A cofibration category is saturated if it satisfies the following equivalent conditions:
- Every map which becomes an isomorphism in the homotopy category is already a weak equivalence.
- The weak equivalences are closed under retracts.
- The weak equivalences satisfy the two-out-of-six property.
The second condition is one of the axioms of a model category, and most categories with weak equivalences that arise in practice satisfy the last two properties more or less by construction. In fact, I realized I don't know any examples of cofibration categories which are not saturated. So the question is:
What is an example of a non-saturated cofibration category? And, what are its maps which become isomorphisms in the homotopy category that are not already weak equivalences?
I think that "finite spaces" (in a suitable sense) with simple homotopy equivalences as the weak equivalences is supposed to form such an example; but I don't know where to find the details.
Edit: By a "cofibration category" I mean any of
a "catégorie dérivable à droite" in the sense of Cisinski
a "precofibration category" in the sense of Radulescu-Banu
a "cofibration category" in the sense of Baues, without the axiom on fibrant models (though I don't see why this axiom should force the category to be saturated)
I'm not picky about exactly which axioms assume which objects are cofibrant, but I do want the weak equivalences to satisfy two-out-of-three.