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The factorisation lemma (Brown 73, prop. 3 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. 4 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.
Let $\mathcal{C}$ be a category of fibrant objects.
Let
be a product in $\mathcal{C}$. Then $p_{X}$ and $p_{Y}$ are fibrations.
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.
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.
Let $X$ be an object of $\mathcal{C}$. Let
be a product in $\mathcal{C}$. By one of the axioms for a category of fibrant objects, there is a commutative diagram
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.
We have the following.
1) The following diagram in $\mathcal{C}$ commutes.
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.
(factorization lemma)
Let $f : X \to Y$ be an arrow of $\mathcal{C}$. There is a commutative diagram
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.
By one of the axioms for a category of fibrant objects, there is a commutative diagram
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.
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.
By the universal property of a pullback, we deduce that there is an arrow $j : X \to Z$ such that the diagrams
and
in $\mathcal{C}$ commute. By the commutativity of the second of these diagrams, and the fact that the diagram
in $\mathcal{C}$ commutes, the diagram
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
be a product diagram in $\mathcal{C}$. The following diagram in $\mathcal{C}$ is a cartesian square.
Thus the following diagram in $\mathcal{C}$ is a cartesian square.
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.
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.
By the commutativity of the diagram
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.
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.
By Proposition 3, there is a commutative diagram
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.
By the commutativity of the diagram
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
in $\mathcal{C}$, we have that the following diagram in $\mathcal{D}$ commutes.
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.
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.
In other words, $F$ is a homotopical functor.
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.
By the dual nature of model categories, we then get that a left Quillen functor preserves weak equivalences between cofibrant objects.
Let $A \to C \leftarrow B$ be a diagram between fibrant objects in a model category. Then the ordinary pullback $A \times_C^h B$
presents the homotopy pullback of the original diagram.
See the section Concrete constructions at homotopy pullback for more details on this.