By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists of exactly the isomorphisms. By the universal property of $\mathcal{C}[\mathcal{W}^{-1}]$, this is equivalent to saying that $\mathcal{W}$ is exactly the preimage of the class of isomorphisms under some functor.

Saturated classes of morphisms are all over the place, but (in my very limited experience) I've never seen a saturated class which was not defined as the preimages of the isomorphisms under some explicit functor, so that the fact that the class is saturated follows immediately from the definition. (E.g. the weak equivalences in $\mathsf{Top}$ are the preimages of the isomorphisms under the functor $\prod_n \pi_n$; the quasi-isomorphisms of chain complexes are the preimages of the isomorphisms under the homology functor; I also include under this umbrella any class of morphisms defined as the saturation of some smaller class in light of the functor $\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$.) I'd like an example which is naturally defined in a way that does not make the saturation obvious.

I ask this because in the definition of a model category, we hypothesize that the weak equivalences satisfy 2-out-of-3 - a consequence of saturation - and then prove that the weak equivalences are saturated. Similarly, in a homotopical category, we hypothesize that the weak equivalences satisfy 2-out-of-6 - another consequence of saturation - and then prove in light of additional hypotheses that the weak equivalences are saturated. What does this extra work buy us? If saturation is obvious in all interesting examples, why not just hypothesize that the weak equivalences are saturated from the get-go? Is the motivation purely to keep the hypotheses elementary, or to avoid reliance on the existence of a possibly non-locally-small category like $\mathcal{C}[\mathcal{W}^{-1}]$?

A couple of related questions would be: what is an example of a class of morphisms which satisfies 2-out-of-3 or 2-out-of-6 but is not saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists of exactly the isomorphisms. By the universal property of $\mathcal{C}[\mathcal{W}^{-1}]$, this is equivalent to saying that $\mathcal{W}$ is exactly the preimage of the class of isomorphisms under some functor.

Saturated classes of morphisms are all over the place, but (in my very limited experience) I've never seen a saturated class which was not defined as the preimages of the isomorphisms under some explicit functor, so that the fact that the class is saturated follows immediately from the definition. (E.g. the weak equivalences in $\mathsf{Top_{*,\mathrm{conn}}}$ are the preimages of the isomorphisms under the functor $\prod_n \pi_n$; the quasi-isomorphisms of chain complexes are the preimages of the isomorphisms under the homology functor; I also include under this umbrella any class of morphisms defined as the saturation of some smaller class in light of the functor $\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$.) I'd like an example which is naturally defined in a way that does not make the saturation obvious.

I ask this because in the definition of a model category, we hypothesize that the weak equivalences satisfy 2-out-of-3 - a consequence of saturation - and then prove that the weak equivalences are saturated. Similarly, in a homotopical category, we hypothesize that the weak equivalences satisfy 2-out-of-6 - another consequence of saturation - and then prove in light of additional hypotheses that the weak equivalences are saturated. What does this extra work buy us? If saturation is obvious in all interesting examples, why not just hypothesize that the weak equivalences are saturated from the get-go? Is the motivation purely to keep the hypotheses elementary, or to avoid reliance on the existence of a possibly non-locally-small category like $\mathcal{C}[\mathcal{W}^{-1}]$?

A couple of related questions would be: what is an example of a class of morphisms which satisfies 2-out-of-3 or 2-out-of-6 but is not saturated?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists of exactly the isomorphisms. By the universal property of $\mathcal{C}[\mathcal{W}^{-1}]$, this is equivalent to saying that $\mathcal{W}$ is exactly the preimage of the class of isomorphisms under some functor.

Saturated classes of morphisms are all over the place, but (in my very limited experience) I've never seen a saturated class which was not defined as the preimages of the isomorphisms under some explicit functor, so that the fact that the class is saturated follows immediately from the definition. (E.g. the weak equivalences in $\mathsf{Top}$ are the preimages of the isomorphisms under the functor $\prod_n \pi_n$; the quasi-isomorphisms of chain complexes are the preimages of the isomorphisms under the homology functor; I also include under this umbrella any class of morphisms defined as the saturation of some smaller class in light of the functor $\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$.) I'd like an example which is naturally defined in a way that does not make the saturation obvious.

I ask this because in the definition of a model category, we hypothesize that the weak equivalences satisfy 2-out-of-3 - a consequence of saturation - and then prove that the weak equivalences are saturated. Similarly, in a homotopical category, we hypothesize that the weak equivalences satisfy 2-out-of-6 - another consequence of saturation - and then prove in light of additional hypotheses that the weak equivalences are saturated. What does this extra work buy us? If saturation is obvious in all interesting examples, why not just hypothesize that the weak equivalences are saturated from the get-go? Is the motivation purely to keep the hypotheses elementary, or to avoid reliance on the existence of a possibly non-locally-small category like $\mathcal{C}[\mathcal{W}^{-1}]$?

By "saturated class of morphisms" in a category $\mathcal{C}$, I mean a subcategory $\mathcal{W} \subset \mathcal{C}$ such that the image of $\mathcal{W}$ in $\mathcal{C}[\mathcal{W}^{-1}]$ consists of exactly the isomorphisms. By the universal property of $\mathcal{C}[\mathcal{W}^{-1}]$, this is equivalent to saying that $\mathcal{W}$ is exactly the preimage of the class of isomorphisms under some functor.

Saturated classes of morphisms are all over the place, but (in my very limited experience) I've never seen a saturated class which was not defined as the preimages of the isomorphisms under some explicit functor, so that the fact that the class is saturated follows immediately from the definition. (E.g. the weak equivalences in $\mathsf{Top}$ are the preimages of the isomorphisms under the functor $\prod_n \pi_n$; the quasi-isomorphisms of chain complexes are the preimages of the isomorphisms under the homology functor; I also include under this umbrella any class of morphisms defined as the saturation of some smaller class in light of the functor $\mathcal{C} \to \mathcal{C}[\mathcal{W}^{-1}]$.) I'd like an example which is naturally defined in a way that does not make the saturation obvious.

I ask this because in the definition of a model category, we hypothesize that the weak equivalences satisfy 2-out-of-3 - a consequence of saturation - and then prove that the weak equivalences are saturated. Similarly, in a homotopical category, we hypothesize that the weak equivalences satisfy 2-out-of-6 - another consequence of saturation - and then prove in light of additional hypotheses that the weak equivalences are saturated. What does this extra work buy us? If saturation is obvious in all interesting examples, why not just hypothesize that the weak equivalences are saturated from the get-go? Is the motivation purely to keep the hypotheses elementary, or to avoid reliance on the existence of a possibly non-locally-small category like $\mathcal{C}[\mathcal{W}^{-1}]$?

A couple of related questions would be: what is an example of a class of morphisms which satisfies 2-out-of-3 or 2-out-of-6 but is not saturated?