homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
In a category $C$, a class $W\subseteq Mor(C)$ of morphisms is said to satisfy 2-out-of-6 if for any sequence of three composable morphisms
if $w v$ and $v u$ are in $W$, then so are $u$, $v$, $w$, and $w v u$.
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.
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 $\xrightarrow{f} \xrightarrow{1} \xrightarrow{g}$, we find that $g f\in W$. On the other hand, if $f$ and $g f$ are in $W$, then applying 2-out-of-6 to $\xrightarrow{1} \xrightarrow{f} \xrightarrow{g}$, we find that $g\in W$, and similarly if $g$ and $g f$ are in $W$.
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 $\xrightarrow{g} \xrightarrow{f} \xrightarrow{g}$ we find that $g$ and $f$ are in $W$.
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):
Suppose a category with weak equivalences $\mathcal{C}$ 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 $\mathcal{C}$ 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 $w v$ and $v u$ weak equivalences. Factor $v u\colon A\to C$ as $A \xrightarrow{i} C' \xrightarrow{p} C$ where $i$ is a cofibration weak equivalence and $p$ is a weak equivalence with a section] $s\colon C\to C'$. Let $B'$ be the pushout
Since $p i = v u$, we have a unique map $g\colon B' \to C$ such that $g h = p$ and $g k = v$. Define $f = h s$; then $g f = g h s = p s = 1_C$.
Since $i$ is a cofibration weak equivalence, so is $k$. And since $w g k = w v\colon B\to D$ is a weak equivalence, by two-out-of-three, $w g\colon B' \to D$ is also a weak equivalence. But now we have a commutative diagram
exhibiting $w$ as a retract of $w g$ 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 $w v u$.
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 $W$ 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.
Suppose $W$ admits a calculus of fractions. Then $W$ satisfies two-out-of-six if and only if it is saturated.
This is from 7.1.20 of Categories and Sheaves. Suppose $f\colon X\to Y$ becomes an isomorphism in $\mathcal{C}[W^{-1}]$, and represent its inverse by $Y \xrightarrow{g} X' \overset{s}{\leftarrow} X$ with $s\in W$. Then since $g f$ and $s$ represent the same morphism in $\mathcal{C}[W^{-1}]$, there is a morphism $t\colon X'\to X''$ in $W$ such that $t g f = t s$. Since $t s\in W$, it follows by 2-out-of-3 that $g f\in W$.
Now applying this same argument to $g$, we obtain an $h$ such that $h g \in W$. But then by 2-out-of-6, we have $f\in W$ as desired.
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).
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 $\mathcal{C}[W^{-1}]$ can be represented by a zigzag $A \to C \overset{\sim}{\leftarrow} B$ in which $B\xrightarrow{\sim} C$ is a cofibration and a weak equivalence. See Blumberg-Mandell, section 5 for a detailed proof. The idea is that given any zigzag $A \overset{\sim}{\leftarrow} D \to B$, 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 $a\colon A\to B$ becomes an isomorphism in $C[W^{-1}]$, and represent its inverse by $B \xrightarrow{b} C \overset{c}{\leftarrow} A$ with $c$ a cofibration weak equivalence. Since the composite $A \xrightarrow{b a} C \overset{c}{\leftarrow} A$ represents $1_A$, we have $b a \in W$. Consider the following diagram where the squares are pushouts:
All the vertical maps are cofibration weak equivalences, by assumption. Moreover, the bottom map $C\to C'$ is a weak equivalence, since it is the pushout of the weak equivalence $b a$ along the cofibration $c$. And since the zigzag
represents the same morphism as
which represents $1_B$, we have that $B\xrightarrow{b} C \to B'$ 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.
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