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, www.iti.cs.tu-bs.de/TI-INFO/milius/research/wellS.full.pdf
[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, www.paultaylor.eu/ordinals/\#towuti.
[T2] P. Taylor, Practical Foundations of Mathematics, Cambridge University Press, 1999.