* Simplicial groups are Kan @ 2011-09-12 0:30 Michael Barr 2011-09-12 2:29 ` Peter May ` (6 more replies) 0 siblings, 7 replies; 14+ messages in thread From: Michael Barr @ 2011-09-12 0:30 UTC (permalink / raw) To: Categories list 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/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr @ 2011-09-12 2:29 ` Peter May 2011-09-13 14:22 ` William Messing 2011-09-12 5:10 ` Fernando Muro ` (5 subsequent siblings) 6 siblings, 1 reply; 14+ messages in thread From: Peter May @ 2011-09-12 2:29 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list The theorem is due to John Moore, but as far as I remember he never published his proof. It appeared in mimeographed notes entitled ``Seminar on algebraic homotopy theory'', Princeton, 1956. The result is Theorem 17.1 in my 1967 book ``Simplicial objects in algebraic topology'', and the argument there is based on Moore's notes (Moore was my adviser). On 9/11/11 7:30 PM, Michael Barr wrote: > 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/ ] [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 2:29 ` Peter May @ 2011-09-13 14:22 ` William Messing 0 siblings, 0 replies; 14+ messages in thread From: William Messing @ 2011-09-13 14:22 UTC (permalink / raw) To: Peter May; +Cc: Michael Barr, Categories list 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/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr 2011-09-12 2:29 ` Peter May @ 2011-09-12 5:10 ` Fernando Muro 2011-09-12 6:07 ` rlk ` (4 subsequent siblings) 6 siblings, 0 replies; 14+ messages in thread From: Fernando Muro @ 2011-09-12 5:10 UTC (permalink / raw) To: Categories list Théorème 3 in Moore, J. C. "Homotopie des complexes monoïdaux, I" Séminaire Henri Cartan, 7 no. 2, 1954-1955, Exp. No. 18, 8 p. http://archive.numdam.org/article/SHC_1954-1955__7_2_A8_0.pdf On Sun, 11 Sep 2011 20:30:47 -0400 (EDT), Michael Barr wrote: > I know that is a theorem, due I think to John Moore. Can anyone give > me a > pointer to the original article. > > Michael > -- Fernando Muro Universidad de Sevilla, Departamento de Álgebra http://personal.us.es/fmuro [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr 2011-09-12 2:29 ` Peter May 2011-09-12 5:10 ` Fernando Muro @ 2011-09-12 6:07 ` rlk 2011-09-12 6:55 ` Urs Schreiber ` (3 subsequent siblings) 6 siblings, 0 replies; 14+ messages in thread From: rlk @ 2011-09-12 6:07 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list Michael Barr writes: > I know that is a theorem, due I think to John Moore. Can anyone give me a > pointer to the original article. > > Michael The result first appeared in J. C. Moore, Homotopie des complexes monoideaux, I, Seminaire Henri Cartan, 1954-55. See Theorem 3 on p. 18-04. This is available on the web at http://archive.numdam.org/article/SHC_1954-1955__7_2_A8_0.pdf The result became somewhat more widely known as a result of J. C. Moore, Seminar on algebraic homotopy theory, Mimeographed notes, Princeton University, Princeton, N. J., 1956 -- Bob -- Robert L. Knighten 541-296-4528 RLK@knighten.org [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr ` (2 preceding siblings ...) 2011-09-12 6:07 ` rlk @ 2011-09-12 6:55 ` Urs Schreiber 2011-09-12 8:49 ` Tim Porter ` (2 subsequent siblings) 6 siblings, 0 replies; 14+ messages in thread From: Urs Schreiber @ 2011-09-12 6:55 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list On Mon, Sep 12, 2011 at 2:30 AM, Michael Barr <barr@math.mcgill.ca> wrote: > I know that is a theorem, due I think to John Moore. Can anyone give me a > pointer to the original article. According to http://ncatlab.org/nlab/show/simplicial+group this is J. C. Moore, Algebraic homotopy theory, lecture notes, Princeton University, 1955–1956 [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr ` (3 preceding siblings ...) 