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factorization lemma

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Idea

The factorisation lemma (Brown 73, prop. 3 below) is a fundamental tool in the theory of categories of fibrant objects (dually: of cofibrant objects). It mimics one half of the factorisation axioms in a model category in that it asserts that every morphisms may be factored as, in particular, a weak equivalence followed by a fibration.

A key corollary of the factorization lemma is the statement, widely known as Ken Brown’s lemma (prop. 4 below) which says that for a functor from a category of fibrant objects to be a homotopical functors, it is sufficient already that it sends acyclic fibrations to weak equivalences.

For more background, see also at Introduction to classical homotopy theory this lemma.

Factorisation lemma

Let 𝒞\mathcal{C} be a category of fibrant objects.

Fact

Let

Xp XX×Yp YY X \overset{p_{X}}{\leftarrow} X \times Y \overset{p_{Y}}{\rightarrow} Y

be a product in 𝒞\mathcal{C}. Then p Xp_{X} and p Yp_{Y} are fibrations.

Proof

By one of the axioms for a category of fibrant objects, 𝒞\mathcal{C} has a final object 11. We have the following.

1) The following diagram in 𝒞\mathcal{C} is a cartesian square.

X×Y p Y Y p X X 1 \array{ X \times Y & \overset{p_{Y}}{\to} & Y \\ p_{X} \downarrow & & \downarrow \\ X & \to & 1 \\ }

2) By one of the axioms for a category of fibrant objects, the arrows Y1Y \to 1 and X1X \to 1 are fibrations.

By one of the axioms for a category of fibrant objects, it follows from 1) and 2) that p Xp_{X} and p Yp_{Y} are fibrations.

Fact

Let XX be an object of 𝒞\mathcal{C}. Let

Xp 0X×Xp 1X X \overset{p_{0}}{\leftarrow} X \times X \overset{p_{1}}{\rightarrow} X

be a product in 𝒞\mathcal{C}. By one of the axioms for a category of fibrant objects, there is a commutative diagram

X c X I Δ e X×X \array{ X & \overset{c}{\to} & X^I \\ & \underset{\Delta}{\searrow} & \downarrow e \\ & & X \times X }

in 𝒞\mathcal{C} in which cc is a weak equivalence, and in which ee is a fibration.

The arrow e 0:X IXe_0 : X^I \to X given by p 0ep_0 \circ e is a trivial fibration. The arrow e 1:X IXe_1 : X^I \to X given by p 1ep_1 \circ e is a trivial fibration.

Proof

We have the following.

1) The following diagram in 𝒞\mathcal{C} commutes.

X c X I id X e 0 X \array{ X & \overset{c}{\to} & X^I \\ & \underset{id_X}{\searrow} & \downarrow e_{0} \\ & & X }

2 By one of the axioms for a category of fibrant objects, id Xid_X is a weak equivalence.

By one of the axioms for a category of fibrant objects, we deduce from 1), 2), and the fact that cc is a weak equivalence, that e 0e_{0} is a weak equivalence.

An entirely analogous argument demonstrates that e 1e_{1} is a weak equivalence.

Proposition

(factorization lemma)

Let f:XYf : X \to Y be an arrow of 𝒞\mathcal{C}. There is a commutative diagram

X j Z f g Y \array{ X & \overset{j}{\to} & Z \\ & \underset{f}{\searrow} & \downarrow g \\ & & Y }

in 𝒞\mathcal{C} such that the following hold.

1) The arrow g:ZYg : Z \to Y is a fibration.

2) There is a trivial fibration r:ZXr : Z \to X such that the following diagram in 𝒞\mathcal{C} commutes.

X j Z id r X \array{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X }
Proof

By one of the axioms for a category of fibrant objects, there is a commutative diagram

Y c Y I Δ e Y×Y \array{ Y & \overset{c}{\to} & Y^I \\ & \underset{\Delta}{\searrow} & \downarrow e \\ & & Y \times Y }

in 𝒞\mathcal{C} in which cc is a weak equivalence, and in which ee is a fibration.

Since ee is a fibration, there is, by one of the axioms for a category of fibrant objects, a cartesian square in 𝒞\mathcal{C} as follows.

Z u 0 Y I u 1 e X×Y f×id Y×Y \array{ Z & \overset{u_{0}}{\to} & Y^I \\ u_{1} \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y }

Let g:ZYg : Z \to Y be p Yu 1p_{Y} \circ u_{1}, where p Y:X×YYp_{Y} : X \times Y \to Y is the projection arrow.

Since ee is a fibration, we have, by one of the axioms for a category of fibrant objects, that u 1u_{1} is a fibration. By Fact 1, the arrow p Yp_{Y} is a fibration. Since a composition of fibrations in a category of fibrant objects is a fibration, we deduce that gg is a fibration.

The following diagram in 𝒞\mathcal{C} commutes.

X cf Y I (id,f) e X×Y f×id Y×Y \array{ X & \overset{c \circ f}{\to} & Y^I \\ (id, f) \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y \\ }

By the universal property of a pullback, we deduce that there is an arrow j:XZj : X \to Z such that the diagrams

X f Y j c Z u 0 Y I \array{ X & \overset{f}{\to} & Y \\ j \downarrow & & \downarrow c \\ Z & \underset{u_{0}}{\to} & Y^I \\ }

and

X j Z (id,f) u 1 X×Y \array{ X & \overset{j}{\to} & Z \\ & \underset{(id, f)}{\searrow} & \downarrow u_{1} \\ & & X \times Y }

in 𝒞\mathcal{C} commute. By the commutativity of the second of these diagrams, and the fact that the diagram

X id×f X×Y id p X X \array{ X & \overset{id \times f}{\to} & X \times Y \\ & \underset{id}{\searrow} & \downarrow p_{X} \\ & & X }

in 𝒞\mathcal{C} commutes, the diagram

X j Z id r X \array{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X }

in 𝒞\mathcal{C} commutes.

