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Joshua Moerman 12 years ago
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11 changed files with 2099 additions and 45 deletions
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  1. 4
      make
  2. BIN
      presentation2/images/cat_contrafunctor.pdf
  3. 205
      presentation2/images/cat_contrafunctor.svg
  4. BIN
      presentation2/images/cat_functor.pdf
  5. 283
      presentation2/images/cat_functor.svg
  6. BIN
      presentation2/images/cat_th.pdf
  7. 215
      presentation2/images/cat_th.svg
  8. BIN
      presentation2/images/simplicial_abgrp.pdf
  9. 1260
      presentation2/images/simplicial_abgrp.svg
  10. 171
      presentation2/presentation.tex
  11. 6
      presentation2/symbols.tex

4
make
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@ -13,8 +13,8 @@ Presentation2) pdflatex "../presentation2/presentation.tex" || exit 1
pdflatex "../presentation2/presentation.tex" || exit 1
mv presentation.pdf ../
;;
Symbols) pdflatex "../thesis/symbols.tex" || exit 1
pdflatex "../thesis/symbols.tex" || exit 1
Symbols) pdflatex "../presentation2/symbols.tex" || exit 1
pdflatex "../presentation2/symbols.tex" || exit 1
scp symbols.pdf moerman@stitch.science.ru.nl:~/symbols.pdf
ssh moerman@stitch.science.ru.nl 'pdf2svg symbols.pdf symbols.svg'
scp moerman@stitch.science.ru.nl:~/symbols.svg ../

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presentation2/presentation.tex
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@ -18,7 +18,7 @@
\input{../thesis/preamble}
\graphicspath{ {../presentation2/images/} {../thesis/images/} }
\title{Dold-Kan correspondentie
\title{De Dold-Kan correspondentie
\huge $$ \Ch{\Ab} \simeq \sAb $$}
\author{Joshua Moerman}
\institute[Radboud Universiteit Nijmegen]{Begeleid door Moritz Groth}
@ -31,14 +31,16 @@
\titlepage
\end{frame}
\begin{frame}
\begin{frame}{Categorie\"en}
Een \emph{categorie} $\cat{C}$ bestaat uit
\vspace{5cm}\td{plaatje}
\begin{center}
\includegraphics{cat_th}
\end{center}
met compositie $-\circ-$, zodat
met \emph{compositie} $-\circ-$, zodat
\begin{itemize}
\item er is een identiteit $\id_c: C \to C$ en
\item er is een \emph{identiteit} $\id_c: C \to C$ en
\item compositie is associatief.
\end{itemize}
\end{frame}
@ -52,15 +54,27 @@
\item[$\Ab$]
objecten: abelse groepen \\
pijlen: groupshomomorfismes
\item[$\underline{4}$] \td{diagram}
\item[$\cat{\underline{4}}$]
\tikz[baseline=-0.5ex]{
\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
\ast_1 \& \ast_2 \\
\ast_3 \& \ast_4 \\
};
\path[->] (m-1-1) edge node[font=\small, auto] {$ a $} (m-1-2);
\path[->] (m-1-1) edge node[font=\small, auto] {$ f $} (m-2-1);
\path[->] (m-1-2) edge node[font=\small, auto] {$ b $} (m-2-2);
\path[->] (m-2-1) edge node[font=\small, auto] {$ g $} (m-2-2);
} \hspace{1cm} met $ba = gf$.
\end{itemize}
\end{frame}
\begin{frame}
\frametitle{Functors}
Een \emph{functor} $F$ is een functie van een categorie $\cat{C}$ naar $\cat{D}$ op objecten \'en pijlen.
Een \emph{functor} $F: \cat{C} \to \cat{D}$ is een functie op objecten \'en pijlen.
\vspace{3cm}\td{plaatje}
\begin{center}
\includegraphics[scale=0.9]{cat_functor}
\end{center}
Zodat
\begin{itemize}
@ -79,7 +93,7 @@
Voor een functie $f: V \to W$ definieer
\begin{gather*}
\Z[f]: \Z[V] \to \Z[W] \\
\Z[f](\phi) = \sum_v \phi(v) e_{f(v)}.
\Z[f](\phi) = \sum_v \phi(v) \chi_{\{f(v)\}}.
\end{gather*}
\bigskip
@ -89,7 +103,31 @@
\begin{frame}
\frametitle{Voorbeeld functor}
\td{Commuterend diagram}
Definieer $F: \cat{\underline{4}} \to \Ab$ als volgt:
$$ F(\ast_1) = F(\ast_2) = F(\ast_3) = F(\ast_4) = \Z $$
en op pijlen:
\begin{align*}
F(f)(n) = 4n & & F(g)(n) = 3n \\
F(a)(n) = 6n & & F(b)(n) = 2n.
