As this thesis considers different categories, each with its own homotopy theory, it is natural to use Quillen's formalism of model categories. Not only gives this the right definition of the associated homotopy category, it also gives existence of lifts and lifts of homotopies.
\newcommand{\W}{\mathfrak{W}}
\newcommand{\Fib}{\mathfrak{Fib}}
\newcommand{\Cof}{\mathfrak{Cof}}
@ -87,6 +89,8 @@ This means that once we choose weak equivalences and fibrations for a category $
\titem Small object argument
}
Of course the most important model category is the one of topological spaces. We will be interested in the standard model structure on topological spaces, which has weak homotopy equivalences as weak equivalences. Equally important is the model category of simplicial sets.
\Example{top-model-structure}{
The category $\Top$ of topological spaces admits a model structure as follows.
\begin{itemize}
@ -115,7 +119,119 @@ In this thesis we often restrict to $1$-connected spaces. The full subcategory $
have no coequalizer and respectively no equalizer in $\Top_r$.
}
\todo{Define homotopy category}
\subsection{Homotopies}
So far we have only seen equivalences between objects of the category. We can, however, also define homotopy relations between maps (as we are used to in $\Top$). There are two such construction, which will coincide on nice objects. We will only state the definitions and important results. One can find proofs of these results in \cite{dwyer}. Throughout this section we silently work with a fixed model category $\cat{C}$.
\newcommand{\cylobj}[1]{Cyl_{#1}}
\newcommand{\pathobj}[1]{Path_{#1}}
\newcommand{\lhtpy}{{\sim^{l}}}
\newcommand{\rhtpy}{{\sim^{r}}}
\newcommand{\lhtpycl}{{\pi^l}}
\newcommand{\rhtpycl}{{\pi^r}}
\Definition{cylinder_object}{
A \Def{cylinder object} for an object $A$ is an object $\cylobj{A}$ together with maps:
$$ A \coprod A \tot{i}\cylobj{A}\we^{p} A, $$
which factors the folding map $\id_A +\id_A: A \coprod A \to A$ (note that we use MC1 here). The cylinder object is called
\begin{itemize}
\item\emph{good} if $i$ is a cofibration and
\item\emph{very good} if in addition $p$ is a fibration.
\end{itemize}
}
\Notation{cylinder_maps}{
The map $i$ consists of two factors, which we will denote $i_0$ and $i_1$.
}
Note that we do not require cylinder objects to be functorial. There can also be more than one cylinder object for $A$. Cylinder objects can now be used to define left homotopies.
\Definition{left_homotopy}{
Two maps $f, g: A \to X$ are \Def{left homotopic} if there exists a cylinder object $\cylobj{A}$ and a map $H: \cylobj{A}\to X$ such that $H \circ i_0= f$ and $H \circ i_1= g$.
We will call $H$ a \Def{left homotopy} for $f$ to $g$ and write $f \lhtpy r$. Moreover, the homotopy is called good (resp. very good) is the cylinder object is good (resp. very good).
}
Note that the relation need not be transitive: consider $f \lhtpy g$ and $g \lhtpy h$, then these homotopies may be defined on different cylinder objects and in general we cannot relate the cylinder objects. However for nice domains $\lhtpy$ will be an equivalence relation.
\Lemma{left_homotopy_eqrel}{
If $A$ is cofibrant, then $\lhtpy$ is an equivalence relation on $\Hom_\cat{C}(A, X)$.
}
\Definition{left_homotopy_classes}{
We will denote the set of \Def{left homotopy classes} as
$$\lhtpycl(A, X)=\Hom_\cat{C}(A, X)/\lhtpy', $$
where $\lhtpy'$ is the equivalence relation generated by $\lhtpy$.
}
\Lemma{left_homotopy_properties}{
We have the following properties
\begin{itemize}
\item If $A$ is cofibrant and $p: X \to Y$ a trivial fibration, then
\item If $X$ is fibrant, $f \lhtpy g: B \to X$ and we have a map $h: A \to B$, then
$$ fh \lhtpy gh. $$
\end{itemize}
}
Of course there is a completely dual definition of right homotopy, in terms of path objects. All of the above also applies (but in a dual way).
\Definition{path_object}{
A \Def{path object} for an object $X$ is an object $\pathobj{X}$ together with maps:
$$ X \we^{i}\pathobj{X}\tot{p} X \times X, $$
which factors the diagonal map $(\id_X, \id_X): X \to X \times X$. The path object is called
\begin{itemize}
\item\emph{good} if $p$ is a fibration and
\item\emph{very good} if in addition $i$ is a cofibration.
\end{itemize}
}
\Notation{cylinder_maps}{
The map $p$ consists of two factors, which we will denote $p_0$ and $p_1$.
}
\Definition{right_homotopy}{
Two maps $f, g: A \to X$ are \Def{right homotopic} if there exists a path object $\pathobj{X}$ and a map $H: A \to\pathobj{X}$ such that $p_0\circ H = f$ and $p_1\circ H = g$.
We will call $H$ a \Def{right homotopy} for $f$ to $g$ and write $f \rhtpy r$. Moreover, the homotopy is called good (resp. very good) is the path object is good (resp. very good).
}
\Lemma{right_homotopy_eqrel}{
If $X$ is fibrant, then $\rhtpy$ is an equivalence relation on $\Hom_\cat{C}(A, X)$.
}
\Definition{right_homotopy_classes}{
We will denote the set of \Def{left homotopy classes} as
$$\rhtpycl(A, X)=\Hom_\cat{C}(A, X)/\rhtpy', $$
where $\rhtpy'$ is the equivalence relation generated by $\rhtpy$.
}
\Lemma{right_homotopy_properties}{
We have the following properties
\begin{itemize}
\item If $X$ is fibrant and $i: A \to B$ a trivial cofibration, then
\item If $A$ is cofibrant, $f \rhtpy g: A \to X$ and we have a map $h: X \to Y$, then
$$ hf \rhtpy hg. $$
\end{itemize}
}
The two notions (left resp. right homotopy) agree on nice objects. Hence in this case we can speak of homotopic maps.
\Lemma{homotopy}{
Let $f, g: A \to X$ be two maps and $A$ cofibrant and $X$ fibrant, then
$$ f \lhtpy g \iff f \rhtpy g. $$
}
\Definition{homotopy}{
In the above case we say that $f$ and $g$ are \Def{homotopic}, this is denoted by $f \sim g$. Furthermore we can define the set of homotopy classes as:
$$[A, X]=\Hom_\cat{C}(A, X)/\sim. $$
A map $f: A \to X$ between cofibrant-fibrant objects is said to have a \Def{homotopy inverse} if there exists $g: X \to A$ such that $fg \sim\id$ and $gf \sim\id$. We will also call $f$ a \Def{strong homotopy equivalence}.
}
\Lemma{weak_strong_homotopy}{
Let $f: A \to B$ be a map between cofibrant-fibrant objects, then:
$$ f \text{ is a weak equivalence }\iff f \text{ is a strong equivalence }. $$
}
\subsection{Quillen pairs}
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.
Joshua Moerman
10 years ago