@ -5,14 +5,14 @@ Before we will introduce the two categories $\Ch{\Ab}$ and $\sAb$, let us begin
\subsection{Categories}
\todo{CT: Where to start...?}
\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}, 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 $\cat{C}$,
\item\emph{(Composition)}
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)}
$f\circ(g \circh)=(f\circ g)\circh$, and
$h\circ(g \circf)=(h\circ g)\circf$, 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}
@ -32,7 +32,7 @@ As the notation suggests maps should be thought of as functions. Which is also t
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}
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
\begin{itemize}
\item for $f: A \to B$, we have $F_1(f): F_0(A)\to F_0(B)$,
\item$F_1(\id_A)=\id_{F_0(A)}$ and
@ -48,11 +48,11 @@ Note that the composition of two functors is again a functor, and that we always
Given a category $\cat{C}$ and two objects $A, B \in\cat{C}$ we would like to know when those objects are regarded as the same, according to the category. This will be the case when there is an isomorphism between the two.
\begin{definition}
A map $f: A \to B$ in a category $\cat{C}$ is an \emph{isomorphism} if there is a map $g: B \to A$ such that:
$$ f \circ g =\id_B \text{and} g \circ f =\id_A.$$
A map $f: A \to B$ in a category $\cat{C}$ is an \emph{isomorphism} if there is a map $g: B \to A$ such that
$$ f \circ g =\id_B \quad\text{and}\quad g \circ f =\id_A.$$
\end{definition}
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, ie. the group homomorphisms which are both injective and surjective.
Isomorphisms in $\Ab$ are exactly the isomorphisms which we know, i.e. the group homomorphisms 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 $GF =\id_\cat{C}$.
@ -69,7 +69,7 @@ Note that an isomorphism between to categories is now also defined. Two categori
};
\path[->]
(m-1-1) edge node[auto] {$\phi_c $} (m-1-2)
(m-2-1) edge node[auto] {$\phi_c' $} (m-2-2)
(m-2-1) edge node[auto] {$\phi_{c'}$} (m-2-2)
(m-1-1) edge node[auto] {$ F(f)$} (m-2-1)
(m-1-2) edge node[auto] {$ G(f)$} (m-2-2);
\end{tikzpicture}
@ -84,17 +84,17 @@ Note that an isomorphism between to categories is now also defined. Two categori
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 a isomorphism between two functors is a natural transformation which is an isomorphism pointwise. Such a natural transformation is called a 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}
\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:
$$ FG =\id_\cat{D}\text{and}\id_\cat{C}= GF. $$
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
With the notion of isomorphisms between functors we can generalize this, and only require a natural isomorphism instead of equality.
\begin{definition}
An \emph{equivalence} between two 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 there are natural isomorphisms:
@ -104,10 +104,9 @@ With the notion of isomorphisms between functors we can generalize this, and onl
\subsection{Adjunctions}
\begin{definition}
An adjunction between two 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 there is a natural bijection:
$$\Hom{\cat{D}}{FX}{Y}\iso\Hom{\cat{C}}{X}{GY}$$
for any $X \in\cat{D}$ and $Y \in\cat{C}$.
$F$ is called the left-adjoint and $G$ the right-adjoint.
An \emph{adjunction} between two categories $\cat{C}$ and $\cat{D}$ consists of two functors $F:\cat{C}\to\cat{D}$ and $G: \cat{D}\to\cat{C}$ together with a natural bijection
for any $X \in\cat{C}$ and $Y \in\cat{D}$. The functor $F$ is called the \emph{left adjoint} and $G$ the \emph{right adjoint}.
\end{definition}
Now there are different definitions of adjunctions, which are equivalent. We will not prove that these are equivalent. One can find the proof in for example in the books of Mac Lane \cite{maclane} or Awodey \cite{awodey}. A particular nice one is the following:
@ -135,44 +134,43 @@ Now there are different definitions of adjunctions, which are equivalent. We wil
Note that by considering the identity map $\id : G(A)\to G(A)$ in $\cat{C}$, we get a uniquely determined map $\overline{\id}:FG(A)\to A$. This map $FG(A)\to A$ is in fact natural in $A$, this natural transformation is called the \emph{co-unit}:
$$\eps: FG \to\id. $$
It can be shown that an equivalence $F: \cat{C}\tot{\simeq}\cat{D}$ is both a left and right-adjoint. We sketch the proof of $F$ being a left-adjoint. Clearly we already have the natural transformation $\eta: \id_\cat{C}\to GF$. To construct $\overline{f}$ from $f: S \to G(A)$ we can apply the functor $F$, to get $F(S)\to FG(A)$, using the other natural isomorphism we get $F(S)\to FG(A)\to A$. We leave the details to the reader.
