From 932be55ab2ad8007e3482b68dfd33f4689191452 Mon Sep 17 00:00:00 2001 From: Joshua Moerman Date: Wed, 8 Oct 2014 21:14:30 +0200 Subject: [PATCH] Adds more about model cats --- thesis/images/Model_Cats_Pushouts.png | Bin 0 -> 30538 bytes thesis/notes/Model_Categories.tex | 99 +++++++++++++++++--------- thesis/references.bib | 8 +++ thesis/thesis.tex | 1 - 4 files changed, 72 insertions(+), 36 deletions(-) create mode 100644 thesis/images/Model_Cats_Pushouts.png diff --git a/thesis/images/Model_Cats_Pushouts.png b/thesis/images/Model_Cats_Pushouts.png new file mode 100644 index 0000000000000000000000000000000000000000..19c50a8ac9a137bbdf49a26164d144fa1bab4928 GIT binary patch literal 30538 zcmce-WmH>F)IN&41&UK7xRc=S!M#Nar4S$#iU&)floksRptwtMD^7tzi@RHa;+jHn zC{FqFd*A0hMi``C!@t)-`qzd}Sn$V;zA z_lNqiiO%VnxwopjtE+>X_oIk`VQ4I<2NDz!Wi^r(yP5xCgMq<{p{1t$%>UPZHX@7Z zc}sil-r<61x-OS0D?b~d8Vx&ZytuPHRcf9;uJShDM_Sr8KC?F3YJR@Ay|_&*{_4kk 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ef58349..b3c921b 100644 --- a/thesis/notes/Model_Categories.tex +++ b/thesis/notes/Model_Categories.tex @@ -7,11 +7,11 @@ \newcommand{\Cof}{\mathfrak{Cof}} \begin{definition} - A \emph{(closed) model category} is a category $\cat{C}$ together with three subcategories: + A \Def{model category} is a category $\cat{C}$ together with three subcategories: \begin{itemize} - \item the class of weak equivalences $\W$, - \item the class of fibrations $\Fib$ and - \item the class of cofibrations $\Cof$, + \item the class of \Def{weak equivalences} $\W$, + \item the class of \Def{fibrations} $\Fib$ and + \item the class of \Def{cofibrations} $\Cof$, \end{itemize} such that the following five axioms hold: \begin{itemize} @@ -19,45 +19,47 @@ \item[MC2] If $f$, $g$ and $fg$ are maps such that two of them are weak equivalences, then so it the third. This is called the \emph{2-out-of-3} property. \item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}. \item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$, - \begin{center} - \begin{tikzpicture} - \matrix (m) [matrix of math nodes]{ - A & X \\ - B & Y \\ - }; - - \path[->] (m-1-1) edge (m-1-2); - \path[->] (m-2-1) edge (m-2-2); - \path[->] (m-1-1) edge node[auto] {$i$} (m-2-1); - \path[->] (m-1-2) edge node[auto] {$p$} (m-2-2); - - \end{tikzpicture} - \end{center} - - there exist a lift $h: B \to Y$ if either - \begin{itemize} - \item[a)] $i \in \W$ or - \item[b)] $p \in \W$. - \end{itemize} + \begin{center} + \begin{tikzpicture} + \matrix (m) [matrix of math nodes]{ + A & X \\ + B & Y \\ + }; + + \path[->] (m-1-1) edge (m-1-2); + \path[->] (m-2-1) edge (m-2-2); + \path[->] (m-1-1) edge node[auto] {$i$} (m-2-1); + \path[->] (m-1-2) edge node[auto] {$p$} (m-2-2); + + \end{tikzpicture} + \end{center} + + there exist a lift $h: B \to Y$ if either + \begin{itemize} + \item[a)] $i \in \W$ or + \item[b)] $p \in \W$. + \end{itemize} \item[MC5] Any map $f : A \to B$ can be factored in two ways: - \begin{itemize} - \item[a)] as $f = pi$, where $i \in \Cof \cap \W$ and $p \in \Fib$ and - \item[b)] as $f = pi$, where $i \in \Cof$ and $p \in \Fib \cap \W$. - \end{itemize} + \begin{itemize} + \item[a)] as $f = pi$, where $i \in \Cof \cap \W$ and $p \in \Fib$ and + \item[b)] as $f = pi$, where $i \in \Cof$ and $p \in \Fib \cap \W$. + \end{itemize} \end{itemize} \end{definition} -\begin{notation} For brevity +\Notation{model-cats-arrows}{ + For brevity \begin{itemize} \item we write $f: A \fib B$ when $f$ is a fibration, \item we write $f: A \cof B$ when $f$ is a cofibration and \item we write $f: A \we B$ when $f$ is a weak equivalence. \end{itemize} -\end{notation} + Furthermore