Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, let us begin by recalling some basic category theory. The reader who is already familiar with these concepts, is invited to skip this section. We will introduce the notions of categories, functors, isomorphisms, natural transformations, equivalences, adjunctions and the Yoneda lemma.
Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, let us begin by recalling some basic category theory. The reader who is already familiar with these concepts, is invited to skip this section. We will introduce the notions of categories, functors, isomorphisms, natural transformations, equivalences, adjunctions and the Yoneda lemma.
We will briefly define categories an functors to fix the notation. We will not provide many examples or intuition in these concepts. For a more elaborated exposition one should have a read in \cite{awodey} or \cite{maclane}. The more complicated definitions will be discussed in a bit more detail.
\subsection{Categories}
\subsection{Categories}
\todo{CT: Where to start...?}
\begin{definition}
\begin{definition}
A \emph{category}$\cat{C}$ consists of a collection of \emph{objects}, and for each two objects $A$ and $B$ in $\cat{C}$there is a set of \emph{maps} from $A$ to $B$, notated as $\Hom{\cat{C}}{A}{B}$, such that
A \emph{category}$\cat{C}$ consists of a collection of \emph{objects}, a set of \emph{maps}$\Hom{\cat{C}}{A}{B}$ for each two objects $A, B \in\cat{C}$and a binary operator named \emph{composition}$-\circ-:\Hom{\cat{C}}{B}{C}\times\Hom{\cat{C}}{A}{B}$ such that
\begin{itemize}
\begin{itemize}
\item\emph{(Identity)}
\item$\circ$ is associative, i.e. $h \circ(g \circ f)=(h \circ g)\circ f$ and
$\id_A \in\Hom{\cat{C}}{A}{A}$ for all $A$ in $\cat{C}$,
\item there exists an neutral element $\id_A \in\Hom{\cat{C}}{A}{A}$ for all $A$ in $\cat{C}$, i.e.
\item\emph{(Composition)}
$$\id_B \circ f = f = f \circ\id_A. $$
for any $f \in\Hom{\cat{C}}{A}{B}$ and $g \in\Hom{\cat{C}}{B}{C}$ we have $g \circ f \in\Hom{\cat{C}}{A}{C}$,
\item\emph{(Associativity)}
$h \circ(g \circ f)=(h \circ g)\circ f$, and
\item\emph{(Identity law)}
$\id_B \circ f = f = f \circ\id_A$ for all $f \in\Hom{\cat{C}}{A}{B}$.
\end{itemize}
\end{itemize}
\end{definition}
\end{definition}
Note that the collection of objects may be a proper class instead of a set, however we will notate $A \in\cat{C}$ if $A$ is an object of $\cat{C}$. And instead of writing $f \in\Hom{\cat{C}}{A}{B}$, we write $f: A \to B$.
Instead of writing $f \in\Hom{\cat{C}}{A}{B}$ we write $f: A \to B$, as many categories have functions as maps. There is a category $\Set$ of sets with functions, a category $\Ab$ of abelian groups with group homomorphisms, a category $\Top$ of topological spaces and continuous maps, and many more.
As the notation suggests maps should be thought of as functions. Which is also the case in many categories, as objects are often sets with an additional structure and maps are functions ``preserving that structure''.
\begin{example}
The category $\Set$ has as its objects sets, and as maps it has ordinary functions. Of course we then have the identity function $\id_X(x)= x$ and composition as usual.
\end{example}
\begin{example}
The category $\Ab$ has as objects abelian groups, and the maps between two objects are exactly the group homomorphisms. We know that the identity function is indeed a group homomorphism, and composing two group homomorpisms, gives indeed a new group homomorphism.
\end{example}
In fact many mathematical structures can be organized in a category, there is a category $\cat{Ring}$ of rings and ring homomorphisms, $\cat{Vect}$ for $\R$-vector spaces and $\R$-linear maps, $\cat{Set_{fin}}$ of finite sets, $\Top$ of topological spaces and continuous functions, etc. Of course we would also like to express relations between categories. For example every abelian group is also a set, and a group homomorphism is also a function. This idea can be formalized by the notion of a functor.
\begin{definition}
\begin{definition}
A \emph{functor}$F$ from a category $\cat{C}$ and to a category $\cat{D}$ consists of a function $F_0$ from the objects of $\cat{C}$ to the objects of $\cat{D}$ and a function $F_1$ from maps in $\cat{C}$ to maps in $\cat{D}$, such that
A \emph{functor}$F$ from a category $\cat{C}$ and to a category $\cat{D}$ consists of a function $F_0$ from the objects of $\cat{C}$ to the objects of $\cat{D}$ and a function $F_1$ from maps in $\cat{C}$ to maps in $\cat{D}$, such that
@ -40,7 +25,8 @@ In fact many mathematical structures can be organized in a category, there is a
\end{itemize}
\end{itemize}
We normally do not write the index of $F_0$ or $F_1$, instead we write $F$ for both functions.
