Subject: categories: [Fwd: Schreier theory discussion]

From: jim stasheff <[email-masked]>
Date: Tue, 22 Nov 2005 13:43:07 -0500

-------- Original Message --------
Subject: 	Schreier theory discussion
Date: 	Tue, 22 Nov 2005 18:01:42 +0100 (CET)
From: 	Johannes Huebschmann <[email-masked]>
To: 	[email-masked], [email-masked], [email-masked],
[email-masked]
CC: 	Johannes Huebschmann <[email-masked]>

Dear Friends and Colleagues

A few addenda to the Schreier theory discussion etc. which you might find
interesting. (I am reacting to a message I received via J. Stasheff.)

1) As pointed out by R. Brwon, an approach to non-abelian cohomology
may be phrased in terms of crossed modules.
There is a notion more general than crossed modules, that of
"crossed pair" which I introduced in the paper

Group extensions, crossed pairs, and an eight term exact sequence,
J. reine angew. Math. 321 (1981), 150--172.

Crossed pairs may be used, for example, to explore group extensions, in
particular, to examine differentials in the spectral sequence of a group extension.
I worked this out in the paper

Automorphisms of group extensions and differentials in the
Lyndon--Hochschild--Serre spectral sequence,
J. of  Algebra 72 (1981), 296--334.

I discovered later that the idea of crossed pair was in the literature
before, under the name "pseudo module" in the 50's.

Crossed pairs arise under other circumstances as well,
for example in the Galois theory of Azumaya algebras.
I have known this all my scientific life but never found the time to write
it up properly.

2) I have worked out a rigorous approach to lattice gauge theory
in the paper

Extended moduli spaces, the Kan construction, and lattice gauge theory
Topology 38 (1999), 555--596.

In this paper, I discretize a space of based gauge equivalence classes of
connections in terms of a combinatorial structure, and the resulting
object is a COSIMPLICIAL space. The geometric realization of this space
is G-equivariantly weakly homotopy equivalent to the space of based gauge
equivalence classes of connections, where G refers to the structure group
of the corresponding principal bundle.

On such a space, for example, path integrals are well defined.
As an illustration I worked out a calculation of the Chern-Simons
invariant for lens spaces. The calculation involves identities among
relations, a notion which arises in the structure theory of crossed
modules.

Best regards

Johannes

HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
http://math.univ-lille1.fr/~huebschm

TEL. (33) 3 20 43 41 97
     (33) 3 20 43 42 33 (secretariat)
     (33) 3 20 43 48 50 (secretariat)
Fax  (33) 3 20 43 43 02

e-mail [email-masked]

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-------- Original Message --------
<table border="0" cellpadding="0" cellspacing="0">
  <tbody>
    <tr>
      <th align="right" nowrap="nowrap" valign="baseline">Subject: </th>
      <td>Schreier theory discussion</td>
    </tr>
    <tr>
      <th align="right" nowrap="nowrap" valign="baseline">Date: </th>
      <td>Tue, 22 Nov 2005 18:01:42 +0100 (CET)</td>
    </tr>
    <tr>
      <th align="right" nowrap="nowrap" valign="baseline">From: </th>
      <td>Johannes Huebschmann
<a class="moz-txt-link-rfc2396E" href="mailto:[email-masked]">&lt;[email-masked]&gt;</a></td>
    </tr>
    <tr>
      <th align="right" nowrap="nowrap" valign="baseline">To: </th>
      <td><a class="moz-txt-link-abbreviated" href="mailto:[email-masked]">[email-masked]</a>, <a class="moz-txt-link-abbreviated" href="mailto:[email-masked]">[email-masked]</a>,
<a class="moz-txt-link-abbreviated" href="mailto:[email-masked]">[email-masked]</a>, <a class="moz-txt-link-abbreviated" href="mailto:[email-masked]">[email-masked]</a></td>
    </tr>
    <tr>
      <th align="right" nowrap="nowrap" valign="baseline">CC: </th>
      <td>Johannes Huebschmann
<a class="moz-txt-link-rfc2396E" href="mailto:[email-masked]">&lt;[email-masked]&gt;</a></td>
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<pre>Dear Friends and Colleagues

A few addenda to the Schreier theory discussion etc. which you might find
interesting. (I am reacting to a message I received via J. Stasheff.)

1) As pointed out by R. Brwon, an approach to non-abelian cohomology
may be phrased in terms of crossed modules.
There is a notion more general than crossed modules, that of
"crossed pair" which I introduced in the paper

Group extensions, crossed pairs, and an eight term exact sequence,
J. reine angew. Math. 321 (1981), 150--172.

Crossed pairs may be used, for example, to explore group extensions, in
particular, to examine differentials in the spectral sequence of a group extension.
I worked this out in the paper

Automorphisms of group extensions and differentials in the
Lyndon--Hochschild--Serre spectral sequence,
J. of  Algebra 72 (1981), 296--334.

I discovered later that the idea of crossed pair was in the literature
before, under the name "pseudo module" in the 50's.

Crossed pairs arise under other circumstances as well,
for example in the Galois theory of Azumaya algebras.
I have known this all my scientific life but never found the time to write
it up properly.

2) I have worked out a rigorous approach to lattice gauge theory
in the paper

Extended moduli spaces, the Kan construction, and lattice gauge theory
Topology 38 (1999), 555--596.

In this paper, I discretize a space of based gauge equivalence classes of
connections in terms of a combinatorial structure, and the resulting
object is a COSIMPLICIAL space. The geometric realization of this space
is G-equivariantly weakly homotopy equivalent to the space of based gauge
equivalence classes of connections, where G refers to the structure group
of the corresponding principal bundle.

On such a space, for example, path integrals are well defined.
As an illustration I worked out a calculation of the Chern-Simons
invariant for lens spaces. The calculation involves identities among
relations, a notion which arises in the structure theory of crossed
modules.

Best regards

Johannes

HUEBSCHMANN Johannes
Professeur de Mathematiques
USTL, UFR de Mathematiques
UMR 8524 Laboratoire Paul Painleve
F-59 655 Villeneuve d'Ascq Cedex  France
<a class="moz-txt-link-freetext" href="http://math.univ-lille1.fr/~huebschm">http://math.univ-lille1.fr/~huebschm</a>

TEL. (33) 3 20 43 41 97
     (33) 3 20 43 42 33 (secretariat)
     (33) 3 20 43 48 50 (secretariat)
Fax  (33) 3 20 43 43 02

e-mail <a class="moz-txt-link-abbreviated" href="mailto:[email-masked]">[email-masked]</a>

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