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Moved labels beneath captions, added picture of DELTA

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Joshua Moerman 11 years ago
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6 changed files with 630 additions and 25 deletions
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  1. BIN
      images/delta_cat_geom.pdf
  2. 599
      images/delta_cat_geom.svg
  3. 2
      thesis/2_ChainComplexes.tex
  4. 29
      thesis/3_SimplicialAbelianGroups.tex
  5. 6
      thesis/4_Constructions.tex
  6. 19
      thesis/symbols.tex

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images/delta_cat_geom.pdf
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images/delta_cat_geom.svg
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2
thesis/2_ChainComplexes.tex
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@ -59,9 +59,9 @@ We now have enough tools to define the singular chaincomplex of a space $X$.
This might seem a bit complicated, but we can pictures this in an intuitive way, as in figure~\ref{fig:singular_chaincomplex3}. And we see that the boundary operators really give the boundary of an $n$-simplex. To see that this indeed is a chaincomplex we have to proof that the composition of two such operators is the zero map.
\begin{figure}
\label{fig:singular_chaincomplex3}
\includegraphics{singular_chaincomplex3}
\caption{The boundary of a 2-simplex}
\label{fig:singular_chaincomplex3}
\end{figure}
\todo{Ch: Proposition: $C(X) \in \Ch{\cat{Ab}}$}

29
thesis/3_SimplicialAbelianGroups.tex
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@ -1,8 +1,9 @@
\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.
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, luckily it can still be visualised in pictures, then we will derive the explicit definition.
\subsection{Abstract definition}
\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}
@ -12,30 +13,40 @@ There are two special kinds of maps in $\DELTA$, the so called \emph{face} and \
$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1, \text{ and}$$
$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
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. \todo{sAb: Epi-mono factorization of $\DELTA$} So in a sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as follows.
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. \todo{sAb: Epi-mono factorization of $\DELTA$} So in a sense, these are all the maps we need to consider. We can now picture the category $\DELTA$ as in figure~\ref{fig:delta_cat}.
\begin{figure}[h!]
\label{fig:delta_cat}
\includegraphics{delta_cat}
\caption{The category $\DELTA$ with the face and degeneracy maps.}
\label{fig:delta_cat}
\end{figure}
Althoug this is a very abstract definition, a more geometric intuition can be given. In $\DELTA$ we can regard $[n]$ as an abstract version of the $n$-simplex $\Delta^n$. The maps face maps $\delta_i$ are then exactly maps which point out how we can embed $\Delta^n$ in $\Delta^{n+1}$. This is shown in figure~\ref{fig:delta_cat_geom}. This picutre shows the images of the face maps, for example the image of $\delta_3$ from $[2]$ to $[3]$ is the set $\{0,1,2\}$, which is the bottom face of the tetrahedron. The degeneracy maps are harder to visualize, one can think of them as collapsing maps, where two points are identified with eachother. \todo{sAb: how to draw $\sigma_i$?}
\begin{figure}
\includegraphics{delta_cat_geom}
\caption{The category $\DELTA$ with the face maps shown in a geometric way.}
\label{fig:delta_cat_geom}
\end{figure}
This category $\DELTA$ will act as a protoype for these kind of geometric structures in other categories. This leads to the following definition.
\begin{definition}
An simplicial abelian group $A$ is a covariant functor:
$$A: \DELTA^{op} \to \Ab.$$
(Or equivalently a contravariant functor $A: \DELTA \to \Ab.$)
An simplicial abelian group $A$ is a contravariant functor:
$$A: \DELTA \to \Ab.$$
(Or equivalently a covariant functor $A: \DELTA^{op} \to \Ab.$)
\end{definition}
So the category of all simplicial abelian groups, $\sAb$, is the functor category $\Ab^{\DELTA^{op}}$, where morphisms are natural transformations. 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 $A: \DELTA^{op} \to Ab$. 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}
\includegraphics{simplicial_abelian_group}
\caption{A simplicial abelian group.}
\label{fig:simplicial_abelian_group}
\end{figure}
Althoug this is a very abstract definition, a more geometric intuition can be given. In $\DELTA$ we can regard $[n]$ as an abstract version of the $n$-simplex $\Delta^n$. The maps face maps $\delta_i$ are then exactly maps which point out how we can embed $\Delta^n$ in $\Delta^{n+1}$. \todo{sAb: add pictures, along the lines of Friedman} The degeneracy maps are harder to visualize, one can think of them as collapsing maps, where two points are identified with eachother. \todo{sAb: how to draw $\sigma_i$?}
\subsection{Explicit definition}
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?}
@ -66,7 +77,7 @@ Because a simplicial abelien group $A$ is a contravariant functor, these equatio
\end{align}
\end{definition}
It is already indicated that a functor from $\DELTA^{op}$ to $\Ab$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So gives this a way of restoring the first definition from this one. Conversely, we can apply functorialty to obtain the second definition from the first. So these definitions are the same \todo{sAb: is it ok not to prove this?}.
It is already indicated that a functor from $\DELTA^{op}$ to $\Ab$ is determined when the images for the face and degeneracy maps in $\DELTA$ are provided. So gives this a way of restoring the first definition from this one. Conversely, we can apply functorialty to obtain the second definition from the first. So these definitions are the same \todo{sAb: is it ok not to prove this?}. So from now on we will denote $A([n])$ by $A_n$, $A(\sigma_i)$ by $\sigma^i$ and $A(\delta_i)$ by $\delta^i$, whenever we have a simplicial abelien group $A$.
When using a simplicial abelian group to construct another object, it is often handy to use this second definition, as it gives you a very concrete objects to work with. On the other hand, constructing this might be hard (as you would need to provide a lot of details), in this case we will often use the more abstract definition.

6
thesis/4_Constructions.tex
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@ -4,10 +4,10 @@
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:
$$\del_{n-1} = A(\delta_0) - A(\delta_1) + \ldots + (-1)^n A(\delta_n) \text{ for every } n > 0.$$
Given a simplicial abelian group $A$, we have a family of abelian groups $A_n$. We define a grouphomomorphism $\del_{n-1} : A_n \to A_{n-1}$ as:
$$\del_{n-1} = \delta^0 - \delta^1 + \ldots + (-1)^n \delta^n \text{ for every } n > 0.$$
\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.
Using $A_n$ as the family of abelian groups and the maps $\del_n$ 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 $\del_n \circ \del_{n+1} = 0$.

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thesis/symbols.tex
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@ -15,20 +15,15 @@
\input{../thesis/preamble}
\title{Dold-Kan Correspondence}
\author{Joshua Moerman}
\begin{document}
\maketitle
\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}
$$ [0] \to [1] \to [2] \to [3] \to \ldots $$
$$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
$$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
% For basic categorical picture of simplicial objects
% $$ [0] \to [1] \to [2] \to [3] \to \ldots $$
% $$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
% $$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
% $$ A_0 \to A_1 \to A_2 \to A_3 $$
$$ A_0 \to A_1 \to A_2 \to A_3 $$
% For geometric picture of simplicial objects
$$ 0 \tot{\delta_0} 1 \tot{\delta_1} 2 \tot{\delta_2} 3 \tot{\delta_3} \cdots $$
\end{document}