an effective descent morphism). Since then, using the concept of Kan extension, I have been able to improve my results. For instance, I have shown that the nth fundamental group functor is in fact a right pointwise Kan extension of a special kind and I have found as a result (not anymore as a definition) that it can be computed using n-fold projective presentations. During my talk, I will present this new work.
Date:  2014-09-30
Start Time:   14:30
Speaker:  Mathieu Duckerts-Antoine (CMUC)
Institution:  CMUC
Place:  Sala 5.5
Research Groups: -Algebra, Logic and Topology
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