Master thesis on Rational Homotopy Theory
https://github.com/Jaxan/Rational-Homotopy-Theory
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101 lines
3.4 KiB
101 lines
3.4 KiB
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\section{Model categories}
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\label{sec:model_cats}
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\newcommand{\W}{\mathfrak{W}}
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\newcommand{\Fib}{\mathfrak{Fib}}
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\newcommand{\Cof}{\mathfrak{Cof}}
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\begin{definition}
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A \emph{(closed) model category} is a category $\cat{C}$ together with three subcategories:
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\begin{itemize}
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\item the class of weak equivalences $\W$,
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\item the class of fibrations $\Fib$ and
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\item the class of cofibrations $\Cof$,
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\end{itemize}
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such that the following five axioms hold:
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\begin{itemize}
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\item[MC1] All finite limits and colimits exist in $\cat{C}$.
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\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.
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\item[MC3] All three classes of maps are closed under retracts\todo{Either draw the diagram or define a retract earlier}.
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\item[MC4] In any commuting square as follows where $i \in \Cof$ and $p \in \Fib$,
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\begin{center}
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\begin{tikzpicture}
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\matrix (m) [matrix of math nodes]{
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A & X \\
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B & Y \\
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};
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\path[->] (m-1-1) edge (m-1-2);
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\path[->] (m-2-1) edge (m-2-2);
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\path[->] (m-1-1) edge node[auto] {$i$} (m-2-1);
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\path[->] (m-1-2) edge node[auto] {$p$} (m-2-2);
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\end{tikzpicture}
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\end{center}
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there exist a lift $h: B \to Y$ if either
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\begin{itemize}
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\item[a)] $i \in \W$ or
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\item[b)] $p \in \W$.
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\end{itemize}
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\item[MC5] Any map $f : A \to B$ can be factored in two ways:
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\begin{itemize}
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\item[a)] as $f = pi$, where $i \in \Cof \cap \W$ and $p \in \Fib$ and
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\item[b)] as $f = pi$, where $i \in \Cof$ and $p \in \Fib \cap \W$.
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\end{itemize}
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\end{itemize}
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\end{definition}
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\begin{notation} For brevity
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\begin{itemize}
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\item we write $f: A \fib B$ when $f$ is a fibration,
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\item we write $f: A \cof B$ when $f$ is a cofibration and
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\item we write $f: A \we B$ when $f$ is a weak equivalence.
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\end{itemize}
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\end{notation}
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\begin{definition}
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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.
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\end{definition}
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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:
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\begin{center}
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\begin{tikzpicture}
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\matrix (m) [matrix of math nodes]{
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\cat{0} & & X \\
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& X^{cof} & \\
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};
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\path[->] (m-1-1) edge (m-1-3);
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\path[right hook->] (m-1-1) edge (m-2-2);
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\path[->>] (m-2-2) edge node[auto] {$ \simeq $} (m-1-3);
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\end{tikzpicture}\quad
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\begin{tikzpicture}
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\matrix (m) [matrix of math nodes]{
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X & & \cat{1} \\
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& X^{fib} & \\
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};
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\path[->] (m-1-1) edge (m-1-3);
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\path[right hook->] (m-1-1) edge node[auto] {$ \simeq $} (m-2-2);
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\path[->>] (m-2-2) edge (m-1-3);
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\end{tikzpicture}
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\end{center}
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\TODO{Maybe some basic propositions (refer to Dwyer \& Spalinski):
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\titem Over/under category (or simply pointed objects)
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\titem If a map has LLP/RLP wrt fib/cof, it is a cof/fib
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\titem Fibs are preserved under pullbacks/limits
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\titem Cofibrantly generated mod. cats.
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\titem Small object argument
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}
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\todo{Define homotopy category}
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\subsection{Quillen pairs}
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In order to relate model categories and their associated homotopy categories we need a notion of maps between them. We want the maps such that they induce maps on the homotopy categories.
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\todo{Definition etc}
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