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factorization lemma

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Model category theory

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Homotopy theory

Contents

Idea

The factorisation lemma is a fundamental tool in categories of fibrant objects (dually: of cofibrant objects). It mimics the factorisation axioms in a model category.

Factorisation lemma

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

Lemma

Given any product

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

in 𝒞\mathcal{C}, the projections 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.

Lemma

Given an object XX in 𝒞\mathcal{C}, let X IX^I be a path space object for XX and let

d=(d 0,d 1):X IX×X d = (d_0, d_1) : X^I \twoheadrightarrow X \times X

denote the canonical fibration. The morphisms d 0:X IXd_0 : X^I \to X and d 1:X IXd_1 : X^I \to X are both trivial fibrations.

Proof

We have the following.

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

X i X I id X d 0 X \array{ X & \overset{i}{\hookrightarrow} & X^I \\ & \underset{id_X}{\searrow} & \downarrow d_{0} \\ & & X }

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

By the 2-out-of-3 axiom for a category of fibrant objects, we deduce from 1), 2), and the fact that cc is a weak equivalence, that d 0d_{0} is a weak equivalence.

An entirely analogous argument demonstrates that d 1d_{1} is a weak equivalence.

Lemma

(Fibrant resolution of a morphism). Let f:XYf : X \to Y a morphism in 𝒞\mathcal{C}. There exists a canonical fibration g:X× YY IYg : X \times_Y Y^I \twoheadrightarrow Y which factors through a trivial fibration s:X× YY IXs: X \times_Y Y^I \stackrel{\sim}{\twoheadrightarrow} X. Here Y IY^I is a path space object for YY.

X× YY I s X g f Y \array{ X \times_Y Y^I & \overset{s}{\to} & X \\ & \underset{g}{\searrow} & \downarrow{f} \\ & & Y }
Proof

Let d=(d 0,d 1):Y IY×Yd = (d_0, d_1) : Y^I \twoheadrightarrow Y \times Y be the canonical fibration. Let s:X× YY IXs : X \times_Y Y^I \stackrel{\sim}{\twoheadrightarrow} X denote the base change of d 0d_0 along ff; this is a trivial fibration because d 0d_0 is by lemma 2, and trivial fibrations are stable under base change.

Let gg denote the composite

g:X× YY IY Id 1Y. g : X \times_Y Y^I \to Y^I \stackrel{d_1}{\twoheadrightarrow} Y.

One can see that this is a fibration by observing that it is the same as the composite

X× YY IX× YY×Y=X×YX×e=X X \times_Y Y^I \to X \times_Y Y \times Y = X \times Y \to X \times e = X

where ee is the final object of CC. Here, the first morphism id X× Ydid_X \times_Y d is a fibration because it is a base change of the fibration dd; the second is a fibration because it is a base change of the fibration YeY \to e (YY is fibrant).

Proposition

(Factorization lemma). Any morphism f:XYf : X \to Y in 𝒞\mathcal{C} admits a factorization as a weak equivalence ii followed by a fibration pp, such that ii is right inverse to a trivial fibration.

X i X^ f p Y \array{ X & \overset{i}{\to} & \hat X \\ & \underset{f}{\searrow} & \downarrow p \\ & & Y }
Proof

Let p:X^=X× YY IYp : \hat X = X \times_Y Y^I \twoheadrightarrow Y be a fibrant resolution of ff as in Lemma 3, so that there is a commutative diagram

X^ s X p f Y \array{ \hat X & \overset{s}{\to} & X \\ & \underset{p}{\searrow} & \downarrow{f} \\ & & Y }

Let j:YY Ij : Y \stackrel{\sim}{\to} Y^I denote the canonical weak equivalence. Since the square

X id X X jf f Y I d 0 Y \array{ X & \overset{id_X}{\to} & X \\ \downarrow{j f} & & \downarrow{f} \\ Y^I & \overset{d_0}{\to} & Y }

commutes, one gets an induced morphism i=(id X,jf):XX^i = (id_X, j f) : X \to \hat X by the universal property of the pullback, which by definition has left inverse ss and makes the diagram

X i X^ f p Y \array{ X & \overset{i}{\to} & \hat X \\ & \underset{f}{\searrow} & \downarrow p \\ & & Y }

commute.

Ken Brown’s lemma

Corollary

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 the factorization lemma, 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.

Homotopy pullbacks

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

For instance page 4 of

Revised on April 14, 2014 00:39:23 by Adeel Khan (132.252.63.38)