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.