Added new chapter (not sure about the title, though) and some more bits

thesis/1_CategoryTheory.tex

@@ -39,6 +39,7 @@ In fact almost any mathematical structure can be described as a category, we hav
 	\end{itemize}
 	We normally do not write the index of $F_0$ or $F_1$, instead we wrtie $F$ for both functions.
 \end{definition}
+\todo{CT: contravariant functor}
 
 \begin{exlemma}
 	There is a category $\cat{Cat}$ with categories as objects, and functors as maps.

thesis/3_SimplicialAbelianGroups.tex

@@ -1,13 +1,15 @@
 \section{Simplicial Abelian Groups}
 \label{sec:Simplicial Abelian Groups}
 
+There are generally two definitions of a \emph{simplicial abelian group}, an abstract one and a very explicit one. We will start with the abstract one, and immediately show in pictures what the explicit definition looks like.
+
 \begin{definition}
 	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
 \end{definition}
 
 There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \emph{degeneracy} maps, defined as (resp.):
 
-$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
+$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1, \text{ and}$$
 $$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
 
 for each $n \in \N$. The nice things about these maps is that every map in $\DELTA$ can be decomposed to a composition of these maps. So in a certain sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
@@ -18,9 +20,9 @@ for each $n \in \N$. The nice things about these maps is that every map in $\DEL
 	\caption{The category $\DELTA$ with the face and degeneracy maps.}
 \end{figure}
 
-\todo{sAb: Epi-mono factorization}
+\todo{sAb: Epi-mono factorization of $\DELTA$}
 
-Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $F: \DELTA^{op} \to Ab$. And hence we can picture a simplicial abelian group as follows.
+Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because the face and degeneracy maps give all the maps in $\DELTA$ it is sufficient to define images of $\delta_i$ and $\sigma_i$ in order to define a functor $F: \DELTA^{op} \to Ab$. And hence we can picture a simplicial abelian group as done in figure~\ref{fig:simplicial_abelian_group}. Comparing this to figure~\ref{fig:delta_cat} we see that the arrows are reversed, because $A$ is a contravariant functor.
 
 \begin{figure}
 	\label{fig:simplicial_abelian_group}
@@ -28,7 +30,7 @@ Now the category $\sAb$ is defined as the category $\Ab^{\DELTA^{op}}$. Because
 	\caption{A simplicial abelian group.}
 \end{figure}
 
-Of course the maps $\delta_i$ and $\sigma_i$ satisfy certain equations, these are the so called \emph{simplicial equations}.
+Of course the maps $\delta_i$ and $\sigma_i$ in $\DELTA$ satisfy certain equations, these are the so called \emph{simplicial equations}.
 \todo{sAb: Is \emph{simplicial equations} really a thing?}
 
 \begin{lemma}
@@ -44,4 +46,11 @@ Of course the maps $\delta_i$ and $\sigma_i$ satisfy certain equations, these ar
 \begin{proof}
 	By writing out the definitions given above.
 \end{proof}
+
+Because a simplicial abelien group $A$ is a contravariant functor, these equations (which only consist of compositions and identities) also hold in its image. For example the first equation would look like: $ A(\delta_i)A(\delta_j) = A(\delta_{j-1})A(\delta_i) $ for $ i < j$ (again note that $A$ is contravariant, and hence composition is reversed). This can be used for a explicit definition of simplicial abelien groups. In this definition a simplicial abelian group $A$ consists of a family abelian groups $(A_n)_{n}$ together with face and degeneracy maps (which are grouphomomorphisms) such that the simplicial equations hold.
+
+\subsection{Other simplicial objects}
+Of course the abstract definition of simplicial abelian group can easilty be generalized to other categories. For example $\Set^{\DELTA^{op}} = \sSet$ is the category of simplicial sets.
+\todo{sAb: as example do the free abelian group pointwise}
+
 \todo{sAb: Say a bit more (because Mueger will not like this)}

thesis/4_Constructions.tex

@@ -0,0 +1,21 @@
+\section{Constructions}
+\label{sec:Constructions}
+
+Comparing chain complexes and simplicial abelian groups, we see a similar structure. Both objects consists of a sequence of abelian groups, with maps in between. At first sight simplicial abelian groups have more structure, because there are maps in both directions. It is not clear how to make degeneracy maps given a chain complex, in fact it is already unclear how to define more maps (the face maps) out of one (the boundary one). Constructing a chain complex from a simplicial abelian group on the other hand seems doable.
+
+\subsection{Unnormalized chain complex}
+Given a simplicial abelian group $A$, we have a family of abelian groups $A([n])_n$. We define a grouphomomorphism $\del_{n-1} : A([n]) \to A([n-1])$ as:
+$$\del_{n-1} = A(\delta_0) - A(\delta_1) + \ldots + (-1)^n A(\delta_n) \text{ for every } n > 0.$$
+\begin{lemma}
+	Using $A([n])_n$ as the family of abelian groups and the maps $(\del_n)_n$ as boundary maps gives a chain complex.
+\end{lemma}
+\begin{proof}
+	We already have a collection of abelian groups together with maps, so the only thing to proof is $\del_n \circ \del_{n+1} = 0$.
+
+	\todo{C: insert calculation with sums}
+
+	So indeed this is a chain complex.
+\end{proof}
+
+This construction gives a functor $C : \sAb \to \Ch{\Ab}$. And in fact we already used it in the construction of the singular chaincomplex, where we defined the boundary maps as $\del(\sigma) = \sigma \circ \delta^0 - \sigma \circ \delta^1 + \ldots + (-1)^{n+1} \sigma \circ \delta^{n+1}$ (on generators). The terms $\sigma \circ \delta^i$ are the maps given by the $\mathbf{Hom}$-functor from $\Top$ to $\Set$, in fact this $\mathbf{Hom}$-functor can be used to get a functor $Sing : \Top \to \sSet$, applying the free abelain group pointwise give a functor $\Z^\ast : \sSet \to \sAb$, and finally using the functor $C$ gives the singular chain complex.
+\todo{C: is this a nice thing to add?}

thesis/DoldKan.tex

@@ -42,6 +42,9 @@ In the first section some definitions from category theory are given, because we
 \newpage
 \input{../thesis/3_SimplicialAbelianGroups}
 
+\newpage
+\input{../thesis/4_Constructions}
+
 \newpage
 \listoftodos
 % \nocite{*}

thesis/preamble.tex

@@ -11,6 +11,8 @@
 \newcommand{\Ab}{\cat{Ab}}
 \newcommand{\sAb}{\cat{sAb}}
 \newcommand{\Set}{\cat{Set}}
+\newcommand{\sSet}{\cat{sSet}}
+\newcommand{\Top}{\cat{Top}}
 \newcommand{\DELTA}{\cat{\Delta}}
 \newcommand{\Ch}[1]{\mathbf{Ch}(#1)}
 \newcommand{\Hom}[3]{\mathbf{Hom}_{#1}(#2, #3)}