Vaughn's point is a good one, but not quite apropos of Tom's notion of self-similarity. In the category of ordered sets the final co-algebra of the Pavlovic-Pratt functor is, indeed, the half-open real interval, [0,1). But Tom was in the category of topological spaces. To make [0,1) into a self-similar _space_ one must reinterpret their functor so that the new point of the one-point compactification of each copy of X is identified with the bottom of the next copy. So what interesting space was first described as the final co-algebra of a functor? (I think that locution succeeds in avoiding such things as constant functors.) I would suggest the one-point compactification of the countable discrete set. (The functor, of course, is 1 + X.) Peter 24-Nov-2004 20:03:48 -0400,2672;000000000000-00000000
Peter Freyd wrote:
Vaughn's point is a good one, but not quite apropos of Tom's notion of self-similarity. In the category of ordered sets the final co-algebra of the Pavlovic-Pratt functor is, indeed, the half-open real interval, [0,1). But Tom was in the category of topological spaces. To make [0,1) into a self-similar _space_ one must reinterpret their functor so that the new point of the one-point compactification of each copy of X is identified with the bottom of the next copy.
the title of tom's series of papers suggests that he wants to develop a General Theory of Self-Similarity, not just a topological theory. and in a general theory, the various decompositions of [0,1) into the copies of [0,1) and so on --- would qualify as self-similarity in most people's eyes. i didn't read tom's papers yet, but it seems to me that the essence of self-similarity in general, and the conceptual value of all of our reconstructions of the reals using self-similarity, is not so much in their topology or order, but in their coalgebra, capturing the infinite subdivisions. and finally, to pop up a level, i am a bit surprised with tom's line of reasoning: "it's not topology, so it shouldn't be mentioned", tacitly supported by peter's posting. i was under the impression that the authors in all sciences consider it to be a matter of good taste to be generous in references and acknowledgements, and not frugal; to be inclusive and not exclusive. people provide as many references as they can, to let the readers track their ideas, and make the connections. useful papers don't cite just their parent papers, but also their grandparent papers, and brothers and sisters. especially when the parents so clearly and generously recognize their debts, as peter's original postings did. -- dusko 25-Nov-2004 14:18:16 -0400,1422;000000000000-00000000
Dusko Pavlovic wrote:
i was under the impression that the authors in all sciences consider it to be a matter of good taste to be generous in references and acknowledgements, and not frugal; to be inclusive and not exclusive. people provide as many references as they can, to let the readers track their ideas, and make the connections. useful papers don't cite just their parent papers, but also their grandparent papers, and brothers and sisters. especially when the parents so clearly and generously recognize their debts, as peter's original postings did.
-- dusko
not all authors - unfortunately though in many cases it may be a lack of knowledge/scholarship generosity and scholarship in references is to be applauded but acknowledgements in a certian science can be so profuse naming everybody the author had a slighly related conversation with there by diluting the major influences another extreme in tha same science is to give a bibliogrpahy without titles this has multiple disadvantages at one conference to a mixed audience (math and ...) the math organizer insisted references include title to which a participant repsonded what if you don't know the title!!! jim
25-Nov-2004 14:18:40 -0400,3095;000000000000-00000000
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
This thread reminds me of John Rhodes' review of the classic, "The Algebraic Theory of Semigroups, Vol. I" by Clifford and Preston. This exceptional account of the state of semigroup theory in the early 60s contained more than copious citations, to authors for published works, to colleagues for ideas, and even to the extent that solutions to problems were cited. Rhodes, his acerbic wit at hand, commented on this incredible documentation of where even the simplest ideas had originated, was led to comment, "it's surprising that the authors didn't cite Gutenberg for the type." Mike Mislove On Nov 24, 2004, at 7:09 PM, jim stasheff wrote:
Dusko Pavlovic wrote:
i was under the impression that the authors in all sciences consider it to be a matter of good taste to be generous in references and acknowledgements, and not frugal; to be inclusive and not exclusive. people provide as many references as they can, to let the readers track their ideas, and make the connections. useful papers don't cite just their parent papers, but also their grandparent papers, and brothers and sisters. especially when the parents so clearly and generously recognize their debts, as peter's original postings did.
-- dusko
not all authors - unfortunately though in many cases it may be a lack of knowledge/scholarship
generosity and scholarship in references is to be applauded but acknowledgements in a certian science can be so profuse naming everybody the author had a slighly related conversation with there by diluting the major influences
another extreme in tha same science is to give a bibliogrpahy without titles this has multiple disadvantages at one conference to a mixed audience (math and ...) the math organizer insisted references include title to which a participant repsonded what if you don't know the title!!!
jim
=============================================== Professor Michael Mislove Phone: +1 504 862-3441 Department of Mathematics FAX: +1 504 865-5063 Tulane University URL: http://www.math.tulane.edu/~mwm New Orleans, LA 70118 USA =============================================== 26-Nov-2004 14:25:47 -0400,1826;000000000000-00000000
Michael Mislove wrote:
This thread reminds me of John Rhodes' review of the classic, "The Algebraic Theory of Semigroups, Vol. I" by Clifford and Preston. This exceptional account of the state of semigroup theory in the early 60s contained more than copious citations, to authors for published works, to colleagues for ideas, and even to the extent that solutions to problems were cited. Rhodes, his acerbic wit at hand, commented on this incredible documentation of where even the simplest ideas had originated, was led to comment, "it's surprising that the authors didn't cite Gutenberg for the type."
i agree, it is easy to exaggerate with citations, and with many other things, in all kinds of ways. one could cite gutenberg out of a scholarly zeal, or just to emphasize that they only use the shoulders of giants. but such generalities aside, we had a much smaller issue here: there is this very interesting research in self-similarity, and it is based on the theorem that [0,1] is a final coalgebra. on the other hand, vaughan and i had written some papers promoting the idea that the reals form a final coalgebra, and felt that we were a bit closer to the topic than gutenberg to semigroups. but enough of that. it is interesting that mandelbrot is talking about the real interval in coalgebraic terms. there are many such examples. continued fractions are also a familiar instance of a coalgebraic expansion. i don't think that we are really discovering the coalgebraic nature of self-similar and analytic objects; just giving it a categorical formulation. but math analysis consists of lots of coalgebraic constructions, often just thinly disguised. -- dusko 30-Nov-2004 08:03:55 -0400,2441;000000000000-00000000
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.)
How about the Cantor set (as the final coalgebra for the endofunctor of Top defined by X |--> X + X )? Tom 24-Nov-2004 20:03:48 -0400,1045;000000000000-00000000
participants (6)
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Dusko Pavlovic -
jim stasheff -
Michael Mislove -
Peter Freyd -
Tom Leinster -
Vaughan Pratt