Question re lax crossed modules

Question re lax crossed modules

[David Roberts]

I have been looking at categorical groups a little and was wondering
what a lax crossed module is. A search through various databases has
turned up nothing. It would seem that they should be like crossed
modules but only satisfy a weakened equivariance property.

Any pointers toward a definition would be great.


------------------------------------------------------------------------
--
David Roberts
School of Mathematical Sciences
University of Adelaide SA 5005
------------------------------------------------------------------------
--
droberts@maths.adelaide.edu.au
www.maths.adelaide.edu.au/~droberts

Re: Question re lax crossed modules

[Tim Porter]

Dear All,
There is an interesting old result on fibrations that may shed light on
this.  It is well known  that if $F\to E\to B$ is a fibration of pointed
spaces then $\pi_1 F \to \pi_1 E$ is a crossed module and in fact any
crossed module can be  given in this form. Now one tries to prove it.
First the action: take a loop   g in E and another a in F. concatentate
to get  gag^{-1}.  This is a loop in E whose image in the base,B, is
null homotopic.  Pick a null homotopy that does this.  Lift it to a
homotopy in E starting at gag^{-1} using the homotopy lifting property
of the fibration.  Evaluate the other end of the lift.  This is a loop
in F. The corresponding element of \pi_1 F is the result of acting on
the class of a by the class of g.
Note the way the action is determined up to homotopy.  The verification
that the rules work up to homotopy is left as an exercise.

I learnt this from a paper by Eric Friedlander, who attributed it to
Deligne.  I suspect it is already essentially in Whitehead's
Combinatorial Homotopy II paper or Peter Hilton's lovely little book on
Homotopy Theory.

It suggests a `homotopy everything' version of crossed module, not just
a lax one.  Its advantage is that it clearly links up the structure with
the quite classical topological version of fibrations and so should be
adaptable to other situations.

Hope this helps.

Tim



David Roberts wrote:

> I have been looking at categorical groups a little and was wondering
> what a lax crossed module is. A search through various databases has
> turned up nothing. It would seem that they should be like crossed
> modules but only satisfy a weakened equivariance property.
>
> Any pointers toward a definition would be great.
>
>
> ------------------------------------------------------------------------
> --
> David Roberts
> School of Mathematical Sciences
> University of Adelaide SA 5005
> ------------------------------------------------------------------------
> --
> droberts@maths.adelaide.edu.au
> www.maths.adelaide.edu.au/~droberts
>
>
>

Re: Question re lax crossed modules

[jim stasheff]

Tim Porter wrote:

> Dear All,
> There is an interesting old result on fibrations that may shed light on
> this.  It is well known  that if $F\to E\to B$ is a fibration of pointed
> spaces then $\pi_1 F \to \pi_1 E$ is a crossed module and in fact any
> crossed module can be  given in this form. Now one tries to prove it.
> First the action: take a loop   g in E and another a in F. concatentate
> to get  gag^{-1}.  This is a loop in E whose image in the base,B, is
> null homotopic.  Pick a null homotopy that does this.  Lift it to a
> homotopy in E starting at gag^{-1} using the homotopy lifting property
> of the fibration.  Evaluate the other end of the lift.  This is a loop
> in F. The corresponding element of \pi_1 F is the result of acting on
> the class of a by the class of g.
> Note the way the action is determined up to homotopy.  The verification
> that the rules work up to homotopy is left as an exercise.
>
> I learnt this from a paper by Eric Friedlander, who attributed it to
> Deligne.  I suspect it is already essentially in Whitehead's
> Combinatorial Homotopy II paper or Peter Hilton's lovely little book on
> Homotopy Theory.
>
> It suggests a `homotopy everything' version of crossed module, not just
> a lax one.


There is a closely related homotopy everything version which I bet can
be adapted to this problem.

Consider \Omega B, the based loop space and F as the fibre over
that base point. Then an argument like that above gives
and action of \Omega B on F which satisfies the usual rules for an
actin only up to homotopy
Easiest way to say it is the adjoint map \MOmega B --> Aut F
meaning the self homotopy equivalences of F
is a strongly homotopy associative map of monids
(if you use Moore loops) or of A_\infty spaces.

It should be in
 "Parallel transport in fibre spaces," Bol. Soc. Mat. Mexicana (1968),
68-86.

or

"Associated fibre spaces," Michigan Math. Journal 15 (1968), 457-470.

Hope that helps

jim

>   Its advantage is that it clearly links up the structure with
> the quite classical topological version of fibrations and so should be
> adaptable to other situations.
>
> Hope this helps.
>
> Tim
>