model category

for ∞-groupoids

# Contents

## 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

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

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

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

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

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

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

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

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{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

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

That establishes the claim.

## Corollaries

###### Corollary (“Ken Brown’s Lemma”)

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

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

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 $A \times_C^h B$

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

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

For instance page 4 of

Revised on January 19, 2014 09:48:25 by Peter Le Fanu Lumsdaine (91.103.36.0)