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 @@ + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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 @@ + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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 @@ + + + + + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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 @@ + + + + + + image/svg+xml + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + 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