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.
\item[MC1] All finite limits and colimits exist in $\cat{C}$.
\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 $p \in\Fib$,
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:
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}.
This means that once we choose the weak equivalences and the fibrations for a category $\cat{C}$, the cofibrations are determined, and vice versa. The classes of fibrations behave nice with respect to pullbacks and dually cofibrations behave nice with pushouts:
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.
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.
\item Cofibrations: the smallest class of maps containing $S^{n-1}\cof D^n$ which is closed under transfinite compositions, pushouts, coproducts and retracts.
In this thesis we often restrict to $1$-connected spaces. The full subcategory $\Top_1$ of $1$-connected spaces satisfies MC2-MC5: the 2-out-of-3 property, retract property and lifting properties hold as we take the \emph{full} subcategory, factorizations exist as the middle space is $1$-connected as well. Both products and coporducts exist. However $\Top_1$ does not have all limits and colimits.
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
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 $p_1\circ H = g$.
For the first diagram, assume it has a coequalizer $S$ in $\Top_r$. Consider $S^{r+1}\in\Top_r$ and its northern and southern hemisphere $D^{r+1}_+, D^{r+1}_-\subset S^{r+1}$. We have a map $I \to S^{r+1}$ sending the interval to the equator, hence sending the interval to both hemispheres. This map induces maps from $S$ as follows
\cimage[scale=0.5]{Topr_No_Coequalizer_1}
The pullback in $\Top$ (not in $\Top_r$!) of this diagram is the circle $S^1$ and we have a map $S \to S^1$. We can also consider the coqualizer of the original diagram in $\Top$, which is also $S^1$ to obtain a map $S^1\to S$. Checking all the diagrams one concludes that $S \iso S^1\not\in\Top_r$.
For the second diagram, assume it has an equalizer $E$ in $\Top_r$ with its map $e: E \to I$. Define two maps $i_0, i_1: \ast\to I$ sending the unique element to $0$ and $1$ respectively. Clearly we get induced maps $j_0, j_1: \ast\to E$ such that $j_0(\ast)\neq j_1(\ast)$. By connectedness, there is a path $p: I \to E$ from $j_0(\ast)$ to $j_1(\ast)$. We then have the following properties $e(p(x))\in\{0, 1\}$, $e(p(0))=0$ and $e(p(1))=1$. These properties contradict continuity of $e \circ p: I \to I$. We conclude that no such $E$ exists in $\Top_r$.
}
Despite the absence of these (co)limits, the category $\Top_1$ still has products (namely cartesian products), sums (wedge products) and a terminal object.
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:
$$ f \text{ is a weak equivalence }\iff f \text{ is a strong equivalence }. $$
Given a model category, we wish to construct a category in which the weak equivalences become actual isomorphisms. From an abstract perspective, this would be a \emph{localization} of categories. To be precise, if we have a category $\cat{C}$ with weak equivalences $\W$, we want a functor $\gamma: \cat{C}\to\Ho(\cat{C})$ such that
\begin{itemize}
\item for every $f \in\W$, the map $\gamma(f)$ is an isomorphism and
\item$\Ho(\cat{C})$ is universal with this property. This means that for every $\psi$ sending weak equivalences to isomorphisms, we get:
For arbitrary categories and classes of weak equivalences, such a localization need not exist. But when we have a model category, we can always construct $\Ho(\cat{C})$.
\Definition{homotopy-category}{
The \Def{homotopy category $\Ho(\cat{C})$} of a model category $\cat{C}$ is defined with
In \cite{dwyer} it is proven that this indeed defines a localization of $\cat{C}$ with respect to $\W$. It is good to note that $\Ho(\Top)$ does not depend on the class of cofibrations or fibrations.
Note that whenever we have a full subcategory $\cat{C'}\subset\cat{C}$, where $\cat{C}$ is a model category, there is a subcategory of the homotopy category: $\Ho(\cat{C'})\subset\Ho(\cat{C})$. There is no need for a model structure on the subcategory.