\end{align*}
\begin{columns}
\begin{column}{0.5\textwidth}
\tikz[baseline=-0.5ex]{
\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
\Z \& \Z \\
\Z \& \Z \\
};
\path[->] (m-1-1) edge node[font=\small, auto] {$ \times 6 $} (m-1-2);
\path[->] (m-1-1) edge node[font=\small, auto] {$ \times 4 $} (m-2-1);
\path[->] (m-1-2) edge node[font=\small, auto] {$ \times 2 $} (m-2-2);
\path[->] (m-2-1) edge node[font=\small, auto] {$ \times 3 $} (m-2-2);
}
\end{column}
\begin{column}{0.5\textwidth}
Compositie is behouden, want het diagram commuteert.
\end{column}
\end{columns}
\end{frame}
\begin{frame}
@ -104,42 +142,55 @@
\item Functor $\sim$ Diagrammen.
\end{itemize}
$F$ is \emph{contravariant} (notatie $F: \cat{C}^{op} \to \cat{D}$) als \\
\td{plaatje}
\bigskip\pause
$F$ is \emph{contravariant} (notatie $F: \cat{C}^{op} \to \cat{D}$) als
\begin{columns}
\begin{column}{0.7\textwidth}\includegraphics[scale=0.8]{cat_contrafunctor}\end{column}
\begin{column}{0.3\textwidth}\small $F(g \circ f) = F(g) \circ F(f)$.\end{column}
\end{columns}
\end{frame}
\begin{frame}
\frametitle{D\'e categorie van mijn scriptie}
\frametitle{Belangrijke categorie in mijn scriptie}
\begin{itemize} \item[$\DELTA$]
heeft als objecten $[n] = \{0, \ldots, n\}$, $n\in\N$ \\
en als pijlen monotoon stijgende functies.
objecten: $[n] = \{0, \ldots, n\}$, $n\in\N$ \\
pijlen: monotoon stijgende functies.
\end{itemize}
\bigskip
\only<1>{\begin{example}
Voor elke $n$ zijn er pijlen
Voor elke $n \in \N$ zijn er pijlen
\end{example}}
\only<2->{\begin{lemma}
Elke pijl in $\DELTA$ is een compositie van
\end{lemma}}
\begin{itemize}
\item $\delta_i$\td{Definitie hier}
\item $\sigma_i$
\item $\delta_i: [n] \mono [n+1]$ slaat $i$ over \hfill ($0 \leq i \leq n$)
\item $\sigma_i: [n+1] \epi [n]$ bereik $i$ twee keer \hfill ($0 \leq i < n$)
\end{itemize}
\visible<3>{
Dus $\DELTA = \cdots$\td{Diagram hier}
Dus $\DELTA = \vcenter{\hbox{\includegraphics{delta_cat}}}$
}
\end{frame}
\begin{frame}
\frametitle{D\'e categorie van mijn scriptie}
\frametitle{Belangrijke categorie in mijn scriptie}
$\DELTA = \cdots$\td{Plaatje hier}
$\DELTA = \vcenter{\hbox{\includegraphics[scale=0.8]{delta_cat_geom}}}$
\pause\bigskip
\begin{lemma}
Cosimpliciale gelijkheden\td{dingetjes}
De \emph{cosimpliciale vergelijkingen} gelden:
\small
\begin{align*}
\delta_j\delta_i &= \delta_i\delta_{j-1}, \hspace{1.5cm} \textnormal{ if } i < j,\\
\sigma_j\delta_i &= \delta_i\sigma_{j-1}, \hspace{1.5cm} \textnormal{ if } i < j,\\
\sigma_j\delta_j &= \sigma_j\delta_{j+1} = \id,\\
\sigma_j\delta_i &= \delta_{i-1}\sigma_j, \hspace{1.5cm} \textnormal{ if } i > j+1,\\
\sigma_j\sigma_i &= \sigma_i\sigma_{j+1}, \hspace{1.5cm} \textnormal{ if } i \leq j.
\end{align*}
\end{lemma}
\end{frame}
@ -149,7 +200,22 @@
\bigskip
\visible<2->{
$ A := $\td{diagram}
$$ A :=
\begin{tikzpicture}[baseline=-0.5ex]
\matrix (m) [matrix of math nodes, ampersand replacement=\&, row sep=2em, column sep=2em] {
A_0 \& A_1 \& A_2 \& \cdots \\
};
\draw [raise line=-5, <-] (m-1-1) -> node[font=\small, above] {$ A(\delta_0) $} (m-1-2);
\draw [raise line=5, <-] (m-1-1) -> node[font=\small, below] {$ A(\delta_1) $} (m-1-2);
\foreach \r in {0} \draw [raise line=\r, ->] (m-1-1) -> (m-1-2);
\foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3);
\foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3);
\foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4);
\foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4);
\end{tikzpicture}$$
}
\end{center}
\end{frame}
@ -160,8 +226,17 @@
\item[Objecten] \emph{Simpliciaal abelse groepen} $A$ \\
preciezer: functoren $A: \DELTA^{op} \to \Ab$
\item[Pijlen] \emph{Natuurlijke transformaties} \\
preciezer: $\phi: A \to B$ bestaat uit $\phi_n: A_n \to B_n$ zodat
\vspace{2cm}\td{diagram}
preciezer: $\phi: A \to B$ bestaat uit $\phi_n: A_n \to B_n$ zodat \\
\tikz[baseline=-0.5ex]{
\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
A_n \& A_m \\
B_n \& B_m \\
};
\path[->] (m-1-1) edge node[font=\small, auto] {$ A(f) $} (m-1-2);
\path[->] (m-1-1) edge node[font=\small, auto] {$ \phi_n $} (m-2-1);
\path[->] (m-1-2) edge node[font=\small, auto] {$ \phi_m $} (m-2-2);
\path[->] (m-2-1) edge node[font=\small, auto] {$ B(f) $} (m-2-2);
} \hspace{1cm} voor alle $f:[m] \to [n]$.