We normally do not write the index of $F_0$ or $F_1$, instead we write $F$ for both functions.
\end{definition}
\end{definition}
\todo{CT: contravariant functor}
For a category $\cat{C}$ we denote the \emph{opposite} category by $\cat{C}^{op}$. The opposite category consists of the same objects, but the maps and composition are reversed. A \emph{contravariant functor}$F$ from $\cat{C}$ to $\cat{D}$ is a functor $F: \cat{C}^{op}\to\cat{D}$.
Note that the composition of two functors is again a functor, and that we always have an identity functor, sending each object to itself and each map to itself. This gives rise to a category $\cat{Cat}$ of \emph{small} categories. Note that we need some kind of \emph{smallness} to avoid set-theoretical issues, because we require the collection of maps between objects to be a set, whereas the collection of objects is not necessarily a set. However we will not be interested in these set-theoretic issues, and hence skip the definition of small.
Note that the composition of two functors is again a functor, and that we always have an identity functor, sending each object to itself and each map to itself. This gives rise to a category $\cat{Cat}$ of \emph{small} categories. Note that we need some kind of \emph{smallness} to avoid set-theoretical issues, because we require the collection of maps between objects to be a set, whereas the collection of objects is not necessarily a set. However we will not be interested in these set-theoretic issues, and hence skip the definition of small.
@ -77,16 +63,14 @@ Note that an isomorphism between to categories is now also defined. Two categori
commutes for any map $f: c \to c'$ and any objects $c, c' \in\cat{C}$.
commutes for any map $f: c \to c'$ and any objects $c, c' \in\cat{C}$.
\end{definition}
\end{definition}
\begin{lemma}
For any two categories $\cat{C}$ and $\cat{D}$ we can form a category with functors $F: \cat{C}\to\cat{D}$ as objects and natural transformations as maps. This category is called the \emph{functor category} and is denoted by $\cat{D}^\cat{C}$.
For any two categories $\cat{C}$ and $\cat{D}$ we can form a category with functors $F: \cat{C}\to\cat{D}$ as objects and natural transformations as maps. This category is called the \emph{functor category} and is denoted by $\cat{D}^\cat{C}$.
\end{lemma}
\begin{proof}
We refer to Mac Lane \cite{maclane} or Awodey \cite{awodey}.
\end{proof}
This now also gives a notion of isomorphisms between functors. It can be easily seen that an isomorphism between two functors is a natural transformation which is an isomorphism pointwise. Such a natural transformation is called a \emph{natural isomorphism}.
This now also gives a notion of isomorphisms between functors. It can be easily seen that an isomorphism between two functors is a natural transformation which is an isomorphism pointwise. Such a natural transformation is called a \emph{natural isomorphism}.
\todo{CT: Hom-functor}
For any category $\cat{C}$ we can define the $\mathbf{Hom}$-functor. It assigns to two objects in $\cat{C}$ the set of maps between them, i.e.
We will show that it indeed gives a functor in the first argument, a similar proof works for the second argument. Let $f: A' \to A$ be a map in $\cat{C}$ and $g \in\Hom{\cat{C}}{A}{B}$, then $g \circ f \in\Hom{\cat{C}}{A'}{B}$. Hence the assignment $g \mapsto g \circ f$ is a map from $\Hom{\cat{C}}{A}{B}$ to $\Hom{\cat{C}}{A'}{B}$. Note that the direction of the map if reversed. Using associativity and identity it is easily checked that this assignment is functorial.
\subsection{Equivalence}
\subsection{Equivalence}
Recall that an \emph{isomorphism} between categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C}\to\cat{D}$ and $G: \cat{D}\to\cat{C}$ such that
Recall that an \emph{isomorphism} between categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C}\to\cat{D}$ and $G: \cat{D}\to\cat{C}$ such that
@ -165,7 +165,7 @@ The above construction defines a functor $C: \Top \to \Ch{\Ab}$ (we will not pro
With Figure~\ref{fig:singular_homology} we indicate what $H^\text{sing}_1$ measures. In the first space $X$ we see a $1$-cycle $\sigma_1-\sigma_2+\sigma_3$ which is also a boundary, because we can define a map $\tau: \Delta^2\to X$ such that $\del(\tau)=\sigma_1-\sigma_2+\sigma_3$, hence we conclude that $0=[\sigma_1-\sigma_2+\sigma_3]\in H^\text{sing}_1(X)$. So this $1$-cycle is not interesting in homology. In the space $X'$ however there is a hole, which prevents a $2$-simplex like $\tau$ te exist, hence $0\neq[\sigma_1-\sigma_2+\sigma_3]\in H^\text{sing}_1(X')$. This example shows that in some sense this functor is capable of detecting holes in a space.
