A coalgebra (A,a) of an endofunctor H is wellfounded 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 powerset functor, this is exactly the usual definition of wellfounded 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 wellfounded 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 wellpointed coalgebras. These are the pointed coalgebras having no proper subobject and no proper quotient. And the initial algebra consists of all wellpointed coalgebras that are wellfounded. 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, Wellpointed coalgebras, preprint 2011, www.iti.cs.tubs.de/TIINFO/milius/research/wellS.full.pdf [O] G. Osius, Categorical set theory: a characterization of the category of sets, J. Pure Appl. Algebra 4 (1974), 79119. [T1] P. Taylor, Towards a unified treatment of induction I: the general recursion theorem, preprint 199596, www.paultaylor.eu/ordinals/\#towuti. [T2] P. Taylor, Practical Foundations of Mathematics, Cambridge University Press, 1999.
