K.S. Brown has shown that much of abstract homotopy theory can be carried out in the setting of Brown (co)fibration categories [MR0341469]. The decisive property, immediate from the axioms, is that any morphism can be factored into a cofibration, followed by a weak equivalence.
H.-J. Baues [MR0985099], Cisinski [MR2729017] and Radulescu-Banu [arxiv.org/abs/math/0610009] then followed a similar path. Now someone wants to convince me that this is the proper setting for abstract homotopy theory. To begin with, I do like the simplicity of the axioms. Still, I'd like to be convinced of the practical necessity of this approach. Therefore my question:
Are there examples of Brown (co)fibration categories that are not already Quillen model categories?
More precisely, does there exist a pair (C,W) consisting of a category C and a subset W in Mor C (weak equivalences) such that (C,W) can be equipped with the structure of a Brown fibration (or cofibration) category, but not with the structure of a Quillen model category?
I would particularly be interested in examples in which the lifting axioms of Quillen are the obstacle. If they fail to be a Quillen model category just because they lack limits or colimits, I would be less enthusiastic.
On the other hand, I would also welcome examples in which it is relatively easy to show that they are Brown, but relatively hard that they are Quillen.
I would also welcome good general stability properties. For instance, in the Brown case there are no big problems if you want to enlarge the set of weak equivalences, as long as the larger set satisfies (2 of 3) and as long as the resulting larger set of acyclic cofibrations is stable under pushouts (incision).
To summarise, I want to be able to exclaim: "Good that we have the Brown apparatus!"
Best Answer
Consider an abelian category $A$ (or, more generally, an exact category in the sense of Quillen), then the category of complexes of $A$ is a category of cofibrant objects with the quasi-isomorphisms as weak equivalences and the degreewise (admissible) monomorphisms as cofibrations. This is not a Quillen model category in general (if you restrict your attention to bounded complexes, Quillen lifting axioms correspond exactly to the fact that there are enough injectives in $A$, which might fail in general, as can be seen by contemplating the opposite category of the category of sheaves over a sufficiently general topological space). Also, even in the case when we have enough injectives, it is also possible to consider degreewise split monomorphisms as cofibrations (keeping the same weak equivalences), and then, Quillen axioms fail unless quasi-isomorphisms are all chain homotopy equivalences (in the case of an abelian category, this means that $A$ is semi-simple). These examples are instances of a more general situation: consider a category of cofibrant objects $C$ with class of weak equivalences $W$. Then, for any class $S$ of maps of $C$, one can define a new class of maps $W(S)$ as the smallest one which contains $W\cup S$ and which satisfies the following properties: it has the two out of three property, and the class of cofibrations which are in $W(S)$ is closed under pushouts and finite sums. The good news are that $C$ is still a category of cofibrant objects with the same cofibrations but with $W(S)$ as class of weak equivalences.
This process is exactly what you need to define the notion of quasi-isomorphism of complexes of an exact category: starting from the category of bounded complexes with degreewise split monomorphisms and chain homotopy equivalences as weak equivalences (which is then a Quillen model category modulo the existence of finite (co) limits), one gets quasi-isomorphisms as the class $W(S)$ where $S$ consists of maps $X\to 0$, where $X$ runs over the family of complexes associated to admissible short exact sequences $$0\to A\to B\to C\to 0$$ There is a non-abelian version of this construction: consider a (small) category $C$. We may then consider the category $s(C)$ of simplicial objects in the free completion of $C$ by finite sums. Then, considering the termwise split monomorphisms, there is a smallest class of maps $W$ such that $s(C)$ is a category of cofibrant objects with $W$ as weak equivalences and such that any simplicial homotopy equivalence is in $W$. The $(\infty,1)$-category (obtained by considering the Dwyer-Kan localization of $C$ by $W$) corresponds to the free completion of $C$ by finite homotopy colimits; for instance, for $C$ the terminal category, $s(C)$ is simply the homotopy theory of finite simplicial sets. If you do the same construction by replacing $C$ by its completion under small sums and by replacing $W(S)$ by its closure under small sums and realizations, then you obtain the homotopy theory of cofibrant simplicial presheaves over $C$ (for the projective model structure), except that you didn't use any complicated tool to define it (no small object argument, no lifting theorem, in fact, you don't need to know the model category of simplicial sets at all). Of course, this apparent simplification comes at a price: you don't know how to construct homotopy limits in this language. The advantage is that you already are able to speak of homotopy colimits, so that you may use this to understand how to construct homotopy theoretic structures (e.g. Quillen model categories).
Another nice example of a category of cofibrant objects which is not a model category is the category of finite CW-complexes. One may argue that this is a subcategory of the category of cofibrant objects of a Quillen model category. But this is in fact always the case: any category of (co)-fibrant objects can be embedded very nicely in a proper simplicial model category; see Theorems 3.2, 3.10 and 3.25 and Remark 3.13 of my paper Invariance de la K-théorie par équivalences dérivées, J. K-theory 6 (2010).