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factorization lemma

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Homotopy theory

Contents

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.

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

and

X j Z id×f u 1 X×Y \array{ X & \overset{j}{\to} & Z \\ & \underset{id \times 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 FF, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant 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 April 6, 2016 14:25:06 by Urs Schreiber (89.204.155.161)