Back in the mid 60s, I read a paper (or maybe a preprint) by John Moore showing that simplicial groups are Kan complexes. As I look through MathSciNet, I can find nothing like that among Moore's publications. Does anyone know if this was published or does anyone have a copy? Or can anyone give me a reference for the fact? Michael
Michael Barr writes:
Back in the mid 60s, I read a paper (or maybe a preprint) by John Moore showing that simplicial groups are Kan complexes. As I look through MathSciNet, I can find nothing like that among Moore's publications. Does anyone know if this was published or does anyone have a copy? Or can anyone give me a reference for the fact?
Michael
To answer the last question first the fact is Lemma 3.4 (credited to Moore) in Paul G. Goerss, John F. Jardine Simplicial Homotopy Theory Birkhauser (1999) I believe Moore published it as part of his outline of simplicial theory in @incollection {MR0111027, AUTHOR = {Moore, John C.}, TITLE = {Semi-simplicial complexes and {P}ostnikov systems}, BOOKTITLE = {Symposium internacional de topolog\'\i a algebraica International symposium on algebraic topology}, PAGES = {232--247}, PUBLISHER = {Universidad Nacional Aut\'onoma de M\'exico and UNESCO, Mexico City}, YEAR = {1958}, MRCLASS = {55.00}, MRNUMBER = {MR0111027 (22 \#1894)}, MRREVIEWER = {J. A. Zilber}, } and that the notes you reference were from his 1956 Princeton Seminar on Algebraic Homotopy Theory. I have long since lost my copy of the notes, and cannot find a copy of the above paper, so these last two have to be considered speculative memories. -- Bob -- Robert L. Knighten Robert@Knighten.org
Michael-- The earliest published proof I know of is in Peter May's classic Simplicial Objects in Algebraic Topology, D. Van Nostrand Mathematical Studies #11 (1967) on pages 67-68. He credits John Moore and references mimeographed Princeton notes of 1956: Seminar on Algebraic Homotopy Theory. I don't know if the proof May gives is the one Moore gave but I suspect it is. Don
Michael Barr wrote:
Back in the mid 60s, I read a paper (or maybe a preprint) by John Moore showing that simplicial groups are Kan complexes. As I look through MathSciNet, I can find nothing like that among Moore's publications. Does anyone know if this was published or does anyone have a copy? Or can anyone give me a reference for the fact?
Michael
Dear Mike and everyone, There is a proof in Peter May's little book and some indication in the survey article by Curtis (but his formaule do not work). The nice thing about the result is not that it is true but that it is possible to construct formulae for the fillers EXPLICITLY from the information on the horns. Heiner Kamps and myself used this approach in our book and it seemed to us better to introduce the problem and then get the reader to investigate the algorithmic idea rather than giving the solution as the latter does not tell you where the formulae came from. Robert Knighten's reference to being in Moore's seminar notes is the earliest I have seen it as well. Tim Porter
There is a proof in Peter May's little book and some indication in the survey article by Curtis (but his formaule do not work). Tim, do you mean that the proof Curtis gives is erroneous? It is important for me to know what exactly is wrong there. You see, with Pirashvili we have a note (in Georgian Math. J. 9 (2002) 71-74, and still online at http://arxiv.org/abs/math.AT/0106143) about a generalization of that proof to any Maltsev varieties. Although we needed a slight modification, till this moment I was sure that the original formulæ were also fine. Btw, there was also another proof by Carboni, Kelly and Pedicchio in Appl. Categ. Struct. 1 (1993) 385-421, working for any Maltsev categories. Mamuka Jibladze
Dear Colleagues, The question Michael asked has already been answered by several people. However I would like to comment on Tim's "EXPLICITLY": A simplicial group is a simplicial object in the category of groups, which is a Mal'tsev category (some people use wrong spelling "Mal'cev"), and - as I know from Aurelio Carboni - every simplicial object in a Mal'tsev category is a Kan complex. This also tells me that the formulae Tim is mentioning should in fact involve just the Mal'tsev operation m (recall that for groups one can put m(x,y,z) = x-y+z for the m, in the additive notation). There must be something like this in Jonathan Smith's "Mal'tsev varieties". George Janelidze Prof T Porter wrote:
Dear Mike and everyone,
There is a proof in Peter May's little book and some indication in the survey article by Curtis (but his formaule do not work). The nice thing about the result is not that it is true but that it is possible to construct formulae for the fillers EXPLICITLY from the information on the horns. Heiner Kamps and myself used this approach in our book and it seemed to us better to introduce the problem and then get the reader to investigate the algorithmic idea rather than giving the solution as the latter does not tell you where the formulae came from.
Robert Knighten's reference to being in Moore's seminar notes is the earliest I have seen it as well.
Tim Porter
participants (6)
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Donovan Van Osdol -
George Janelidze -
Mamuka Jibladze -
Michael Barr -
Prof T Porter -
Robert Knighten