Homotopical categories, the 2-out-of-6 property, and saturation

Question

A homotopical category is a cateogry with a class $\mathcal W$ of arrows called weak equivalences which satisfies the 2-out-of-6 property.

The nlab article shows a deep connection between $\mathcal W$ having the 2-out-of-6 property and $\mathcal W$ being saturated - under mild conditions, the two notions coincide. For instance:

Theorem: Suppose $\mathcal W$ admits a calculus of fractions. Then $\mathcal W$ has the 2-out-of-6 proprety iff it is saturated.

Thus, for nice enough classes of arrows, one could equivalently define homotopical categories as categories with saturated classes of arrows. In light of the fact homotopical categories are supposed to capture the essence of weak equivalences, it seems like saturation is actually an extremely deep and important underlying property in homotopy theory.

1. Why are homotopical categories defined via the 2-out-of-6 property instead of via saturated classes of arrows?
2. Why is saturation such an important property? Intuitively, I see it as saying that localizing at the weak equivalences does not perturb any of the arrows outside $\mathcal W$.
3. Where can I explicitly see saturation at play in homotopy theory?

Any additional insights would be gladly accepted.

Answer 1

Here's an answer, with the caveat that I'm only a student just starting to study these things myself. I hope people will call me out if I say stuff that's inaccurate or misleading.

Ad (1)

There are cases where it is easy to see that the weak equivalences are saturated. This is the case if the weak equivalences are explicitly defined to be the morphisms inverted by some particular functor -- for example:

• Quasi-isomorphisms of chain complexes are defined to be the morphisms inverted by the homology functor.
• Weak equivalences of sufficiently nice pointed, path-connected spaces can be defined to be the morphisms inverted by the product of the homotopy group functors.
• Homotopy equivalences are defined to be the morphisms inverted when you mod out by the congruence identifying homotopic maps. Similarly for pointed homotopy equivalences, chain homotopy equivalences, and equivalences of categories.

But if we perturb these definitions a little, we get examples where there is no functor that obviously inverts exactly the weak equivalences. For example, weak equivalences of spaces are "almost" defined to be the morphisms inverted by a product of homotopy group functors, except that a weak equivalence has to be an isomorphism on homotopy groups at all basepoints, and it doesn't seem to be possible to express this in terms of a functor.

This is partly why so much machinery like model categories and homotopical categories gets built up: to yield easy-to-check criteria guaranteeing the saturatedness of a class of morphisms.

Ad (2)

I wouldn't overthink this one. Homotopy theory is all about studying what happens when you see certain morphisms ("weak equivalences") as isomorphisms. For these purposes, it makes no difference to replace the weak equivalences with their saturation if they are not saturated already, because morphisms in the saturation automatically look like isomorphisms from this viewpoint too.

Ad (3)

I think my general sense is that in practice, if it's natural to localize at a given class of morphisms, that class of morphisms is probably already saturated. It just may be difficult to show. But perhaps there are examples of abstract constructions in the theory of relative categories / categories with weak equivalences / homotopical categories / model categories / ... where one might input one or more categories with saturated weak equivalences and output a category with non-saturated weak equivalences.