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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{factorization lemma} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{model_category_theory}{}\paragraph*{{Model category theory}}\label{model_category_theory} [[!include model category theory - contents]] \hypertarget{homotopy_theory}{}\paragraph*{{Homotopy theory}}\label{homotopy_theory} [[!include homotopy - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{factorisation_lemma}{Factorisation lemma}\dotfill \pageref*{factorisation_lemma} \linebreak \noindent\hyperlink{ken_browns_lemma}{Ken Brown's lemma}\dotfill \pageref*{ken_browns_lemma} \linebreak \noindent\hyperlink{computing_a_homotopy_pullback_by_means_of_an_ordinary_pullback}{Computing a homotopy pullback by means of an ordinary pullback}\dotfill \pageref*{computing_a_homotopy_pullback_by_means_of_an_ordinary_pullback} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} The \emph{factorisation lemma} (\hyperlink{Brown73}{Brown 73}, prop. \ref{TheFactorizationLemma} below) is a fundamental tool in the theory of [[categories of fibrant objects]] ([[formal dual|dually]]: [[cofibration category|of cofibrant objects]]). It mimics one half of the \emph{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 \emph{Ken Brown's lemma} (prop. \ref{KenBrownLemma} 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 \emph{[[Introduction to Stable homotopy theory -- P|Introduction to classical homotopy theory]]} \href{Introduction+to+Stable+homotopy+theory+--+P#FactorizationLemma}{this lemma}. \hypertarget{factorisation_lemma}{}\subsection*{{Factorisation lemma}}\label{factorisation_lemma} Let $\mathcal{C}$ be a [[category of fibrant objects]]. \begin{prop} \label{}\hypertarget{}{} Let \begin{displaymath} X \overset{p_{X}}{\leftarrow} X \times Y \overset{p_{Y}}{\rightarrow} Y \end{displaymath} be a product in $\mathcal{C}$. Then $p_{X}$ and $p_{Y}$ are fibrations. \end{prop} \begin{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. \begin{displaymath} \itexarray{ X \times Y & \overset{p_{Y}}{\to} & Y \\ p_{X} \downarrow & & \downarrow \\ X & \to & 1 \\ } \end{displaymath} 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. \end{proof} \begin{prop} \label{}\hypertarget{}{} Let $X$ be an object of $\mathcal{C}$. Let \begin{displaymath} X \overset{p_{0}}{\leftarrow} X \times X \overset{p_{1}}{\rightarrow} X \end{displaymath} be a product in $\mathcal{C}$. By one of the axioms for a category of fibrant objects, there is a commutative diagram \begin{displaymath} \itexarray{ X & \overset{c}{\to} & X^I \\ & \underset{\Delta}{\searrow} & \downarrow e \\ & & X \times X } \end{displaymath} 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. \end{prop} \begin{proof} We have the following. 1) The following diagram in $\mathcal{C}$ commutes. \begin{displaymath} \itexarray{ X & \overset{c}{\to} & X^I \\ & \underset{id_X}{\searrow} & \downarrow e_{0} \\ & & X } \end{displaymath} 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. \end{proof} \begin{prop} \label{TheFactorizationLemma}\hypertarget{TheFactorizationLemma}{} \textbf{(factorization lemma)} Let $f : X \to Y$ be an arrow of $\mathcal{C}$. There is a commutative diagram \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{f}{\searrow} & \downarrow g \\ & & Y } \end{displaymath} 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. \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X } \end{displaymath} \end{prop} \begin{proof} By one of the axioms for a category of fibrant objects, there is a commutative diagram \begin{displaymath} \itexarray{ Y & \overset{c}{\to} & Y^I \\ & \underset{\Delta}{\searrow} & \downarrow e \\ & & Y \times Y } \end{displaymath} 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. \begin{displaymath} \itexarray{ Z & \overset{u_{0}}{\to} & Y^I \\ u_{1} \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y } \end{displaymath} 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. \begin{displaymath} \itexarray{ X & \overset{c \circ f}{\to} & Y^I \\ (id, f) \downarrow & & \downarrow e \\ X \times Y & \underset{f \times id}{\to} & Y \times Y \\ } \end{displaymath} By the universal property of a pullback, we deduce that there is an arrow $j : X \to Z$ such that the diagrams \begin{displaymath} \itexarray{ X & \overset{f}{\to} & Y \\ j \downarrow & & \downarrow c \\ Z & \underset{u_{0}}{\to} & Y^I \\ } \end{displaymath} and \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{(id, f)}{\searrow} & \downarrow u_{1} \\ & & X \times Y } \end{displaymath} in $\mathcal{C}$ commute. By the commutativity of the second of these diagrams, and the fact that the diagram \begin{displaymath} \itexarray{ X & \overset{id \times f}{\to} & X \times Y \\ & \underset{id}{\searrow} & \downarrow p_{X} \\ & & X } \end{displaymath} in $\mathcal{C}$ commutes, the diagram \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X } \end{displaymath} 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 \begin{displaymath} Y \overset{p_{0}}{\leftarrow} Y \times Y \overset{p_{1}}{\rightarrow} Y \end{displaymath} be a product diagram in $\mathcal{C}$. The following diagram in $\mathcal{C}$ is a cartesian square. \begin{displaymath} \itexarray{ X \times Y & \overset{f \times id }{\to} & Y \times Y \\ p_{X} \downarrow & & \downarrow p_{0} \\ X & \underset{f}{\to} & Y \\ } \end{displaymath} Thus the following diagram in $\mathcal{C}$ is a cartesian square. \begin{displaymath} \itexarray{ Z & \overset{u_{0}}{\to} & Y^I \\ r \downarrow & & \downarrow p_{0} \circ e \\ X & \underset{f}{\to} & Y \\ } \end{displaymath} 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. \end{proof} \begin{remark} \label{}\hypertarget{}{} 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. \end{remark} \begin{remark} \label{}\hypertarget{}{} By the commutativity of the diagram \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X } \end{displaymath} 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. \end{remark} \hypertarget{ken_browns_lemma}{}\subsection*{{Ken Brown's lemma}}\label{ken_browns_lemma} \begin{prop} \label{KenBrownLemma}\hypertarget{KenBrownLemma}{} 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. \end{prop} \begin{proof} By Proposition 3, there is a commutative diagram \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{w}{\searrow} & \downarrow g \\ & & Y } \end{displaymath} 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. \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X } \end{displaymath} By the commutativity of the diagram \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{w}{\searrow} & \downarrow g \\ & & Y } \end{displaymath} 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 \begin{displaymath} \itexarray{ X & \overset{j}{\to} & Z \\ & \underset{id}{\searrow} & \downarrow r \\ & & X } \end{displaymath} in $\mathcal{C}$, we have that the following diagram in $\mathcal{D}$ commutes. \begin{displaymath} \itexarray{ F(X) & \overset{F(j)}{\to} & F(Z) \\ & \underset{id}{\searrow} & \downarrow F(r) \\ & & F(X) } \end{displaymath} 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. \begin{displaymath} \itexarray{ F(X) & \overset{F(j)}{\to} & F(Z) \\ & \underset{F(w)}{\searrow} & \downarrow F(g) \\ & & F(Y) } \end{displaymath} 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. \end{proof} \begin{remark} \label{}\hypertarget{}{} In other words, $F$ is a [[homotopical functor]]. \end{remark} \begin{remark} \label{}\hypertarget{}{} If $C$ is the full subcategory of fibrant objects in a [[model category]], then this corollary asserts that a [[Quillen adjunction|right Quillen functor]] $G$, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant objects. \end{remark} \begin{remark} \label{}\hypertarget{}{} By the dual nature of [[model categories]], we then get that a [[Quillen adjunction|left Quillen functor]] preserves weak equivalences between cofibrant objects. \end{remark} \hypertarget{computing_a_homotopy_pullback_by_means_of_an_ordinary_pullback}{}\subsection*{{Computing a homotopy pullback by means of an ordinary pullback}}\label{computing_a_homotopy_pullback_by_means_of_an_ordinary_pullback} \begin{cor} \label{}\hypertarget{}{} Let $A \to C \leftarrow B$ be a diagram between fibrant objects in a [[model category]]. Then the ordinary [[pullback]] $A \times_C^h B$ \begin{displaymath} \itexarray{ A \times_C^h B &\to& C^I \\ \downarrow && \downarrow \\ A \times B &\to& C \times C } \end{displaymath} presents the [[homotopy pullback]] of the original diagram. \end{cor} See the section \emph{\href{http://ncatlab.org/nlab/show/homotopy+pullback#ConcreteConstructions}{Concrete constructions}} at \emph{[[homotopy pullback]]} for more details on this. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item 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 $G$. \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Kenneth Brown]], page 4 of \emph{[[BrownAHT|Abstract Homotopy Theory and Generalized sheaf Cohomology]]}, 1973. \end{itemize} A version in the setup of $\infty$-cosmoi is Lemma 2.1.6 in \begin{itemize}% \item Emily Riehl, Dominic Verity, \emph{Fibrations and Yoneda’s lemma in an $\infty$-cosmos}, J. Pure Appl. Alg. \textbf{221}:3 (2017), 499-564 \href{http://arxiv.org/abs/1506.05500}{arxiv/1506.05500} \end{itemize} [[!redirects ken brown's lemma]] [[!redirects Ken Brown's lemma]] \end{document}