Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, we will first look at some basic category theory. If one is already familier with these concepts, he or she can skip this section. We will introduce the notions of categories, functors, isomorphims, 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.
\subsection{Categories}
\begin{definition}
A \emph{category}$\cat{C}$ consists of a collection \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}, 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:
\begin{itemize}
\item\emph{(Identity)}
$\id_A \in\Hom{\cat{C}}{A}{A}$ for all $A$ in $\cat{C}$,
@ -19,16 +19,16 @@ Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, we will first
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 $f: A \to B$.
As the notation suggests maps can be thought of as functions, which is also the case in many examples.
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 grouphomomorphisms. We know that the identity function is indeed a grouphomomorphism, and composing two grouphomomorpisms, gives indeed a new grouphomomorphism.
The category $\Ab$ has as objects abelian groups, and the maps between two objects are exactly the grouphomomorphisms. We know that the identity function is indeed a grouphomomorphism, and composing two grouphomomorpisms, gives indeed a new grouphomomorphism.
\end{example}
In fact almost any mathematical structure can be described as a category, we have:$\cat{Ring}$for rings, $\cat{Vect}$ for $\R$-vectorspaces, $\cat{Set_{fin}}$for finite sets, $\cat{Poset}$ for posets, etc. Of course we would also like to express relations between categories, for example every abelian group is also a set. This idea can be formulated by the notion of a functor.
In fact many mathematical structures can be organized in a category, there is a category$\cat{Ring}$of rings and ringhomomorphisms, $\cat{Vect}$ for $\R$-vectorspaces 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}
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:
@ -51,7 +51,7 @@ Given a category $\cat{C}$ and two objects $A, B \in \cat{C}$ we would like to k
$$ f \circ g =\id_B \text{ and } g \circ f =\id_A.$$
\end{definition}
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. the grouphomomorphisms which are both injective and surjective.
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. the grouphomomorphisms which are both injective and surjective.
For example the cyclic group $\Z_4$ and the klein four-group $V_4$ are not isomorphic in $\Ab$, but if we regard only the sets $\Z_4$ and $V_4$, then they are (because there is a bijection). So it is good to note that whether two objects are isomorphic really depends on the category we are working in.
Note that an isomorphism between to categories is now also defined. Two categories $\cat{C}$ and $\cat{D}$ are isomorphic if there are functors $F$ and $G$ such that $ FG =\id_\cat{D}$ and $GF =\id_\cat{C}$.
A chain complex $C$ is a collection of abelian groups $C_n$ together with boundary operators $\del_n: C_{n+1}\to C_n$, such that $\del_n \circ\del_{n+1}=0$. The collections of all such objects will be denoted by $\Ch{\cat{Ab}}$.
A \emph{(non-negative) chain complex}$C$ is a collection of abelian groups $C_n$ together with group homomorphisms $\del_n: C_{n+1}\to C_n$, which we call \emph{boundary homomorphisms}, such that $\del_n \circ\del_{n+1}=0$.
\end{definition}
In other words a chain complex is the following diagram.
$$\cdots\to C_4\to C_3\to C_2\to C_1\to C_0$$
Thus graphically a chain complex $C$ can be depicted by the following diagram:
$$\cdots\to C_4\to C_3\to C_2\to C_1\to C_0.$$
There are many variants to thie notion. For example, there are also unbounded chain complexes with an abelian group for each $n \in\Z$ instead of $\N$. In this thesis we will only need chain complexes in the sense of the definition above. Hence we will simply call them chain complexes, instead of non-negative chain complexes.
Of course we can make this more general by taking for example $R$-modules instead of abelian groups. We will later see which kind of algebraic objects make sense to use in this definition \todo{Ch: Will I discuss ab. cat. ?}. The boundary operators give rise to certain subgroups, because all groups are abelian, subgroups are normal subgroups.
\begin{definition}
Given a chain complex $C$ we define the following subgroups:
\begin{itemize}
\item$Z_n(C)=\ker(\del: C_n \to C_{n-1})\nsubgrp C_n$, and
@ -24,15 +26,18 @@ Of course we can make this more general by taking for example $R$-modules instea
It follows from $\del_n \circ\del_{n+1}=0$ that $\im(\del: C_{n+1}\to C_n)$ is a subset of $\ker(\del: C_n \to C_{n-1})$. Those are exactly the abelian groups $B_n(C)$ and $Z_n(C)$, so $ B_n(C)\nsubgrp Z_n(C)$.
\end{proof}
\begin{definition}
Given a chain complex $C$ we define the \emph{$n$-th homology group}$H_n(C)$:
Given a chain complex $C$ we define the \emph{$n$-th homology group}$H_n(C)$ for each $n \in\N$ as:
$$ H_n(C)= Z_n(C)/ B_n(C).$$
\end{definition}
\todo{CC: Chain maps}
\todo{CC: $H_n$ as a functor}
\subsection{The singular chain complex}
In order to see why we are interested in the construction of homology groups, we will look at an example from algebraic topology. We will see that homology gives a nice invariant for spaces. So we will form a chain complex from a topological space $X$. In order to do so, we first need some more notions.
\begin{definition}
The topological space $\Delta^n$ is called the \emph{topological $n$-simplex} and is defined as:
$$\Delta^n =\{x \in\R^{n+1}\I x_i \geq0\text{ and } x_0+\ldots+ x_n =1\}.$$
In this thesis we will look at a correspondence which was discovered by A. Dold and D. Kan independently, hence it is called the \emph{Dold-Kan correspondence}. Abstractly it is the following equivalence of categories:
$$\Ch{\Ab}\simeq\sAb$$
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. In particular this correspondence also gives a isomorphism between homology groups (on the left hand side) and homotopy groups (on the right hand side). A bit more precise:
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:
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N$$
where $N: \sAb\to\Ch{\Ab}$ is one half of the equivalence.
Joshua Moerman
11 years ago