Let r:ZXr : Z \to X be p Xu 1p_{X} \circ u_{1}, where p X:X×YXp_{X} : X \times Y \to X is the projection arrow.

Let

Yp 0Y×Yp 1Y Y \overset{p_{0}}{\leftarrow} Y \times Y \overset{p_{1}}{\rightarrow} Y

be a product diagram in 𝒞\mathcal{C}. The following diagram in 𝒞\mathcal{C} is a cartesian square.

X×Y f×id Y×Y p X p 0 X f Y \array{ X \times Y & \overset{f \times id }{\to} & Y \times Y \\ p_{X} \downarrow & & \downarrow p_{0} \\ X & \underset{f}{\to} & Y \\ }

Thus the following diagram in 𝒞\mathcal{C} is a cartesian square.

Z u 0 Y I r p 0e X f Y \array{ Z & \overset{u_{0}}{\to} & Y^I \\ r \downarrow & & \downarrow p_{0} \circ e \\ X & \underset{f}{\to} & Y \\ }

By Fact 2, the arrow p 0ep_{0} \circ e is a trivial fibration. By one of the axioms for a category of fibrant objects, we deduce that rr is a trivial fibration.

Remark

That rr is a fibration can be demonstrated in exactly the same way as that gg is a fibration. It is to prove the stronger assertion that rr is a trivial fibration that the argument with which we concluded the proof is needed.

Remark

By the commutativity of the diagram

X j Z id r X \array{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X }

and the fact that rr is a weak equivalence, we have, by one of the axioms for a category of fibrant objects, that jj is a weak equivalence.

Ken Brown’s lemma

Proposition

Let 𝒞\mathcal{C} be a category of fibrant objects. Let 𝒟\mathcal{D} be a category with weak equivalences. Let F:CDF : C \to D be a functor with the property that, for every arrow ff of 𝒞\mathcal{C} which is a trivial fibration, we have that F(f)F(f) is a weak equivalence.

Let w:XYw : X \to Y be an arrow of 𝒞\mathcal{C} which is a weak equivalence. Then F(w)F(w) is a weak equivalence.

Proof

By Proposition 3, there is a commutative diagram

X j Z w g Y \array{ X & \overset{j}{\to} & Z \\ & \underset{w}{\searrow} & \downarrow g \\ & & Y }

in 𝒞\mathcal{C} such that the following hold.

1) The arrow g:ZYg : Z \to Y is a fibration.

2) There is a trivial fibration r:ZXr : Z \to X such that the following diagram in 𝒞\mathcal{C} commutes.

X j Z id r X \array{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X }

By the commutativity of the diagram

X j Z w g Y \array{ X & \overset{j}{\to} & Z \\ & \underset{w}{\searrow} & \downarrow g \\ & & Y }

and the fact that both jj and ww are weak equivalences, we have that gg is a weak equivalence, by one of the axioms for a category of fibrant objects.

By assumption, we thus have that F(g):F(Z)F(Y)F(g) : F(Z) \to F(Y) is a weak equivalence.

The following hold.

1) By the commutativity of the diagram

X j Z id r X \array{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X }

in 𝒞\mathcal{C}, we have that the following diagram in 𝒟\mathcal{D} commutes.

F(X) F(j) F(Z) id F(r) F(X) \array{ F(X) & \overset{F(j)}{\to} & F(Z) \\ & \underset{id}{\searrow} & \downarrow F(r) \\ & & F(X) }

2) Since rr is a trivial fibration, we have by assumption that F(r)F(r) is a trivial fibration. In particular, F(r)F(r) is a weak equivalence.

3) By one of the axioms for a category with weak equivalences, we have that id:F(X)F(X)id : F(X) \to F(X) is a weak equivalence.

By 1), 2), 3) and one of the axioms for a category with weak equivalences, we have that F(j)F(j) is a weak equivalence.

The following diagram in 𝒞\mathcal{C} commutes.

F(X) F(j) F(Z) F(w) F(g) F(Y) \array{ F(X) & \overset{F(j)}{\to} & F(Z) \\ & \underset{F(w)}{\searrow} & \downarrow F(g) \\ & & F(Y) }

Since F(j)F(j) and F(r)F(r) are weak equivalences, we conclude, by one of the axioms for a category with weak equivalences, that F(f)F(f) is a weak equivalence.

Remark

In other words, FF is a homotopical functor.

Remark

If CC is the full subcategory of fibrant objects in a model category, then this corollary asserts that a right Quillen functor GG, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant objects.

Remark

By the dual nature of model categories, we then get that a left Quillen functor preserves weak equivalences between cofibrant objects.

Computing a homotopy pullback by means of an ordinary pullback

Corollary

Let ACBA \to C \leftarrow B be a diagram between fibrant objects in a model category. Then the ordinary pullback A× C hBA \times_C^h B

A× C hB C I A×B C×C \array{ A \times_C^h B &\to& C^I \\ \downarrow && \downarrow \\ A \times B &\to& C \times C }

presents the homotopy pullback of the original diagram.

See the section Concrete constructions at homotopy pullback for more details on this.

Examples

  • For GG an ∞-group object in CC with delooping BG\mathbf{B}G, applying the factorization lemma to the point inclusion *BG* \to \mathbf{B}G yields a morphism *EGpBG* \stackrel{\simeq}{\to} \mathbf{E}G \stackrel{p}{\to} \mathbf{B}G. This exhibits a universal principal ∞-bundle for GG.

References

Revised on March 13, 2017 10:35:07 by Urs Schreiber (46.183.103.8)