From 23f00e9d9bd18976dc2ed6b9ab49b941a5180d53 Mon Sep 17 00:00:00 2001 From: CDaut Date: Thu, 20 Oct 2022 15:34:07 +0200 Subject: [PATCH] Bidualabbildung --- chapters/nat_trafo.tex | 50 ++++++++++++++++++++++++++++++++++++++--- main.pdf | Bin 323097 -> 327830 bytes 2 files changed, 47 insertions(+), 3 deletions(-) diff --git a/chapters/nat_trafo.tex b/chapters/nat_trafo.tex index 3331eb8..bc446d0 100644 --- a/chapters/nat_trafo.tex +++ b/chapters/nat_trafo.tex @@ -10,8 +10,8 @@ \end{tikzcd} Funktoren.\\ Eine natürliche Transformation (nat. Trafo.) $\eta$ von $\mathcal{F}$ nach $\mathcal{G}$ ist eine Sammlung von Morphismen wie folgt:\\ - $\forall x\in\ob\mathscr{C}$ gilt $\mathcal{F}(x)\xrightarrow{\eta_x}\mathcal{G}(x)$\\ - $x\mapsto\eta_x\in\mor{\mathscr{D}}{\mathcal{F}(x)}{\mathcal{G}(x)}$. Folgendes Diagramm soll also kommutieren:\\ + $\forall x\in\ob\mathscr{C}$ gilt $\mathcal{F}(x)\xrightarrow{\eta_x}\mathcal{G}(x)$, sodass\\ + $\forall f\in\mor{\mathscr{C}}{X}{Y}$ gilt: $\mathcal{G}(f)\circ\eta_x=\eta_y\circ\mathcal{F}(f)$. Folgendes Diagramm soll also kommutieren: \begin{figure}[h] \begin{center} \begin{tikzcd}[sep=large] @@ -25,4 +25,48 @@ \end{center} \caption{Das kommutative Diagramm für eine natürliche Transformation $\eta$.} \end{figure} -\end{definition} \ No newline at end of file +\end{definition} +\begin{example}{Bidualisierung\\} + Gegeben seien Die Funktoren:\\ + $\mathbb{D}:\cat{K-VR}\mapsto\cat{K-VR}$\\ + $V\mapsto V^{*^*}$ und\\ + $id: \cat{K-VR}\mapsto\cat{K-VR}$\\ + $V\mapsto V$\\ + Wir zeigen, dass eine nat. Trafo. $id\xrightarrow{\eta}\mathbb{D}$ existiert.\\ + Dazu definieren wir $\forall V\in\ob\cat{K-VR}:\eta_V:V\mapsto V^{*^*}$ mit $v\mapsto[\alpha\mapsto \alpha(v)]$. + Das folgende Diagramm muss außerdem kommutieren:\\ + \begin{center} + \begin{tikzcd}[sep=large] + V \arrow[d, "\psi"'] \\ W + \end{tikzcd} + \hspace{20mm} + \begin{tikzcd}[sep=large] + V \arrow[r, "\eta_V"] \arrow[d, "\psi"]& V^{*^*} \arrow[d, "\psi^{*^*}"]\\ + W \arrow[r, "\eta_W"] & W^{*^*} + \end{tikzcd} + \end{center} + Desweiteren gilt für die Dual- und Bidualabbildung: $W^*\xrightarrow{f^*}V^*:\alpha\mapsto\alpha\circ f$ und $V^{*^*}\xrightarrow{f^{*^*}}W^{*^*}:\alpha\mapsto\alpha\circ f^*$\\ + Bleibt noch die Kommutativität des oben genannten Diagramms zu zeigen:\\ + Sei $v\in V$ und $\alpha\in W^*$.\\ + \underline{Behauptung:} + \begin{flalign*} + & [(\psi^{*^*}\circ\eta_V)(v)](\alpha) \\ + & =[(\eta_W\circ\psi)(v)](\alpha) + \end{flalign*} + \underline{Beweis:}\\ + \begin{flalign*} + & (\psi^{*^*}(\eta_V(v)))(\alpha) \\ + & =(\eta_V(v)\circ\psi^*)(\alpha) \\ + & =\eta_V(v)(\alpha\circ\psi) \\ + & =(\alpha\circ\psi)(v) + \end{flalign*} + und + \begin{flalign*} + & ((\eta_W\circ\psi)(v))(\alpha) \\ + & = (\eta_W(\psi(v)))(\alpha) \\ + & =\alpha(\psi(v)) \\ + & =(\alpha\circ\psi)(v) + \end{flalign*} + $\implies$ Beh.\qed + +\end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index e974e25e010c1d284323f84b7272ff72d763f45e..a3017b7008a76eab4c30ca0666deb383b8da62cb 100644 GIT binary patch delta 81273 zcmbPvM|fJJ$b|6vnl9so+;UjL`F#0T-y}8t-rh|M^9__xWPPbNnX zRopvuB))#@x^DfSUkeX#rRJp+mrQSe%EY@liE+b57GpDWJ&|y2&#t_C zIzLjq4UTo!>+E6(?WrjB{P?JD=k5D+YKGuWsjY z+1s@pvShA#pnl2awmD~$`osLHzlpQ=au`kDbue*VQb1pez`ut_3jQkXp3ZsHgZtK& zplp?%4T~+-a?kZLo#d*eKChm~B{<+vd&P+8+ zo)pRUaArR4Rg(_q&1L3`W?ZKId-=yYgU)kGf1>yI)c@}mOcz?l$ehjT91&PCSK**( zt^-Gl(khl4sXMl2>cv%MicMm6%9B_;@q+gU-_Pd)V-8>1*Si0I>syXv=?3{4lU#&n zEN)fhKG!STrp>jdQ9?3|=U$4$yf0BN<+px}{9bl@uUvY@VG;RR)6aIR8O_O6P}w-u 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