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.