Contents

model category

for ∞-groupoids

# Contents

## Idea

The factorisation 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 factorisation axioms in a model category in that it asserts that every morphisms 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 functors, 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.

## Factorisation lemma

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

###### Fact

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.

###### Fact

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

By one of the axioms for a category of fibrant objects, there is a commutative diagram

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

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

Since $e$ is a fibration, there is, by one of the axioms for a category of fibrant objects, a cartesian square in $\mathcal{C}$ as follows.

$\array{ Z & \overset{u_{0}}{\to} & Y^I \\ u_{1} \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y }$

Let $g : Z \to Y$ be $p_{Y} \circ u_{1}$, where $p_{Y} : X \times Y \to Y$ is the projection arrow.

Since $e$ is a fibration, we have, by one of the axioms for a category of fibrant objects, that $u_{1}$ is a fibration. By Fact 1, the arrow $p_{Y}$ is a fibration. Since a composition of fibrations in a category of fibrant objects is a fibration, we deduce that $g$ is a fibration.

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

$\array{ X & \overset{c \circ f}{\to} & Y^I \\ (id, f) \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y \\ }$

By the universal property of a pullback, we deduce that there is an arrow $j : X \to Z$ such that the diagrams

$\array{ X & \overset{f}{\to} & Y \\ j \downarrow & & \downarrow c \\ Z & \underset{u_{0}}{\to} & Y^I \\ }$

and

$\array{ X & \overset{j}{\to} & Z \\ & \underset{(id, f)}{\searrow} & \downarrow u_{1} \\ & & X \times Y }$

in $\mathcal{C}$ commute. By the commutativity of the second of these diagrams, and the fact that the diagram

$\array{ X & \overset{id \times f}{\to} & X \times Y \\ & \underset{id}{\searrow} & \downarrow p_{X} \\ & & X }$

in $\mathcal{C}$ commutes, the diagram

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

in $\mathcal{C}$ commutes.

Let $r : Z \to X$ be $p_{X} \circ u_{1}$, where $p_{X} : X \times Y \to X$ is the projection arrow.

Let

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

be a product diagram in $\mathcal{C}$. The following diagram in $\mathcal{C}$ is a cartesian square.

$\array{ X \times Y & \overset{f \times id }{\to} & Y \times Y \\ p_{X} \downarrow & & \downarrow p_{0} \\ X & \underset{f}{\to} & Y \\ }$

Thus the following diagram in $\mathcal{C}$ is a cartesian square.

$\array{ Z & \overset{u_{0}}{\to} & Y^I \\ r \downarrow & & \downarrow p_{0} \circ e \\ X & \underset{f}{\to} & Y \\ }$

By Fact 2, the arrow $p_{0} \circ e$ is a trivial fibration. By one of the axioms for a category of fibrant objects, we deduce that $r$ is a trivial fibration.

###### 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

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 $G$.

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, J. Pure Appl. Alg. 221:3 (2017), 499-564 arxiv/1506.05500