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two-out-of-six property

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Category theory

Homotopy theory

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Definition

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

XuYvZwK X\xrightarrow{u} Y \xrightarrow{v} Z \xrightarrow{w} K

if wvw v and vuv u are in WW, then so are uu, vv, ww, and wvuw v u.

Examples

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 ff and gg are in WW, then applying 2-out-of-6 to f1g\xrightarrow{f} \xrightarrow{1} \xrightarrow{g}, we find that gfWg f\in W. On the other hand, if ff and gfg f are in WW, then applying 2-out-of-6 to 1fg\xrightarrow{1} \xrightarrow{f} \xrightarrow{g}, we find that gWg\in W, and similarly if gg and gfg f are in WW.

Identities and isomorphisms

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

Closure under retracts

The 2-out-of-6 property is closely related to the property that WW 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\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 𝒞\mathcal{C} are closed under retracts, they also satisfy 2-out-of-6.

Proof

Suppose the first three assumptions on the cofibrations, and let

AuBvCwDA \xrightarrow{u} B \xrightarrow{v} C \xrightarrow{w} D

be a sequence of composable maps, with wvw v and vuv u weak equivalences. Factor vu:ACv u\colon A\to C as AiCpCA \xrightarrow{i} C' \xrightarrow{p} C where ii is a cofibration weak equivalence and pp is a weak equivalence with a section] s:CCs\colon C\to C'. Let BB' be the pushout

A i C u h B k B \array{ A & \overset{i}{\to} & C'\\ ^u\downarrow && \downarrow^h\\ B& \underset{k}{\to} & B'}

Since pi=vup i = v u, we have a unique map g:BCg\colon B' \to C such that gh=pg h = p and gk=vg k = v. Define f=hsf = h s; then gf=ghs=ps=1 Cg f = g h s = p s = 1_C.

Since ii is a cofibration weak equivalence, so is kk. And since wgk=wv:BDw g k = w v\colon B\to D is a weak equivalence, by two-out-of-three, wg:BDw g\colon B' \to D is also a weak equivalence. But now we have a commutative diagram

C f B g C w wg w D = D = D\array{C & \overset{f}{\to} & B' & \xrightarrow{g} & C \\ ^w\downarrow && \downarrow^{w g} && \downarrow^w\\ D& \underset{=}{\to} & D & \underset{=}{\to} & D}

exhibiting ww as a retract of wgw g in the arrow category. Thus, by assumption ww is a weak equivalence. By successive applications of two-out-of-three, so are vv, uu, and wvuw 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.

Saturation

The 2-out-of-6 property is also closely related to the property that WW is saturated, in the sense that any morphism which becomes an isomorphism in the localization C[W 1]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 WW admits a calculus of fractions. Then WW 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:XYf\colon X\to Y becomes an isomorphism in 𝒞[W 1]\mathcal{C}[W^{-1}], and represent its inverse by YgXsXY \xrightarrow{g} X' \overset{s}{\leftarrow} X with sWs\in W. Then since gfg f and ss represent the same morphism in 𝒞[W 1]\mathcal{C}[W^{-1}], there is a morphism t:XXt\colon X'\to X'' in WW such that tgf=tst g f = t s. Since tsWt s\in W, it follows by 2-out-of-3 that gfWg f\in W.

Now applying this same argument to gg, we obtain an hh such that hgWh g \in W. But then by 2-out-of-6, we have fWf\in W as desired.

Theorem

Suppose CC 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 WW 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 WW admits a homotopy calculus of left fractions?, and in particular that every morphism in 𝒞[W 1]\mathcal{C}[W^{-1}] can be represented by a zigzag ACBA \to C \overset{\sim}{\leftarrow} B in which BCB\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 ADBA \overset{\sim}{\leftarrow} D \to B, we factor DAD\to A as a cofibration weak equivalence followed by a retraction weak equivalence, then push out the cofibration along DBD\to B and use the section to obtain a map from AA into the pushout.

Now suppose a:ABa\colon A\to B becomes an isomorphism in C[W 1]C[W^{-1}], and represent its inverse by BbCcAB \xrightarrow{b} C \overset{c}{\leftarrow} A with cc a cofibration weak equivalence. Since the composite AbaCcAA \xrightarrow{b a} C \overset{c}{\leftarrow} A represents 1 A1_A, we have baWb a \in W. Consider the following diagram where the squares are pushouts:

A a B b C c B b C B C\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 CCC\to C' is a weak equivalence, since it is the pushout of the weak equivalence bab a along the cofibration cc. And since the zigzag

BbCBB B \xrightarrow{b} C \to B' \overset{\sim}{\leftarrow} B

represents the same morphism as

BbCcAaB B \xrightarrow{b} C \overset{c}{\leftarrow} A \xrightarrow{a} B

which represents 1 B1_B, we have that BbCBB\xrightarrow{b} C \to B' is a weak equivalence. Thus, by 2-out-of-6, bb is a weak equivalence, hence so is aa by 2-out-of-3.

Of course, there is a dual theorem for fibrations.

References

Revised on January 3, 2012 12:31:49 by Urs Schreiber (89.204.130.224)