if and are in , then so are , , , and .
The class of isomorphisms in any category satisfies 2-out-of-6. This case is the archetype of most of the cases in which the property is invoked: 2-out-of-6 is characteristic of morphisms that have a notion of inverse.
The 2-out-of-6 property implies the two-out-of-three property. For on the one hand, if and are in , then applying 2-out-of-6 to , we find that . On the other hand, if and are in , then applying 2-out-of-6 to , we find that , and similarly if and are in .
If satisfies 2-out-of-6 and contains the identities (i.e. is a homotopical category), then contains all isomorphisms. For if has inverse , then applying 2-out-of-6 to we find that and are in .
The 2-out-of-6 property is closely related to the property that is closed under retracts, as a subcategory of the arrow category. For instance, we have the following theorem due to Blumberg-Mandell (stated there in the context of Waldhausen categories):
All pushouts of cofibrations exist.
The pushout of a cofibration that is also a weak equivalence is again a cofibration and a weak equivalence.
Every weak equivalence factors as a weak equivalence that is a cofibration followed by a weak equivalence that is a retraction.
Then if the weak equivalences in are closed under retracts, they also satisfy 2-out-of-6.
Suppose the first three assumptions on the cofibrations, and let
be a sequence of composable maps, with and weak equivalences. Factor as where is a cofibration weak equivalence and is a weak equivalence with a section] . Let be the pushout
Since , we have a unique map such that and . Define ; then .
Since is a cofibration weak equivalence, so is . And since is a weak equivalence, by two-out-of-three, is also a weak equivalence. But now we have a commutative diagram
exhibiting as a retract of in the arrow category. Thus, by assumption is a weak equivalence. By successive applications of two-out-of-three, so are , , and .
Of course, there is a dual theorem for fibrations. Note that the fibrations in a category of fibrant objects satisfy (the duals of) all the above conditions. They are not implied by the axioms for the cofibrations in a Waldhausen category (the factorization axiom is what is missing), but many Waldhausen categories do satisfy them.
The 2-out-of-6 property is also closely related to the property that is saturated, in the sense that any morphism which becomes an isomorphism in the localization is already a weak equivalence. (This is unrelated to the notion of saturated class of maps used in the theory of weak factorization systems.)
Clearly saturation implies 2-out-of-6, but we also have the following two converses.
Suppose admits a calculus of fractions. Then satisfies two-out-of-six if and only if it is saturated.
This is from 7.1.20 of Categories and Sheaves. Suppose becomes an isomorphism in , and represent its inverse by with . Then since and represent the same morphism in , there is a morphism in such that . Since , it follows by 2-out-of-3 that .
Now applying this same argument to , we obtain an such that . But then by 2-out-of-6, we have as desired.
Suppose has a class of “cofibrations” satisfying the properties in Theorem 1, and moreover the pushout of any weak equivalence along a cofibration is a weak equivalence. Then satisfies two-out-of-six if and only if it is saturated (and hence, if and only if it is closed under retracts).
See Blumberg-Mandell for details; an outline follows.
We first observe that admits a homotopy calculus of left fractions?, and in particular that every morphism in can be represented by a zigzag in which is a cofibration and a weak equivalence. See Blumberg-Mandell, section 5 for a detailed proof. The idea is that given any zigzag , we factor as a cofibration weak equivalence followed by a retraction weak equivalence, then push out the cofibration along and use the section to obtain a map from into the pushout.
Now suppose becomes an isomorphism in , and represent its inverse by with a cofibration weak equivalence. Since the composite represents , we have . Consider the following diagram where the squares are pushouts:
All the vertical maps are cofibration weak equivalences, by assumption. Moreover, the bottom map is a weak equivalence, since it is the pushout of the weak equivalence along the cofibration . And since the zigzag
represents the same morphism as
which represents , we have that is a weak equivalence. Thus, by 2-out-of-6, is a weak equivalence, hence so is by 2-out-of-3.
Of course, there is a dual theorem for fibrations.