With Figure~\ref{fig:singular_homology} we indicate what $H^\text{sing}_1$ measures. In the first space $X$ we see a $1$-cycle $\sigma_1-\sigma_2+\sigma_3$ which is also a boundary, because we can define a map $\tau: \Delta^2\to X$ such that $\del(\tau)=\sigma_1-\sigma_2+\sigma_3$, hence we conclude that $0=[\sigma_1-\sigma_2+\sigma_3]\in H^\text{sing}_1(X)$. So this $1$-cycle is not interesting in homology. In the space $X'$ however there is a hole, which prevents a $2$-simplex like $\tau$ te exist, hence $0\neq[\sigma_1-\sigma_2+\sigma_3]\in H^\text{sing}_1(X')$. This example shows that in some sense this functor is capable of detecting holes in a space.
@ -180,9 +180,9 @@ With Figure~\ref{fig:singular_homology} we indicate what $H^\text{sing}_1$ measu
\label{fig:singular_homology}
\label{fig:singular_homology}
\end{figure}
\end{figure}
A direct consequence of being a functor is that homeomorphic spaces have isomorphic singular homology groups. There is even a stronger statement which tells us that homotopy equivalent spaces have isomorphic homology groups. So from a homotopy perspective this construction is nice.
A direct consequence of being a functor is that homeomorphic spaces have isomorphic singular homology groups. There is even a stronger statement which tells us that homotopy equivalent spaces have isomorphic homology groups. So if one is interested in homotopy of a space, then homology already gives some information.
In the remainder of this section we will give the homology groups of some basic spaces. It is hard to calculate these results from the definition above, so generally one proves these results by using theorems from algebraic topology or homological algebra, which are beyond the scope of this thesis. So we simply give these results.
In the remainder of this section we will give the homology groups of some basic spaces. For most spaces it is hard to calculate the homology groups from the definitions above. One generally proves these results by using theorems from algebraic topology or homological algebra, which are beyond the scope of this thesis. The first example can be calculated from the definitions above, however the proof is not included as the example is only included as a motivation.
\begin{example}
\begin{example}
The homology of the one-point space $\ast$ is given by:
The homology of the one-point space $\ast$ is given by:
In this thesis we will look at a correspondence which was discovered by A.~Dold \cite{dold} and D.~Kan \cite{kan} independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories:
In this thesis we will study the Dold-Kan correspondence, a celebrated result which belongs to the field of homological algebra or simplicial homotopy theory. Abstractly, one version of the theorem states that there is an equivalence of categories
$$\Ch{\Ab}\simeq\sAb$$
$$ K: \Ch{\Ab}\simeq\sAb :N, $$
It is interesting because objects on the left hand side are considered to be algebraic of nature, whereas objects on the right are more topological. Objects of either of these categories have important invariants. A more refined statement of this equivalence tells us that there is an isomorphism between homology groups (on the left hand side) and homotopy groups (on the right hand side). A bit more precise:
where $\Ch{\Ab}$ is the category of chain complexes and $\sAb$ is the category of simplicial abelian groups. This theorem was discovered by A.~Dold \cite{dold} and D.~Kan \cite{kan} independently in 1957. Objects of either of these categories have important invariants. A more refined statement of this equivalence tells us that there is an natural isomorphism between homology groups of chain complexes and homotopy groups of simplicial abelian groups. A bit more precise:
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N$$
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N. $$
where $N: \sAb\to\Ch{\Ab}$ is one half of the equivalence.
In the first section some definitions from category theory are given, because we will need them later on. Then in the second section we will discuss the first category involved in the correspondence, $\Ch{\Ab}$, the category of chain complexes. The third section then continues with the second category involved, $\sAb$, especially for this section we will need category theory. Then we will look at the correspondence itself.
In the first section some definitions from category theory are recalled, which are especially important in Sections~\ref{sec:Simplicial Abelian Groups} and \ref{sec:Constructions}. In Section~\ref{sec:Chain Complexes} we will discuss the category of chain complexes and in the end of this section a motivation from algebraic topology will be given for these objects. Section~\ref{sec:Simplicial Abelian Groups} then continues with the second category involved, $\sAb$. This section start with a slightly more general notion and it will be illustrated to have a geometrical meaning. In Section~\ref{sec:Constructions} the correspondence will be defined and proven. In the last section (Section~\ref{sec:Homotopy}) the refined statement will be proven and in the end some more general notes about topology and homotopy will be given, justifying once more the beauty of this correspondence.
Joshua Moerman
12 years ago