It can be shown that an equivalence $F: \cat{C}\tot{\simeq}\cat{D}$ is both a left and rightadjoint. We sketch the proof of $F$ being a leftadjoint. Clearly we already have the natural transformation $\eta: \id_\cat{C}\to GF$. To construct $\overline{f}$ from $f: S \to G(A)$ we can apply the functor $F$, to get $F(S)\to FG(A)$, using the other natural isomorphism we get $F(S)\to FG(A)\to A$. We leave the details to the reader.
The first definition of adjunction is useful when dealing with maps, since it gives an bijection between the $\mathbf{Hom}$-sets. However the second definition is useful when proving a certain construction is part of an adjunction, as shown in the following example.
\begin{example}
\emph{(The free abelian group)} There is an obvious functor $U: \Ab\to\Set$, which sends an abelian group to its underlying set, forgetting the additional structure. It is hence called a \emph{forgetful functor}. This functor is a right-adjoint. The left-adjoint $\Z[-]: \Set\to\Ab$ is given by the \emph{free abelian group}; for a set $S$ define:
\emph{(The free abelian group)} There is an obvious functor $U: \Ab\to\Set$, which sends an abelian group to its underlying set, forgetting the additional structure. It is hence called a \emph{forgetful functor}. This functor is a right adjoint. The left adjoint $\Z[-]: \Set\to\Ab$ is given by the \emph{free abelian group}; for a set $S$ define
$$\Z[S]=\{\phi: S \to\Z\I\text{supp}(\phi)\text{ is finite}\}, $$
where $\text{supp}(\phi)=\{ s \in S \I\phi(s)\neq0\}$. The group structure on $\Z[S]$ is given pointwise. One can think of elements of this abelian group as formal sums, namely:
It is clear that $U(\overline{f})\circi= f$. We will leave the other details (naturality of $\eta$, $\overline{f}$ being a group homomorphism, and uniqueness w.r.t. $U(\overline{f})\circi= f$) to the reader.
It is clear that $U(\overline{f})\circ\eta= f$. We will leave the other details (naturality of $\eta$, $\overline{f}$ being a group homomorphism, and uniqueness w.r.t. $U(\overline{f})\circ\eta= f$) to the reader.
\end{example}
\subsection{Yoneda}
\subsection{The Yoneda lemma}
So far we have only encountered definitions from category theory. However there is a very important lemma by Yoneda. This lemma gives a nice way to construct certain natural transformations.
\begin{definition}
For any category $\cat{C}$, we define a functor $y:\cat{C}\to\Set^{\cat{C}^{op}}$ as follows:
For any category $\cat{C}$, we define a functor $y:\cat{C}\to\Set^{\cat{C}^{op}}$ as follows
$$ y(X)=\Hom{\cat{C}}{-}{X}. $$
The functor $y$ is called the \emph{Yoneda embedding}.
\end{definition}
\begin{lemma}\emph{(The Yoneda lemma)}
Given a functor $F: \cat{C}\to\Set$ and any object $C \in\cat{C}$ there is a bijection:
Given a functor $F: \cat{C}\to\Set$ and any object $C \in\cat{C}$ there is a bijection
$$\mathbf{Nat}(y(C), F)\iso F(C), $$
which is natural in both $F$ and $C$, where $\mathbf{Nat}(G, G')$ denotes the set of natural transformation between $G$ and $G'$, in other words $\mathbf{Nat}=\mathbf{Hom}_{\Set^{\cat{C}^{op}}}$.
\end{lemma}
\todo{CT: Prove Yoneda? I guess not...}
We will use this lemma when we discuss simplicial abelian groups.
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 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:
$$\Ch{\Ab}\simeq\sAb$$
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. 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:
$$\pi_n(A)\iso H_n(N(A))\text{ for all } n \in\N$$
Joshua Moerman
11 years ago