model category

for ∞-groupoids

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

###### Lemma

Given any product

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

in $\mathcal{C}$, the projections $p_{X}$ 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.

###### Lemma

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

$d = (d_0, d_1) : X^I \twoheadrightarrow X \times X$

denote the canonical fibration. The morphisms $d_0 : X^I \to X$ and $d_1 : X^I \to X$ are both trivial fibrations.

###### Proof

We have the following.

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

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

2) By one of the axioms for a category of fibrant objects, $id_X$ 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 $c$ is a weak equivalence, that $d_{0}$ is a weak equivalence.

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

###### Lemma

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

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

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

Let $g$ denote the composite

$g : X \times_Y Y^I \to Y^I \stackrel{d_1}{\twoheadrightarrow} Y.$

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

$X \times_Y Y^I \to X \times_Y Y \times Y = X \times Y \to X \times e = X$

where $e$ is the final object of $C$. Here, the first morphism $id_X \times_Y d$ is a fibration because it is a base change of the fibration $d$; the second is a fibration because it is a base change of the fibration $Y \to e$ ($Y$ is fibrant).

###### Proposition

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

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

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

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

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

$\array{ X & \overset{id_X}{\to} & X \\ \downarrow{j f} & & \downarrow{f} \\ Y^I & \overset{d_0}{\to} & Y }$

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

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

commute.

## Ken Brown’s lemma

###### Corollary

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 the factorization lemma, 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 $F(r)$ are weak equivalences, we conclude, by one of the axioms for a category with weak equivalences, that $F(f)$ 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 $F$, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant objects.

## Homotopy pullbacks

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