a map which is a fibration and a weak equivalence is callend a \Def{trivial fibration}, similarly we have \Def{trivial cofibration}. +} -\begin{definition} - An object $A$ in a model category $\cat{C}$ will be called \emph{fibrant} if $A \to \cat{1}$ is a fibration and \emph{cofibrant} if $\cat{0} \to A$ is a cofibration. -\end{definition} +\Definition{model-cats-fibrant-cofibrant}{ + An object $A$ in a model category $\cat{C}$ will be called \Def{fibrant} if $A \to \cat{1}$ is a fibration and \Def{cofibrant} if $\cat{0} \to A$ is a cofibration. +} Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equivalent fibrant object $X^{fib}$ and by [MC5b] by a weakly equivalent cofibrant object $X^{cof}$, as seen in the following diagram: @@ -86,10 +88,37 @@ Note that axiom [MC5a] allows us to replace any object $X$ with a weakly equival \end{tikzpicture} \end{center} +The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abreviate left lifting property with LLP and right lifting property with RLP. We will not prove these statements, but only expose them because we use them throughout this thesis. One can find proofs in \cite{dwyer, may}. + +\Lemma{model-cats-characterization}{ + Let $\cat{C}$ be a model category. + \begin{itemize} + \item The cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. trivial fibrations. + \item The fibrations in $\cat{C}$ are the maps with a RLP w.r.t. trivial cofibrations. + \item The trivial cofibrations in $\cat{C}$ are the maps with a LLP w.r.t. fibrations. + \item The trivial fibrations in $\cat{C}$ are the maps with a RLP w.r.t. cofibrations. + \end{itemize} +} + +This means that once we choose weak equivalences and fibrations for a category $\cat{C}$, the third class is determined, and vice versa. The classes of fibrations behave nice with respect to pullbacks and dually cofibrations behave nice with pushouts: + +\Lemma{model-cats-pushouts}{ + Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively. + + \cimage{Model_Cats_Pushouts} + + \begin{itemize} + \item If $i$ is a (trivial) cofibrations, so is $j$. + \item If $p$ is a (trivial) fibrations, so is $q$. + \end{itemize} +} + +\Lemma{model-cats-coproducts}{ + Let $\cat{C}$ be a model category. Let $f: A \cof B$ and $g:A' \cof B'$ be two (trivial) cofibrations, then the induced map of the coproducts $f+g: A+A' \to B+B'$ is also a (trivial) cofibration. Dually: the product of two (trivial) fibrations is a (trivial) fibration. +} + \TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski): \titem Over/under category (or simply pointed objects) -\titem If a map has LLP/RLP w.r.t. fib/cof, it is a cof/fib -\titem Fibs are preserved under pullbacks/limits \titem Cofibrantly generated mod. cats. \titem Small object argument } diff --git a/thesis/references.bib b/thesis/references.bib index 4c62975..9a698b8 100644 --- a/thesis/references.bib +++ b/thesis/references.bib @@ -53,6 +53,14 @@ publisher={Providence, RI; American Mathematical Society; 1999} } +@book{hovey, + title={Model categories}, + author={Hovey, Mark}, + number={63}, + year={2007}, + publisher={American Mathematical Soc.} +} + @inproceedings{kreck, title={A quick proof of the rational Hurewicz theorem and a computation of the rational homotopy groups of spheres}, author={Klaus, Stephan and Kreck, Matthias}, diff --git a/thesis/thesis.tex b/thesis/thesis.tex index aa4b5fa..5e7317a 100644 --- a/thesis/thesis.tex +++ b/thesis/thesis.tex @@ -42,7 +42,6 @@ Some general notation: \todo{leave this out, or define somewhere else?} % \listoftodos -\nocite{*} \bibliographystyle{alpha} \bibliography{references}