diff --git a/make b/make
index d00914f..ce9ded2 100755
--- a/make
+++ b/make
@@ -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 ../
diff --git a/presentation2/images/cat_contrafunctor.pdf b/presentation2/images/cat_contrafunctor.pdf
new file mode 100644
index 0000000..8e053c5
Binary files /dev/null and b/presentation2/images/cat_contrafunctor.pdf differ
diff --git a/presentation2/images/cat_contrafunctor.svg b/presentation2/images/cat_contrafunctor.svg
new file mode 100644
index 0000000..9d57fa0
--- /dev/null
+++ b/presentation2/images/cat_contrafunctor.svg
@@ -0,0 +1,205 @@
+
+
+
+
diff --git a/presentation2/images/cat_functor.pdf b/presentation2/images/cat_functor.pdf
new file mode 100644
index 0000000..d70e9df
Binary files /dev/null and b/presentation2/images/cat_functor.pdf differ
diff --git a/presentation2/images/cat_functor.svg b/presentation2/images/cat_functor.svg
new file mode 100644
index 0000000..cdd83ff
--- /dev/null
+++ b/presentation2/images/cat_functor.svg
@@ -0,0 +1,283 @@
+
+
+
+
diff --git a/presentation2/images/cat_th.pdf b/presentation2/images/cat_th.pdf
new file mode 100644
index 0000000..26bac1e
Binary files /dev/null and b/presentation2/images/cat_th.pdf differ
diff --git a/presentation2/images/cat_th.svg b/presentation2/images/cat_th.svg
new file mode 100644
index 0000000..9716c3d
--- /dev/null
+++ b/presentation2/images/cat_th.svg
@@ -0,0 +1,215 @@
+
+
+
+
diff --git a/presentation2/images/simplicial_abgrp.pdf b/presentation2/images/simplicial_abgrp.pdf
new file mode 100644
index 0000000..03227dd
Binary files /dev/null and b/presentation2/images/simplicial_abgrp.pdf differ
diff --git a/presentation2/images/simplicial_abgrp.svg b/presentation2/images/simplicial_abgrp.svg
new file mode 100644
index 0000000..9798600
--- /dev/null
+++ b/presentation2/images/simplicial_abgrp.svg
@@ -0,0 +1,1260 @@
+
+
diff --git a/presentation2/presentation.tex b/presentation2/presentation.tex
index 8611dcd..c1b0507 100644
--- a/presentation2/presentation.tex
+++ b/presentation2/presentation.tex
@@ -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}
diff --git a/presentation2/symbols.tex b/presentation2/symbols.tex
index c051f30..96b756d 100644
--- a/presentation2/symbols.tex
+++ b/presentation2/symbols.tex
@@ -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}
\ No newline at end of file