Peter F: So what interesting space was first described as the final co-algebra of a functor? [...] I would suggest the one-point compactification of the countable discrete set. (The functor, of course, is 1 + X.) Tom L: How about the Cantor set (as the final coalgebra for the endofunctor of Top defined by X |--> X + X )?
I'd love to know the date Peter has in mind for his candidate. Meanwhile these are both nice candidates. In the interest (inter alia :) of settling Peter's question in an orderly fashion, I'd be happy to serve as a clearing house for dates for spaces first presented as a final coalgebra of a functor Since Peter's question didn't impose Tom's implicit assumption that the functor be on Top (so to speak) (and moreover Peter's original example of a finitary functor for the continuum was not on Top but on Pos+-, posets with top and bottom), I'll accept functors on any category. Obviously the question has its specializations to people's favorite categories, I'll handle just the general list and let others worry about the preferred-category lists. I can prime the pump with the candidates I'm aware of, ordered by date of description as a final coalgebra. Contributions can take the form either of an earlier publication date for a candidate on the list (with reference of course), or a space not yet on the list. CANDIDATE DATE REFERENCE Baire space Mar 99 Pavlovic&Pratt, CMCS'99 The continuum Mar 99 Pavlovic&Pratt, CMCS'99 Cantor space Mar 99 Pavlovic&Pratt, CMCS'99 Wilson space Dec 99 Peter's categories posting, 12/23/99 1-pt comp'n of discrete N Nov 04 Peter's categories posting, 11/23/04 The n-th item on this list for n<5 can be described as the continuum with n-1 copies of each rational. As a generalization of Wilson space (which itself can be regarded as an extension of Cantor space), this list (less the discrete N example) can be extended further via the finitary functor X;(n-3);X where ; is poset concatenation and n-3 denotes the chain with n-3 elements, for n >= 3. Thus Turkey Space (today being Thanksgiving in North America), as the continuum with each rational in quadruplicate, would be described by the functor X;2;X, and so on with X;3;X etc. describing spaces that hopefully will rename nameless. Vaughan 25-Nov-2004 15:59:20 -0400,3875;000000000000-00000000