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

$f:X\stackrel{{i}_{\simeq }}{\to }\stackrel{^}{X}\stackrel{p}{\to }Y\phantom{\rule{thinmathspace}{0ex}},$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 $Y\stackrel{\simeq }{\to }{Y}^{I}\stackrel{\left({d}_{0},{d}_{1}\right)}{\to }Y×Y$ be a path space object for $Y$. Let $\stackrel{^}{X}:={Y}^{I}{×}_{Y}X$ be the pullback of $f$ along one of its legs, to get the diagram

$\begin{array}{ccc}\stackrel{^}{X}& \to & X\\ ↓& & {↓}^{f}\\ {Y}^{I}& \stackrel{{d}_{1}}{\to }& Y\\ {}^{{d}_{0}}↓\\ Y\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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:

$p:\stackrel{^}{X}\to {Y}^{I}\stackrel{{d}_{0}}{\to }Y\phantom{\rule{thinmathspace}{0ex}}.$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

$\begin{array}{ccccc}\stackrel{^}{X}& \stackrel{}{\to }& X×Y& \stackrel{{p}_{1}}{\to }& X\\ ↓& & {↓}^{\left(f,\mathrm{Id}\right)}& & {↓}^{f}\\ {Y}^{I}& \stackrel{\left({d}_{0},{d}_{1}\right)}{\to }& Y×Y& \stackrel{{p}_{1}}{\to }& Y\\ {↓}^{{d}_{1}}& {↙}_{{p}_{2}}\\ Y\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\stackrel{^}{X}\to X×Y$ is a fibration. Similarly, since $Y$ is assumed to be fibrant, also the projection map ${p}_{i}:Y×Y\to Y$ is a fibration.

Since $p$ is therefore the composite

$\begin{array}{rl}p& :\stackrel{^}{X}\to X×Y\stackrel{\left(f,\mathrm{Id}\right)}{\to }Y×Y\stackrel{{p}_{2}}{\to }Y\end{array}$\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 $\stackrel{^}{X}\to X$ 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 $\stackrel{^}{X}\to X$ fitting into a pasting diagram

$\begin{array}{ccccc}X& \to & \stackrel{^}{X}& \to & X\\ {}^{f}↓& & ↓& & {↓}^{f}\\ Y& \to & {Y}^{I}& \stackrel{{d}_{1}}{\to }& Y\\ & {}_{\mathrm{Id}}↘& {}^{{d}_{0}}↓\\ & & Y\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $p:\stackrel{^}{X}\to Y$ from the above proof is a weak equivalence, hence a trivial fibration.

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

$\begin{array}{ccc}F\left(\stackrel{^}{X}\right)& \to & F\left(X\right)\\ ↓& & {↓}^{F\left(f\right)}\\ F\left({Y}^{I}\right)& \stackrel{F\left({d}_{1}\right)}{\to }& F\left(Y\right)\\ {}^{F\left({d}_{0}\right)}↓\\ F\left(Y\right)\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $F\left(f\right)$ 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 $A\to C←B$ be a diagram between fibrant objects in a model category. Then the ordinary pullback $A{×}_{C}^{h}B$

$\begin{array}{ccc}A{×}_{C}^{h}B& \to & {C}^{I}\\ ↓& & ↓\\ A×B& \to & C×C\end{array}$\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

Revised on February 25, 2012 02:40:44 by Stephan Alexander Spahn (79.219.122.169)