Simplicial groups are Kan

Simplicial groups are Kan

[Michael Barr]

I know that is a theorem, due I think to John Moore.  Can anyone give me a
pointer to the original article.

Michael


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Re: Simplicial groups are Kan

[Peter May]

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/ ]



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Re: Simplicial groups are Kan

[Fernando Muro]

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


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Simplicial groups are Kan

[rlk]

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


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Re: Simplicial groups are Kan

[Urs Schreiber]

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


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Re: Simplicial groups are Kan

[Tim Porter]

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






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Re: Simplicial groups are Kan

[Ronnie Brown]

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
>


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RE: Simplicial groups are Kan

[Tierney, Myles]

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


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Re: Simplicial groups are Kan

[William Messing]

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


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Simplicial versus (cubical with connections)

[Marco Grandis]

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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Re: Simplicial versus (cubical with connections)

[Ronnie Brown]

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
>


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Re: Simplicial versus (cubical with connections)

[Jonathan CHICHE 齊正航]

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

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Re: Simplicial groups are Kan

[Ronnie Brown]

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


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Re: Simplicial groups are Kan

[Fred E.J. Linton]

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>

...

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