\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[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[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 $p \in\Fib$,
\item[MC4] In any commuting square as follows where $i \in\Cof$ and $p \in\Fib$,
\begin{center}
\cdiagram{Model_Liftproblem}
\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
there exist a lift $h: B \to Y$ if either
\begin{itemize}
\begin{itemize}
@ -63,30 +50,7 @@
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:
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:
The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abbreviate 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}.
The fourth axiom actually characterizes the classes of (trivial) fibrations and (trivial) cofibrations. We will abbreviate 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}.
@ -105,11 +69,11 @@ This means that once we choose weak equivalences and fibrations for a category $
\Lemma{model-cats-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.
Let $\cat{C}$ be a model category. Consider the following two diagrams where $P$ is the pushout and pullback respectively.
\cimage{Model_Cats_Pushouts}
\cdiagram{Model_Cats_Pushouts}
\begin{itemize}
\begin{itemize}
\item If $i$ is a (trivial) cofibrations, so is $j$.
\item If $i$ is a (trivial) cofibration, so is $j$.
\item If $p$ is a (trivial) fibrations, so is $q$.
\item If $p$ is a (trivial) fibration, so is $q$.
\end{itemize}
\end{itemize}
}
}
@ -146,8 +110,7 @@ In this thesis we often restrict to $1$-connected spaces. The full subcategory $
\Lemma{topr-no-colimit}{
\Lemma{topr-no-colimit}{
Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams
Let $r > 0$ and $\Top_r$ be the full subcategory of $r$-connected spaces. The diagrams
\cimage[scale=0.5]{Topr_No_Coequalizer}
\cdiagram{Topr_No_Coequalizer}
\cimage[scale=0.5]{Topr_No_Equalizer}
have no coequalizer and respectively no equalizer in $\Top_r$.
have no coequalizer and respectively no equalizer in $\Top_r$.
Joshua Moerman
10 years ago
