on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
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.
Let $\mathcal{C}$ be a category of fibrant objects.
Given any product
in $\mathcal{C}$, the projections $p_{X}$ 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.
Given an object $X$ in $\mathcal{C}$, let $X^I$ be a path space object for $X$ and let
denote the canonical fibration. The morphisms $d_0 : X^I \to X$ and $d_1 : X^I \to X$ are both trivial fibrations.
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 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.
(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 $Y$.
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
One can see that this is a fibration by observing that it is the same as the composite
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).
(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.
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
Let $j : Y \stackrel{\sim}{\to} Y^I$ denote the canonical weak equivalence. Since the square
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
commute.
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 the factorization lemma, 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 $F$, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant 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.
For instance page 4 of