nLab factorization lemma

Idea

The factorization lemma (Brown 73, prop. below) is a fundamental tool in the theory of categories of fibrant objects (dually: of cofibrant objects). It mimics one half of the factorization axioms in a model category in that it asserts that every morphism 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. below) which says that for a functor from a category of fibrant objects to be a homotopical functor, 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.

Factorization lemma

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

Proposition

Let

$$X \overset{p_{X}}{\leftarrow} X \times Y \overset{p_{Y}}{\rightarrow} Y$$

be a product in $\mathcal{C}$. Then $p_{X}$ and $p_{Y}$ are fibrations.

Proof

By one of the axioms for a category of fibrant objects, $\mathcal{C}$ has a final object $1$. We have the following.

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

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

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

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

Proposition

Let $X$ be an object of $\mathcal{C}$. Let

$$X \overset{p_{0}}{\leftarrow} X \times X \overset{p_{1}}{\rightarrow} X$$

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

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

in $\mathcal{C}$ in which $c$ is a weak equivalence, and in which $e$ is a fibration.

The arrow $e_0 : X^I \to X$ given by $p_0 \circ e$ is a trivial fibration. The arrow $e_1 : X^I \to X$ given by $p_1 \circ e$ is a trivial fibration.

Proof

We have the following.

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

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

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

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

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

Proposition

(Factorization lemma)

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

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

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

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

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

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

Proof

We can construct the following diagram:

The triangle on the right exists with $c$ a weak equivalence and $e$ a fibration, as categories of fibrant objects have path objects. As $e$ is a fibration, we can pullback along $f \times id$ to get the upper-left pullback square. By the universal property of this pullback square, we can produce a unique $X \overset j \to Z$ which makes the top half of the diagram commute.

We let $g$ be the composite $Z \overset{\pi_1^Z} \to X \times Y \overset{\pi_1^{X \times Y}} \to Y$ so that $g \circ j = f$; these are fibrations due to being a pullback of the fibration $e$ and a product projection (Fact 1), so their composite $g$ is a fibration too.

We let $r$ be the composite $Z \overset{\pi_1^Z} \to X \times Y \overset{\pi_0^{X \times Y}} \to X$ so that $r \circ j = id_X$. As the upper square and lower square are both pullback squares, $r$ is the pullback of the morphism $Y^I \overset{e} \to Y \overset{\pi_0^{Y\times Y}} \to Y$ (which is an acyclic fibration by Fact 2), hence $r$ is also an acyclic fibraiton.

Remark

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

Remark

By the commutativity of the diagram

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

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

Ken Brown’s lemma

Proposition

(Ken Brown’s lemma)

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

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

Proof

By Proposition 3, there is a commutative diagram

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

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

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

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

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

By the commutativity of the diagram

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

and the fact that both $j$ and $w$ are weak equivalences, we have that $g$ is a weak equivalence, by one of the axioms for a category of fibrant objects.

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

The following hold.

1) By the commutativity of the diagram

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

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

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

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

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

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

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

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

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

Remark

In other words, $F$ is a homotopical functor.

Remark

If $C$ is the full subcategory of fibrant objects in a model category, then this corollary asserts that a right Quillen functor $G$, 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 $A \to C \leftarrow B$ be a diagram between fibrant objects in a model category. Then the ordinary pullback $A \times_C^h B$

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

presents the homotopy pullback of the original diagram.

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

Examples

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

References

The factorization lemma is due to

• Kenneth Brown, p. 421 (4 of 41) in: Abstract Homotopy Theory and Generalized Sheaf Cohomology, 1973, p. 4

The corollary commonly known as “Ken Brown’s lemma” does not appear explicitly in Brown 73; it does appear under this name in

• Mark Hovey, Lemma 1.1.12 in: Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (pdf, Google books, ISBN:978-0-8218-4361-1)

Lecture notes:

• here and here in: Introduction to Homotopy Theory

A version in the setup of $\infty$-cosmoi is Lemma 2.1.6 in

• Emily Riehl, Dominic Verity, Fibrations and Yoneda’s lemma in an $\infty$-cosmos, Journal of Pure and Applied Algebra, 221:3, 2017, pp. 499–564 (arXiv:1506.05500)