Subject: Crossed complexes and model categories
From: "Ronnie Brown"
Date: Tue, 23 May 2006 17:30:43 +0100
Tim and I recently asked the list on whether chain complexes over variable
rings formed a proper model category in that pushouts of weak equivalences
by cofibrations were weak equivalences, and got positive answers to this.
The origin of this question was the related question for the category of
crossed complexes.
Marek Golasinski and I proved in ``A model structure for the homotopy
theory of crossed complexes'', {\em
Cah. Top. G\'eom. Diff. Cat}. 30 (1989) 61-82.
that a pushout of homotopy equivalences by a cofibration is a homotopy
equivalence, but left open the corresponding question for weak
equivalences.
In his thesis Orin Sauvageot, STABILISATION DES COMPLEXES CROISES, on the
Hopf Archive, claimed to have proved that crossed complexes form a proper,
cellular category. I have had difficulty with the proof of the proper
condition.
The problem is that a pushout of crossed complexes gives rise to
constructions
more complicated than exact sequences, and involves induced crossed
modules
and induced modules. (See for example a paper RB & C.D.Wensley,
`Computation and
homotopical applications of induced crossed modules', J. Symbolic
Computation 35
(2003) 59-72. math.AT/0209068)
If the result were true, the resulting gluing theorem for weak
equivalences
should be useful for the computation of free crossed resolutions for
constructions
on groups and groupoids. For a use of free crossed resolutions of
group(oid)s, see
for example math.AT/0207037:
they well model CW-complexes which are K(G,1)'s (Whitehead) and also
their cellular
maps (Baues).
All this is part of the use of crossed complexes as a `linear' model of
homotopy types,
including the fundamental group(oid) and its actions, and
all 2-types, using nonabelian crossed modules, but not, say, Whitehead
products in dimensions >1.
It would be good to know the answer, for a book in preparation on crossed
complexes.
By the way, note that we do have the notions of the fundamental crossed
complex of a
simplicial set, and of a simplicial crossed complex (Andy Tonk's thesis),
but we do
not now (for ever?) have a crossed complex notion corresponding to a
simplicial abelian group,
since the structure of a crossed complex is not uniform over dimensions,
unlike chain complexes.
Ronnie Brown
For a survey of crossed complexes and related structures, see
Fields Institute Communications 43 (2004) 101-130. UWB Math Preprint
02.26.