John Moore did in fact publish his proof that simplicial groups are Kan. It is stated as Theorem 3.4 in his paper Semi-Simplicial Complexes And Postnikov Systems (page 242 of the book, Symposium International De Topologia Algebraica, 1956 conference, book published in 1958). Moore refers to the Seminaire Cartan, 1954-55, expose XVIII, where the proof is given in full as Theorem 3 on page 18-04. Bill Messing [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Mike, I believe this theorem first appeared in Moore's 1956 Princeton notes "Seminar on algebraic homotopy theory". Unfortunately I seem to have lost my copy of this, so I can't really verify it, but I'm pretty sure. Myles -----Original Message----- From: Michael Barr [mailto:barr@math.mcgill.ca] Sent: Sun 9/11/2011 8:30 PM To: Categories list Subject: categories: Simplicial groups are Kan I know that is a theorem, due I think to John Moore. Can anyone give me a pointer to the original article. Michael [For admin and other information see: http://www.mta.ca/~cat-dist/ ]

Dear categorists, I would like to comment on Ronnie Brown's message, copied below, insisting on a parallelism that is not often acknowledged, and may 'clarify' - for instance - why simplicial groups somehow behave as 'cubical groups with connections' (see Tonks' paper cited by RB), rather than as 'ordinary cubical groups'. The degeneracies of a simplicial object correspond to the connections (or higher degeneracies) of a cubical one, introduced by Brown and Higgins, more than to the ordinary degeneracies. Formally, this fact can be motivated as follows. Let us start from the cylinder endofunctor I(X) = X x [0, 1] of topological spaces. Its main structure consists of natural transformations of powers of I, derived from (part of) the lattice structure of [0, 1]: - two faces 1 --> I, sending x to (x, 0) OR (x, 1), - a degeneracy I --> 1, sending (x, t) to x, - two connections I^2 --> I, sending (x, t, t') to (x, max(t, t')) OR (x, min(t, t')). Then we collapse the higher face of I (for instance), and we get a cone functor C, with a monad structure: - the lower face of I gives the unit 1 --> C, - the lower connection gives the multiplication C^2 --> C, - the other transformations (including the degeneracy of I) induce nothing. Now the cylinder I, with the above structure (which i [myself, not the cylinder] call a 'diad'), operating on any space, gives a cocubical object with connections, while the monad C gives an augmented cosimplicial object. [[ Addendum. If one wants to take on the parallelism to the singular cubical/ simplicial set of a space X, the construction becomes more involved. One should start from: - the cocubical space I* (with connections) of all standard cubes, produced by the cylinder I on the singleton space; - the augmented cosimplicial space Delta* produced by C on the empty space 0 (taking care that C(0), defined as a pushout, is the singleton, and C^n(0) is the standard simplex of dimension n-1). Then one applies to these structures the contravariant functor Top(-, X) and gets the singular cubical set of X (with connections) OR the singular simplicial set of X (augmented). ]] With best regards Marco Grandis On 12 Sep 2011, at 11:35, Ronnie Brown wrote:

The reference is included in this review *MR1173825 *of the cubical case.

Tonks, A. P. <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/ mathscinet/search/publications.html?pg1=IID&s1=325533>(4-NWAL) <http://0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet// search/institution.html?code=4_NWAL> Cubical groups which are Kan. /J. Pure Appl. Algebra/ <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/journaldoc.html? cn=J_Pure_Appl_Algebra> 81 <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? pg1=ISSI&s1=118323>(1992), <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? pg1=ISSI&s1=118323>no. 1, <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publications.html? pg1=ISSI&s1=118323> 83–87. <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/mscdoc.html? code=55U10,%2818D35,18G30%29><javascript:openWin('http:// unicat.bangor.ac.uk:4550/resserv', 'AMS:MathSciNet', 'atitle=Cubical %20groups%20which%20are% 20Kan&aufirst=A.&auinit=AP&auinit1=A&auinitm=P&aulast=Tonks&coden=JPAA A2&date=1992&epage=87&genre=article&issn=0022-4049&issue=1&pages=83-87 &spage=83&stitle=J.%20Pure%20Appl.%20Algebra&title=Journal%20of% 20Pure%20and%20Applied%20Algebra&volume=81')>

The author shows that group objects in the category of cubical sets with connections [R. Brown and P. J. Higgins, J. Pure Appl. Algebra 21 (1981), no. 3, 233--260; MR0617135 (82m:55015a) <http://0- ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/ publdoc.html?r=1&pg1=CNO&s1=617135&loc=fromrevtext>] satisfy the Kan extension condition. This is a very nice correspondence with the simplicial case [J. C. Moore, in Séminaire Henri Cartan de l'Ecole Normale Supérieure, 1954/1955, Exp. No. 18, Secrétariat Math., Paris, 1955; see MR0087934 (19,438e) <http://0-ams.mpim- bonn.mpg.de.unicat.bangor.ac.uk/mathscinet/search/publdoc.html? r=1&pg1=CNO&s1=87934&loc=fromrevtext>].

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