nLab
two-out-of-six property

Definition

In a category $C$, a class $W\subseteq \mathrm{Mor}(C)$ of morphisms is said to satisfy 2-out-of-6 if for any sequence of three composable morphisms

$$X\stackrel{u}{\to }Y\stackrel{v}{\to }Z\stackrel{w}{\to }K$$X\xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} K

if $wv$ and $vu$ are in $W$, then so are $u$, $v$, $w$, and $wvu$.

Examples

• 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.

• A category equipped with a class of “weak equivalences” containing the identity morphisms and satisfying 2-out-of-6 is called a homotopical category. In particular, this includes any model category.

Relation to other concepts

2-out-of-3

The 2-out-of-6 property implies the two-out-of-three property. For on the one hand, if $f$ and $g$ are in $W$, then applying 2-out-of-6 to $\stackrel{f}{\to }\stackrel{1}{\to }\stackrel{g}{\to }$, we find that $gf\in W$. On the other hand, if $f$ and $gf$ are in $W$, then applying 2-out-of-6 to $\stackrel{1}{\to }\stackrel{f}{\to }\stackrel{g}{\to }$, we find that $g\in W$, and similarly if $g$ and $gf$ are in $W$.

Identities and isomorphisms

If $W$ satisfies 2-out-of-6 and contains the identities (i.e. $C$ is a homotopical category), then $W$ contains all isomorphisms. For if $f$ has inverse $g$, then applying 2-out-of-6 to $\stackrel{g}{\to }\stackrel{f}{\to }\stackrel{g}{\to }$ we find that $g$ and $f$ are in $W$.

Closure under retracts

The 2-out-of-6 property is closely related to the property that $W$ 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):

Theorem

Suppose a category with weak equivalences $𝒞$ has an additional class of maps called cofibrations which satisfy the following properties:

• 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 $𝒞$ are closed under retracts, they also satisfy 2-out-of-6.

Proof

Suppose the first three assumptions on the cofibrations, and let

$$A\stackrel{u}{\to }B\stackrel{v}{\to }C\stackrel{w}{\to }D$$A \xrightarrow{u} B \xrightarrow{v} C \xrightarrow{w} D

be a sequence of composable maps, with $wv$ and $vu$ weak equivalences. Factor $vu:A\to C$ as $A\stackrel{i}{\to }C\prime \stackrel{p}{\to }C$ where $i$ is a cofibration weak equivalence and $p$ is a weak equivalence with a section] $s:C\to C\prime$. Let $B\prime$ be the pushout

$$\begin{array}{ccc}A& \stackrel{i}{\to }& C\prime \\ {}^u\downarrow & & {\downarrow }^h\\ B& \underset{k}{\to }& B\prime \end{array}$$\array{ A & \overset{i}{\to} & C'\\ ^u\downarrow && \downarrow^h\\ B& \underset{k}{\to} & B'}

Since $pi=vu$, we have a unique map $g:B\prime \to C$ such that $gh=p$ and $gk=v$. Define $f=hs$; then $gf=ghs=ps=1_C$.

Since $i$ is a cofibration weak equivalence, so is $k$. And since $wgk=wv:B\to D$ is a weak equivalence, by two-out-of-three, $wg:B\prime \to D$ is also a weak equivalence. But now we have a commutative diagram

$$\begin{array}{ccccc}C& \stackrel{f}{\to }& B\prime & \stackrel{g}{\to }& C\\ {}^w\downarrow & & {\downarrow }^{wg}& & {\downarrow }^w\\ D& \underset{=}{\to }& D& \underset{=}{\to }& D\end{array}$$\array{C & \overset{f}{\to} & B' & \xrightarrow{g} & C \\ ^w\downarrow && \downarrow^{w g} && \downarrow^w\\ D& \underset{=}{\to} & D & \underset{=}{\to} & D}

exhibiting $w$ as a retract of $wg$ in the arrow category. Thus, by assumption $w$ is a weak equivalence. By successive applications of two-out-of-three, so are $v$, $u$, and $wvu$.

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.

