nLab
factorization lemma

Idea

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

Statement

Let $C$ be a category of fibrant objects.

Lemma

Every morphism $f:X\to Y$ in $C$ factors as

$$f:X\stackrel{i_{\simeq }}{\to }\hat{X}\stackrel{p}{\to }Y,$$f : X \stackrel{i_\simeq}{\to} \hat X \stackrel{p}{\to} Y \,,

where

1. $i$ is a right inverse to a trivial fibration, hence in particular a weak equivalence;

2. $p$ is a fibration.

Proof

Let $Y\stackrel{\simeq }{\to }{Y}^I\stackrel{(d_0,d_1)}{\to }Y\times Y$ be a path space object for $Y$. Let $\hat{X}:=Y^I{\times }_{Y}X$ be the pullback of $f$ along one of its legs, to get the diagram

$$\begin{array}{ccc}\hat{X}& \to & X\\ \downarrow & & {\downarrow }^f\\ Y^I& \stackrel{d_1}{\to }& Y\\ {}^{d_0}\downarrow \\ Y\end{array}.$$\array{ \hat X &\to& X \\ \downarrow && \downarrow^{\mathrlap{f}} \\ Y^I &\stackrel{d_1}{\to}& Y \\ {}^{\mathllap{d_0}}\downarrow \\ Y } \,.

Take $p$ to be the composite vertical morphism here:

$$p:\hat{X}\to Y^I\stackrel{d_0}{\to }Y.$$p : \hat X \to Y^I \stackrel{d_0}{\to} Y \,.

To see that this is indeed a fibration, notice that, by the pasting law, the above pullback diagram can be refined to a double pullback diagram as follows

$$\begin{array}{ccccc}\hat{X}& \stackrel{}{\to }& X\times Y& \stackrel{p_1}{\to }& X\\ \downarrow & & {\downarrow }^{(f,\mathrm{Id})}& & {\downarrow }^f\\ Y^I& \stackrel{(d_0,d_1)}{\to }& Y\times Y& \stackrel{p_1}{\to }& Y\\ {\downarrow }^{d_1}& {\swarrow }_{p_2}\\ Y\end{array}.$$\array{ \hat X &\stackrel{}{\to}& X \times Y &\stackrel{p_1}{\to}& X \\ \downarrow && \downarrow^{(f, Id)} && \downarrow^\mathrlap{f} \\ Y^I &\stackrel{(d_0 , d_1) }{\to}& Y \times Y &\stackrel{p_1}{\to}& Y \\ \downarrow^{\mathrlap{d_1}} & \swarrow_{p_2} \\ Y } \,.

Both squares are pullback squares. Since pullbacks of fibrations are fibrations, the morphism $\hat{X}\to X\times Y$ is a fibration. Similarly, since $Y$ is assumed to be fibrant, also the projection map $p_i:Y\times Y\to Y$ is a fibration.

Since $p$ is therefore the composite

$$\begin{array}{rl}p& :\hat{X}\to X\times Y\stackrel{(f,\mathrm{Id})}{\to }Y\times Y\stackrel{p_2}{\to }Y\end{array}$$\begin{aligned} p &: \hat X \to X \times Y \stackrel{(f ,Id)}{\to} Y \times Y \stackrel{p_2}{\to} Y \end{aligned}

of two fibrations, it is itself a fibration.

Next, by the axioms of path space objects in a category of fibrant objects that $d_1:Y^I\to Y$ is a trivial fibration. Since these are stable under pullback, also $\hat{X}\to X$ is a trivial fibration.

But, by the axioms, $Y^I\to Y$ has a right inverse $Y\to Y^I$. By the pullback property this induces a right inverse of $\hat{X}\to X$ fitting into a pasting diagram

$$\begin{array}{ccccc}X& \to & \hat{X}& \to & X\\ {}^f\downarrow & & \downarrow & & {\downarrow }^f\\ Y& \to & Y^I& \stackrel{d_1}{\to }& Y\\ & {}_{\mathrm{Id}}\searrow & {}^{d_0}\downarrow \\ & & Y\end{array}.$$\array{ X &\to& \hat X &\to& X \\ {}^{\mathrlap{f}}\downarrow && \downarrow && \downarrow^{\mathrlap{f}} \\ Y &\to& Y^I &\stackrel{d_1}{\to}& Y \\ & {}_{\mathllap{Id}}\searrow& {}^{\mathllap{d_0}}\downarrow \\ && Y } \,.

That establishes the claim.

Corollaries

Corollary

Let $F:C\to D$ be a functor from a category of fibrant objects to any category with weak equivalences that sends trivial fibrations to weak equivalences. Then this functor necessarily sends all weak equivalences to weak equivalences, hence is a homotopical functor.

Proof

If $f:X\to Y$ is a weak equivalence, then by 2-out-of-3 also the $p:\hat{X}\to Y$ from the above proof is a weak equivalence, hence a trivial fibration.

Apply the functor $F$ to the diagram of the above proof

$$\begin{array}{ccc}F(\hat{X})& \to & F(X)\\ \downarrow & & {\downarrow }^{F(f)}\\ F(Y^I)& \stackrel{F(d_1)}{\to }& F(Y)\\ {}^{F(d_0)}\downarrow \\ F(Y)\end{array}.$$\array{ F(\hat X) &\to& F(X) \\ \downarrow && \downarrow^{\mathrlap{F(f)}} \\ F(Y^I) &\stackrel{F(d_1)}{\to}& F(Y) \\ {}^{\mathllap{F(d_0)}}\downarrow \\ F(Y) } \,.

By the assumption that $F$ preserves trivial fibrations we have that both horizontal morphisms as well as the total vertical morphism and the bottom vertical morphism are weak equivalences. By 2-out-of-3 it then follows that also the top left vertical morphism is a weak equivalence and then finally that $F(f)$ is.

Remark

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

Corollary

Let $A\to C\leftarrow B$ be a diagram between fibrant objects in a model category. Then the ordinary pullback $A{\times }_C^{h}B$

$$\begin{array}{ccc}A{\times }_C^{h}B& \to & C^I\\ \downarrow & & \downarrow \\ A\times B& \to & C\times C\end{array}$$\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 $BG$, applying the factorization lemma to the point inclusion $*\to BG$ yields a morphism $*\stackrel{\simeq }{\to }EG\stackrel{p}{\to }BG$. This exhibits a universal principal ∞-bundle for $G$.

References

For instance page 4 of

• Kenneth Brown, Abstract Homotopy Theory and Generalized sheaf Cohomology .