The category $\Ho(\Top)$ has as objects just topological spaces and homotopy classes between cofibrant replacements (note that every objects is already fibrant). Moreover, if we restrict to the full subcategory of CW complexes, maps are precisely homotopy classes between objects.
Homotopical invariants are often defined as functors on $\Top$. For example we have the $n$-th homotopy group functor $\pi_n: \Top\to\Grp$ and the $n$-th homology group functor $H_n: \Top\to\Ab$. But since they are homotopy invariant, they can be expressed as functors on $\Ho(\Top)$:
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.
We first make an observation. Notice that whenever we have an adjunction $F : \cat{C}\leftadj\cat{D} : G$, finding a lift in the following diagram on the left is equivalent to finding one in the diagram on the right.
\[\xymatrix{
FA \ar[r]\ar[d]& X \ar[d]\\
FB \ar[r]& Y
}\qquad\xymatrix{
A \ar[r]\ar[d]& GX \ar[d]\\
B \ar[r]& GY
}\]
So it should not come as a surprise that adjunctions play an important role in model categories. The useful notion of maps between model categories is the following.
\Definition{quillen-pair}{
An adjunction $F : \cat{C}\leftadj\cat{D} : G$ between model categories is a \Def{Quillen pair} if $F$ preserves cofibrations and $G$ preserves fibrations.
In this case $F$ is the \Def{left Quillen functor} and $G$ is the \Def{right Quillen functor}.
}
Notice that by the lifting properties $(F,G)$ is a Quillen pair if and only if $F$ preserves cofibrations and trivial cofibrations (or dually $G$ preserves fibrations and trivial fibrations). The Quillen pairs are important as they induce functors on the homotopy categories.
\Theorem{quillen-pair-induces-hocat}{
If $(F, G)$ is a Quillen pair, then there an induced adjunction
\[ LF : \Ho(cat{C})\leftadj\Ho(\cat{D}) : RG, \]
where $LF(X)= F(X^{cof})$ and $RG(Y)= G(Y^{fib})$.
}
Such an adjunction between homotopy categories is an equivalence if the unit and counit are isomorphisms in $\Ho(\cat{C})$. This means that the following two maps should be weak equivalences in $\cat{C}$ for all cofibrant $X$ and all fibrant $Y$
\begin{align*}
\unit&: X \to G(F(X)^{fib}) \\
\counit&: F(G(Y)^{cof}) \to Y.
\end{align*}
In this case, such a pair of functors is called a \Def{Quillen equivalence}.
\Example{sset-top-equivalence}{
The geometric realization and singular functor form a Quillen equivalence
\[ |-| : \sSet\leftadj\Top : S(-). \]
}
\section{Homotopy pushouts and pullbacks}
In category theory we know that colimits (and limits) are unique up to isomorphism, and that isomorphic diagrams will have isomorphic colimits (and limits). We would like a similar theory for weak equivalences. Unfortunately the ordinary colimit (or limit) is not homotopically nice. For example consider the following two diagrams, with the obvious maps.
\[\xymatrix{
S^1 \ar[r]\ar[d]& D^2 \\
D^2 &
}\qquad\xymatrix{
S^1 \ar[r]\ar[d]&\ast\\
\ast&
}\]
The diagrams are pointwise weakly equivalent. But the induced map $S^n \to\ast$ on the pushout is clearly not. In this section we will briefly indicate what homotopy pushouts are (and dually we get homotopy pullbacks).
One direct way to obtain a homotopy pushout is by the use of \emph{Reedy categories}\cite{hovey}. In this case the diagram category is endowed with a model structure, which gives a notion of cofibrant diagram. In such diagrams the ordinary pushout is the homotopy pushout.
Consider the following pushout diagram. The if all objects are cofibrant and the map $f$ is a cofibration, then the homotopy pushout is given by the ordinary pushout.
If we combine this lemma with \LemmaRef{htpy-pushout-reedy} we obtain precisely Lemma 5.2.6 in \cite{hovey}. We get similar theorems for the dual case of homotopy pullbacks.
