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Well-founded and well-pointed coalgebras
Description:  A coalgebra (A,a) of an endofunctor H is well-founded if the square Hm.a'=a.m is not a pullback for any proper subcoalgebra m:(A',a')-->(A,A). For a graph, seen as a coalgebra of the power-set functor, this is exactly the usual definition of well-founded graph, as observed by G. Osius [O]. P. Taylor [T1, T2] proved that for set endofunctors preserving inverse images the concepts of initial algebra and final well-founded coalgebra coincide. By using a different approach, it is shown that this coincidence in fact holds for all set endofunctors. For set functors preserving intersections, a new description of the final coalgebra and the initial algebra is presented: the former consists of all well-pointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. And the initial algebra consists of all well-pointed coalgebras that are well-founded.

This talk is based on joint work with J. Adámek, S. Milius and L. Moss (see [AMMS]).

[AMMS] J. Adámek, S. Milius, L. Moss and L. Sousa, Well-pointed coalgebras, preprint 2011,
[O] G. Osius, Categorical set theory: a characterization of the category of sets, J. Pure Appl. Algebra 4 (1974), 79-119.
[T1] P. Taylor, Towards a unified treatment of induction I: the general recursion theorem, preprint 1995-96,\#towuti.
[T2] P. Taylor, Practical Foundations of Mathematics, Cambridge University Press, 1999.

Date:  2012-01-20
Start Time:   15:00
Speaker:  Lurdes Sousa (IP Viseu/CMUC)
Institution:  IP Viseu/CMUC
Place:  Sala 2.5
Research Groups: -Algebra, Logic and Topology
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