2011-09-12 6:55 ` Urs Schreiber @ 2011-09-12 8:49 ` Tim Porter 2011-09-12 9:35 ` Ronnie Brown 2011-09-12 13:00 ` Simplicial groups are Kan Tierney, Myles 6 siblings, 0 replies; 14+ messages in thread From: Tim Porter @ 2011-09-12 8:49 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list Quoting Michael Barr <barr@math.mcgill.ca>: > I know that is a theorem, due I think to John Moore. Can anyone give me a > pointer to the original article. > > Michael > > Dear All, In Curtis's survey article he gives Kan's paper: A combinatorial description of homotopy groups, Ann. Math. 67(1958)288 - 312. Actually I believe that the algorithm that Curtis gives does not work. That in Peter May's book does. Tim [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr ` (4 preceding siblings ...) 2011-09-12 8:49 ` Tim Porter @ 2011-09-12 9:35 ` Ronnie Brown 2011-09-13 15:12 ` Simplicial versus (cubical with connections) Marco Grandis [not found] ` <BDF51495-03DB-4725-8372-094AD1608A11@dima.unige.it> 2011-09-12 13:00 ` Simplicial groups are Kan Tierney, Myles 6 siblings, 2 replies; 14+ messages in thread From: Ronnie Brown @ 2011-09-12 9:35 UTC (permalink / raw) To: Michael Barr; +Cc: Categories list 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=JPAAA2&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>]. Ronnie On 12/09/2011 01:30, Michael Barr wrote: > 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/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Simplicial versus (cubical with connections) 2011-09-12 9:35 ` Ronnie Brown @ 2011-09-13 15:12 ` Marco Grandis [not found] ` <BDF51495-03DB-4725-8372-094AD1608A11@dima.unige.it> 1 sibling, 0 replies; 14+ messages in thread From: Marco Grandis @ 2011-09-13 15:12 UTC (permalink / raw) To: categories 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>]. [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
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* Re: Simplicial versus (cubical with connections) [not found] ` <BDF51495-03DB-4725-8372-094AD1608A11@dima.unige.it> @ 2011-09-13 16:58 ` Ronnie Brown 2011-09-14 7:08 ` Jonathan CHICHE 齊正航 0 siblings, 1 reply; 14+ messages in thread From: Ronnie Brown @ 2011-09-13 16:58 UTC (permalink / raw) To: Marco Grandis; +Cc: categories In response to Marco's interesting points, there is a related way of expressing this: degeneracies in the simplicial theory give simplices with some adjacent faces equal; in the cubical theory, degeneracies give cubes with some opposite faces equal, and never the twain shall meet! The connections \Gamma_i (which arise from the monoid structure max on the unit interval) restore the analogy with simplices, since \Gamma_i x has two adjacent faces the same. The advantage of cubes for our ideas was always the *easy* expression of `algebraic inverses to subdivision' (not so easy simplicially) and the application of this to local-to-global problems. The connections were found from trying to express the notion of `commutative cube'; an account of this search is in the Introduction to `Nonabelian algebraic topology'. The nice surprise was that this extra structure was also what was needed to get equivalences of some algebraic categories (e.g. crossed modules versus double groupoids with connections) so it all fitted together amazingly. For more on these ideas, see Grandis, M. and Mauri, L. Cubical sets and their site. Theory Appl. Categ. {11} (2003) 185--201. Higgins, P.~J. Thin elements and commutative shells in cubical {$\omega$}-categories. Theory Appl. Categ. {14} (2005) 60--74. I have never tried cubical sets without degeneracies but with connections! Ronnie On 13/09/2011 16:12, Marco Grandis wrote: > 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 > [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial versus (cubical with connections) 2011-09-13 16:58 ` Ronnie Brown @ 2011-09-14 7:08 ` Jonathan CHICHE 齊正航 0 siblings, 0 replies; 14+ messages in thread From: Jonathan CHICHE 齊正航 @ 2011-09-14 7:08 UTC (permalink / raw) To: Categories list, Ronnie Brown There is another way to state that the cube category with connections behaves "as well as" the simplex category. Both are strict test categories (as defined by Grothendieck in "Pursuing Stacks"). See http://www.math.jussieu.fr/~maltsin/ps/cubique.pdf. Without connections, the