Subject: Crossed complexes
From: "Ronnie Brown"
Date: Tue, 30 May 2006 21:54:54 +0100
Here I try to answer the questions of Carlos.
1) There is a functor Pi: FTop \to Crs from filtered spaces to crossed
complexes, and the singular functor S: Top \to FTop (skeletal filtration)
is
standard. Then Pi: Top \to Crs can be regarded as a convenient
generalisation, or strengthening, of the singular chains of the universal
cover with a group(oid) of operators.
2) The category Crs has a unit interval object, and is monoidal closed.
Its
homotopy theory is discussed in Brown-Golasinski (as cited) and in
Kamps-Porter `Abstract homotopy and simple homotopy theory', World
Scientific (1996). Its model theory seems satisfactory as far as it goes,
but I do not know if a pushout of a weak equivalence along a cofibration
is
a weak equivalence (pace Sauvageot), and have not studied at all his proof
of cellularity.
3) Pi: Ftop \to Crs satisfies a Generalised van Kampen Theorem (GvKT) so
one
can compute to some extent, even non abelian information in dimension 2.
4) So crossed complexes give a convenient `linear' model of homotopy
types.
For a survey, see math.AT/0212274. You can get more information by using
`crossed differential algebras', i.e. crossed complexes A with m: A
\otimes
A \to A (Baues-Tonks).
5) It would be nice to believe that double crossed complexes model also
the
quadratic information, triple crossed complexes the cubic, ...... This
would
be related to the Loday model, and the Brown-Loday van Kampen theorem
could
allow some computations! In fact, if pressed, I do expect it to be true!
But what is the model structure for double crossed complexes? The mind
boggles
(or at least mine does).
6) An advantage of such *strict* structures is that it is easier to see,
or
imagine, what are colimits, and so to compute from the GvKTs, when they
exist. In this way we found the nonabelian tensor product of groups (whose
bibliography now has 90 items). So we can compute some 3-types using
crossed
squares, but can we do so using other models of 3-types?
7) I do not know how all this is related to Goodwillie's work, except that
I
recall he uses at one point in his work an Ellis -Steiner consequence of
the
Brown-Loday van Kampen Theorem, for some critical group of an n-ad. Recall
Barratt-Whitehead could prove only connectivity, because they had not an
appropriate algebraic gadget to compute more. The critical group is but a
pale shadow of the underlying higher dimensional algebraic structure by
which it is governed.
8) How to relate all this to nonabelian homological algebra? One step is
to
use cohomology with coefficients in a crossed complex, as in
82. (with O. MUCUK), ``Covering groups of non-connected topological
groups
revisited'', {\em Math. Proc. Camb. Phil. Soc}, 115 (1994) 97-110.
Ronnie
> Subject: Re: crossed complexes
> From: carlos@math.unice.fr
> Date: Thu, 25 May 2006 22:00:53 +0200 (CEST)
>
> Hello,
> In Ronnie's message he is asking about the properness condition.
> For Ronnie, does that mean that the model structure is ok?
> (Did you do that in the paper you cited?)
> Does this mean that we have a functor from all homotopy types
> to crossed complexes? Namely, take a simplicial set
> and take the homotopy colimit of the constant functor
> with values *, along this simplicial set, in the model
> category of crossed complexes.
> That would be a sort of linearization of the homotopy in degrees >= 2.
> It might also respond to what is going on with the construction of
> Kapranov and Voevodsky: is it possible that their ``Poincare
> infinity groupoid'' construction actually more or less constructs
> this functor? Also, is this in any way related to the Goodwillie
> tower of the identity functor?
> ---Carlos Simpson
>