From 1cae8473847a10cf3f1b8175e1f8b6f638944e88 Mon Sep 17 00:00:00 2001 From: CDaut Date: Sat, 22 Oct 2022 15:04:48 +0200 Subject: [PATCH] tikzcd is cursed --- chapters/nat_trafo.tex | 49 ++++++++++++++++++++++++++++++++++++++++- main.pdf | Bin 327830 -> 352824 bytes 2 files changed, 48 insertions(+), 1 deletion(-) diff --git a/chapters/nat_trafo.tex b/chapters/nat_trafo.tex index bc446d0..dcfdd5e 100644 --- a/chapters/nat_trafo.tex +++ b/chapters/nat_trafo.tex @@ -69,4 +69,51 @@ \end{flalign*} $\implies$ Beh.\qed -\end{example} \ No newline at end of file +\end{example} +\begin{example}{Determinante\\} + Es seien $\times: R\mapsto R^\times$ und $GL_n: R\mapsto GL_n(R)$ Funktoren von + $\cat{KR1ng}$ nach $\cat{Grp}$. Dann ist die Determinantenabbildung ($det$) eine + Natürliche Transformation $GL_n\mapsto\times$. Dabei ist + $det_2: GL_n(R)\mapsto R^\times$ definiert als $A\mapsto det(A)$. Folgendes kommutative Diagramm + stellt die Situation dar:\\ + \begin{center} + \begin{tikzcd}[sep=large] + R \arrow[d, "f"'] \\ S + \end{tikzcd} + \hspace{20mm} + \begin{tikzcd}[sep=large] + GL_n(R) \arrow[r, "det"] \arrow[d, "g"]{}& R^\times \arrow[d, "f^\times"]\\ + GL_n(S) \arrow[r, "det"] & S^\times + \end{tikzcd} + \end{center} + Wobei $g: A=(a_{ij})\mapsto(f(a_{ij}))$\\ %TODO: Typeset this as on blackboard + Dieses Diagramm kommutiert (ohne Beweis), also ist die Determinante eine natürliche Transformation. +\end{example} +\subsection{Adjunktion von Funktoren} +Seien +\begin{tikzpicture}[baseline=-1mm] + \node (C) at (0,0) {$\mathscr{C}$}; + \node (D) at (2,0) {$\mathscr{D}$}; + + \draw + (C) edge[bend left] node[above]{$\mathcal{F}$} (D) + (D) edge[bend left] node[below]{$\mathcal{G}$} (C) + ; +\end{tikzpicture} +Funktoren.\\ +\begin{center} + \begin{tikzcd}[sep=large] + x\arrow[d, "f"'] & y \arrow[d, "g"']\\ \tilde{x} & \tilde{y} + \end{tikzcd} + \hspace{20mm} + \begin{tikzcd}[sep=large] + \mor{\mathscr{D}}{\mathcal{F}(\tilde{x})}{y} \arrow[r, "\eta_{\tilde{x},y}"] & \mor{\mathscr{C}}{\tilde{x}}{\mathcal{G}(y)}\\ + \mor{\mathscr{D}}{\mathcal{F}(x)}{y} \arrow[r, "\eta_{x,y}"] \arrow[u, "\mor{\mathscr{D}}{\mathcal{F}(x)}{y}"] & \mor{\mathscr{C}}{x}{\mathcal{G}(y)} \arrow[u, "\mor{\mathscr{C}}{f}{\mathcal{G}(y)}"']\\ + \mor{\mathscr{D}}{\mathcal{F}(x)}{\tilde{y}} \arrow[r, "\eta_{x,\tilde{y}}"] \arrow[u, "\mor{\mathscr{D}}{\mathcal{F}(x)}{g}"]& \mor{\mathscr{C}}{x}{\mathcal{G}(\tilde{y})} \arrow[u, "\mor{\mathscr{C}}{x}{\mathcal{G}(g)}"']\\ + \end{tikzcd} +\end{center} +\begin{definition}{Adjunktion von Funktoren\\} + Eine Adjunktion von $\mathcal{F}$ und $\mathcal{G}$ ist ein natürlicher Isomorphismus von\\ + $\mor{\mathscr{D}}{\mathcal{F}(\_)}{\_}\mapsto\mor{\mathscr{C}}{\_}{\mathcal{G}(\_)}$. +\end{definition} + diff --git a/main.pdf b/main.pdf index a3017b7008a76eab4c30ca0666deb383b8da62cb..fb60d8bc53de85827f0ef8f347fccf0cbfa0b01a 100644 GIT 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