\end{itemize}
\end{frame}
@ -171,15 +246,24 @@
\item[Objecten] \emph{Ketencomplexen} $C$ \\
preciezer: collectie abelse groepen $C_n$ en groepshomonorfismes $\del_{n+1}: C_{n+1} \to C_n$ zodat $\del \circ \del = 0$
\item[Pijlen] \emph{Ketenafbeeldingen} \\
preciezer: $\phi: C \to D$ bestaat uit $\phi_n: C_n \to D_n$ zodat
\vspace{2cm}\td{diagram}
preciezer: $\phi: C \to D$ bestaat uit $\phi_n: C_n \to D_n$ zodat \\
\tikz[baseline=-0.5ex]{
\matrix (m) [matrix of math nodes, row sep=2em, column sep=2em, ampersand replacement=\&]{
C_{n+1} \& C_n \\
D_{n+1} \& D_n \\
};
\path[->] (m-1-1) edge node[font=\small, auto] {$ \del $} (m-1-2);
\path[->] (m-1-1) edge node[font=\small, auto] {$ \phi_{n+1} $} (m-2-1);
\path[->] (m-1-2) edge node[font=\small, auto] {$ \phi_n $} (m-2-2);
\path[->] (m-2-1) edge node[font=\small, auto] {$ \del $} (m-2-2);
}
\end{itemize}
\end{frame}
\begin{frame}{$\sAb$ lijkt op $\Ch{\Ab}$}
Simpliciaal abelse groepen:
\begin{center}
\includegraphics{simplicial_set} \\
\includegraphics{simplicial_abgrp} \\
met de 5 vergelijkingen
\end{center}
@ -220,18 +304,10 @@ Ketencomplexen:
(want $\sigma_0 \delta_0 = \id$, dus $A(\delta_0)A(\sigma_0) = \id$)
\end{frame}
\begin{frame}{Belangrijke definities}
\begin{frame}{Definities}
Zij $A \in \sAb$ \\
$x \in A_n$ heet een \emph{$n$-simplex} \\
$x$ is \emph{gedegenereerd} als $x = A(\sigma_i)(y)$ voor een zekere $i$ en $y$.
\bigskip\pause
\begin{lemma}
$\forall x \in A_n$ \\
$\exists !$ surjectie $\beta: [n] \epi [m]$ en\\
niet-gedegenereerde $y \in A_m$ zodat
$$ x = A(\beta)(y). $$
\end{lemma}
$x \in A_n$ is \emph{gedegenereerd} als $x = A(\sigma_i)(y)$ voor een zekere $i$ en $y$.
\end{frame}
\begin{frame}{De juiste constructie}
@ -240,20 +316,31 @@ Ketencomplexen:
N(A)_n &= \bigcap_{i=1}^n \ker(A(\delta_i)) \\
\del &= A(\delta_0)
\end{align*}
\bigskip\pause
\pause
\begin{lemma}
$x \in N(A)_n$ is niet-gedegenereerd.
\end{lemma}
\bigskip
\begin{lemma}
Sterker nog:
$$ A_n = N(A)_n \oplus D_n(A). $$
\centering$ A_n = N(A)_n \oplus D_n(A). $
\end{lemma}
\end{frame}
\begin{frame}{Voorbeeld}
Definieer de volgende simpliciaal abelse groep:
\begin{gather*}
A_n = \Z \\
A(\delta_i) = A(\sigma_i) = \id.
\end{gather*}
\pause
$$ N(A) = \Z \from 0 \from 0 \from \cdots. $$
\end{frame}
\begin{frame}
\begin{center}
$$ N: \sAb \rightleftarrows \Ch{\Ab} :K $$
\pause\bigskip
\Huge Vragen?
\end{center}
\end{frame}

6
presentation2/symbols.tex
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@ -16,7 +16,11 @@
\begin{frame}
$$ \cat{C} \cat{D} $$
$$ A B C X Y Z $$
$$ F(A) F(B) F(C) $$
\small $$ f g g \circ f $$
\small $$ F(f) F(g) $$
\end{frame}
\end{document}