Saturation

The 2-out-of-6 property is also closely related to the property that $W$ is saturated, in the sense that any morphism which becomes an isomorphism in the localization $C[W^{-1}]$ 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.

Theorem

Suppose $W$ admits a calculus of fractions. Then $W$ satisfies two-out-of-six if and only if it is saturated.

Proof

This is from 7.1.20 of Categories and Sheaves. Suppose $f:X\to Y$ becomes an isomorphism in $𝒞[W^{-1}]$, and represent its inverse by $Y\stackrel{g}{\to }X\prime \stackrel{s}{\leftarrow }X$ with $s\in W$. Then since $gf$ and $s$ represent the same morphism in $𝒞[W^{-1}]$, there is a morphism $t:X\prime \to X″$ in $W$ such that $tgf=ts$. Since $ts\in W$, it follows by 2-out-of-3 that $gf\in W$.

Now applying this same argument to $g$, we obtain an $h$ such that $hg\in W$. But then by 2-out-of-6, we have $f\in W$ as desired.

Theorem

Suppose $C$ 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 $W$ satisfies two-out-of-six if and only if it is saturated (and hence, if and only if it is closed under retracts).

Proof

See Blumberg-Mandell for details; an outline follows.

We first observe that $W$ admits a homotopy calculus of left fractions?, and in particular that every morphism in $𝒞[W^{-1}]$ can be represented by a zigzag $A\to C\stackrel{\sim }{\leftarrow }B$ in which $B\stackrel{\sim }{\to }C$ is a cofibration and a weak equivalence. See Blumberg-Mandell, section 5 for a detailed proof. The idea is that given any zigzag $A\stackrel{\sim }{\leftarrow }D\to B$, we factor $D\to A$ as a cofibration weak equivalence followed by a retraction weak equivalence, then push out the cofibration along $D\to B$ and use the section to obtain a map from $A$ into the pushout.

Now suppose $a:A\to B$ becomes an isomorphism in $C[W^{-1}]$, and represent its inverse by $B\stackrel{b}{\to }C\stackrel{c}{\leftarrow }A$ with $c$ a cofibration weak equivalence. Since the composite $A\stackrel{ba}{\to }C\stackrel{c}{\leftarrow }A$ represents $1_A$, we have $ba\in W$. Consider the following diagram where the squares are pushouts:

$$\begin{array}{ccccccc}& & A& \stackrel{a}{\to }& B& \stackrel{b}{\to }& C\\ & & {}^c\downarrow & & \downarrow & & \downarrow \\ B& \underset{b}{\to }& C& \underset{}{\to }& B\prime & \underset{}{\to }& C\prime \end{array}$$\array{ && A & \overset{a}{\to} & B & \xrightarrow{b} & C \\ && ^c\downarrow && \downarrow && \downarrow\\ B & \underset{b}{\to} & C& \underset{}{\to} & B' & \underset{}{\to} & C'}

All the vertical maps are cofibration weak equivalences, by assumption. Moreover, the bottom map $C\to C\prime$ is a weak equivalence, since it is the pushout of the weak equivalence $ba$ along the cofibration $c$. And since the zigzag

$$B\stackrel{b}{\to }C\to B\prime \stackrel{\sim }{\leftarrow }B$$B \xrightarrow{b} C \to B' \overset{\sim}{\leftarrow} B

represents the same morphism as

$$B\stackrel{b}{\to }C\stackrel{c}{\leftarrow }A\stackrel{a}{\to }B$$B \xrightarrow{b} C \overset{c}{\leftarrow} A \xrightarrow{a} B

which represents $1_B$, we have that $B\stackrel{b}{\to }C\to B\prime$ is a weak equivalence. Thus, by 2-out-of-6, $b$ is a weak equivalence, hence so is $a$ by 2-out-of-3.

Of course, there is a dual theorem for fibrations.

References

• William Dwyer, Philip Hirschhorn, Daniel Kan, Jeff Smith, Homotopy Limit Functors on Model Categories and Homotopical Categories

• Andrew Blumberg and Michael Mandell, Algebraic $K$-theory and abstract homotopy theory

• Masaki Kashiwara, Pierre Schapira, Categories and Sheaves