How (non-)unique Is the Choice of Cofibrations?

The question that serves as the title is to be understood in the context of Quillen model categories: having fixed the category of models and the subcategory of weak equivalences, how much ambiguity is there in finding suitable fibration and cofibration classes? Already for simplicial sets, infinitely many different subclasses of the monomorphisms can serve as cofibrations (with the usual weak equivalences) so as to satisfy the axioms. The situation is yet more interesting for sheaf categories, with open questions. However, in a wide class of model categories, if one keeps the weak equivalences fixed, then any two small-generated cofibration classes will give Quillen equivalent model structures. Morally, the Quillen equivalence class of a model category is determined by the underlying category and the subcategory of weak equivalences. Under Quillen’s conception, a category of models for homotopy theory is to come equipped with three distinguished classes of morphisms — cofibrations, weak equivalences and fibrations — of which the weak equivalences alone determine the associated homotopy category. The first example of distinct model categories with the same homotopy category is due to Quillen, and appears in his ‘Homotopical Algebra’. (It is chain complexes, modelling the derived category in two different ways.) Another example was found separately by Bousfield– Kan [9] and Heller [12], on simplicial diagrams. In both of these cases, the underlying category of models for the alternative structures is the same, as are the weak equivalences, but two (co)fibration classes are possible, one properly contained in the other. Here’s a specimen with three cofibration classes, still ordered linearly. Example 0.1. On the category SSet (the cosimplicial spaces of Bousfield–Kan [9]), let the weak equivalences be maps that are ∆-objectwise weak equivalences in SSet . Let cof1 be the class of maps with the left lifting property with respect to all maps that are ∆-objectwise acyclic fibrations in SSet . Let cof2 be the class of monos that induce isomorphisms on the maximal augmentation of the underlying cosimplicial space. Let cof3 be the class of all monomorphisms. Any of these can serve as the class of cofibrations on SSet with the above weak equivalences and one has cof1 $ cof2 $ cof3. Example 1.3 is a homotopy theory “from nature” that demonstrates that there is no a priori bound on the cardinality of possible cofibration classes (even having fixed both the category of models and the weak equivalences, hence, the homotopy category); nor do these classes have to be linearly ordered by inclusion. The following cheap argument shows the same. Let M be a Quillen model category as in Example 0.1, with several possible classes of cofibrations cofi, i ∈ I. Let X be a set (considered as a discrete diagram), and let a map Date: January 27, 2003.