cube category is a test category, but not a strict one, so that the product in the cube category does not reflect the product of homotopy types. This issue vanishes if connections are allowed. Grothendieck explicitly wrote in "Pursuing Stacks" that he believed that, homotopically speaking, any strict test category was "as good as" the simplex category. For instance, he conjectured there that an analog of the Dold-Kan correspondence (which he called Dold- Puppe) holds for every strict test category. (As regards the existence of a Quillen model structure the cofibrations of which are monomorphisms on the presheaf category, and so on, see the introduction to Astérisque 301 by Maltsiniotis and Astérisque 308 by Cisinski.) Best regards, Jonathan Chiche [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* RE: Simplicial groups are Kan 2011-09-12 0:30 Simplicial groups are Kan Michael Barr ` (5 preceding siblings ...) 2011-09-12 9:35 ` Ronnie Brown @ 2011-09-12 13:00 ` Tierney, Myles 6 siblings, 0 replies; 14+ messages in thread From: Tierney, Myles @ 2011-09-12 13:00 UTC (permalink / raw) To: Michael Barr, Categories list 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/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
* Re: Simplicial groups are Kan @ 2011-09-13 1:33 Fred E.J. Linton 0 siblings, 0 replies; 14+ messages in thread From: Fred E.J. Linton @ 2011-09-13 1:33 UTC (permalink / raw) To: Ronnie Brown; +Cc: Categories list Hi, Ronnie, How does a mere mortal get past the gate-keeper lines, BANGOR UNIVERSITY STUDENTS AND STAFF Students and staff of Bangor University login here using your user name and password. User name: Password: if I may be so bold as to ask? (That's for trying to access https://unicat.bangor.ac.uk/validate?url=http%3A%2F%2F0-ams.mpim-bonn.mpg.de.unicat.bangor.ac.uk%3A80%2Fmathscinet%2Fsearch%2Fpubldoc.html?r=1&pg1=CNO&s1=87934&loc=fromrevtext for Moore's original notes.) Cheers, -- Fred ------ Original Message ------ Received: Mon, 12 Sep 2011 08:53:00 PM EDT From: Ronnie Brown <ronnie.profbrown@btinternet.com> To: Michael Barr <barr@math.mcgill.ca>Cc: Categories list <categories@mta.ca> Subject: categories: Re: Simplicial groups are Kan > 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> ... [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
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* Re: Simplicial groups are Kan [not found] <615PimBGm3072S04.1315877592@web04.cms.usa.net> @ 2011-09-13 9:01 ` Ronnie Brown 0 siblings, 0 replies; 14+ messages in thread From: Ronnie Brown @ 2011-09-13 9:01 UTC (permalink / raw) To: Fred E.J. Linton; +Cc: Categories list I'm sorry my previous email contained too much rubbish pasted from Mathscinet. Let me say that the review MR1173825 of the cubical case, by Andrew Tonks, refers to *MR0087934 *Séminaire Henri Cartan de l'Ecole Normale Supérieure, 1954/1955. Algèbres d'Eilenberg-MacLane et homotopie. *"*In den Protokollen 18, 19, 21 betrachtet Moore Monoid-Komplexe." But I don't have a copy of the seminar to check. Ronnie [For admin and other information see: http://www.mta.ca/~cat-dist/ ] ^ permalink raw reply [flat|nested] 14+ messages in thread
end of thread, other threads:[~2011-09-14 7:08 UTC | newest] Thread overview: 14+ messages (download: mbox.gz / follow: Atom feed) -- links below jump to the message on this page -- 2011-09-12 0:30 Simplicial groups are Kan Michael Barr 2011-09-12 2:29 ` Peter May 2011-09-13 14:22 ` William Messing 2011-09-12 5:10 ` Fernando Muro 2011-09-12 6:07 ` rlk 2011-09-12 6:55 ` Urs Schreiber 2011-09-12 8:49 ` Tim Porter 2011-09-12 9:35 ` Ronnie Brown 2011-09-13 15:12 ` Simplicial versus (cubical with connections) Marco Grandis [not found] ` <BDF51495-03DB-4725-8372-094AD1608A11@dima.unige.it> 2011-09-13 16:58 ` Ronnie Brown 2011-09-14 7:08 ` Jonathan CHICHE 齊正航 2011-09-12 13:00 ` Simplicial groups are Kan Tierney, Myles 2011-09-13 1:33 Fred E.J. Linton [not found] <615PimBGm3072S04.1315877592@web04.cms.usa.net> 2011-09-13 9:01 ` Ronnie Brown
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