From 66a2fc6de283e8a2d74c8818d0e16aa3446b43b9 Mon Sep 17 00:00:00 2001 From: CDaut Date: Wed, 28 Sep 2022 13:30:33 +0200 Subject: [PATCH] Hom Functor and more examples --- chapters/Grundlagen.tex | 40 ++++++++++++++++++++++++++++++++++++---- main.pdf | Bin 271324 -> 285495 bytes 2 files changed, 36 insertions(+), 4 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index 0d3372c..4634e37 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -171,14 +171,15 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \path (W) edge node[left]{$f$} (X) - (VW) edge (VX) + (VW) edge node[right]{$f\circ \mathcal{F}$} (VX) (FVW) edge (FVX) (W) edge node[above]{$\mathcal{F}$} (VW) + (X) edge node[above]{$\mathcal{F}\circ f$} (VX) ; \end{tikzpicture} \end{center} Also werden Objekte aus der Definitions- auf Objekte aus der Bildkategorie und Morphismen aus der Definitions- auf Morphismen - aus der Bildkategorie abgebildet.\\ + aus der Bildkategorie abgebildet. (Das Diagramm muss nicht unbedingt kommutieren)\\ Bleibt noch zu zeigen, dass $\mathcal{F}(g\circ f)=\mathcal{F}(g)\circ\mathcal{F}(f)$ gilt. \begin{center} \begin{tikzpicture}[every edge/.style = {draw, -to}] @@ -208,7 +209,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{tikzpicture} \end{center} und - \begin{center} + \begin{center} \begin{tikzpicture}[every edge/.style = {draw, -to}] \node(VW1) at (0,0){$V\times W_1$}; \node(VW2) at (2,0){$V\times W_2$}; @@ -227,6 +228,37 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. ; \end{tikzpicture} \end{center} - Man sieht also, dass $\mathcal{F}(g\circ f)=\mathcal{F}(g)\circ\mathcal{F}(f)$ gilt. + Man sieht also, dass $\mathcal{F}(g\circ f)=\mathcal{F}(g)\circ\mathcal{F}(f)$ gilt. Damit ist $\mathcal{F}$ ein Funktor. +\end{example} +\begin{example}{Der Hom-Funktor (Kovariant)\\} + Es sei $\mathscr{C}$ eine Kategorie, $X\in\ob\mathscr{C}$\\ + Definiere den Funktor $\mor{\mathscr{C}}{X}{\_}: \mathscr{C}\mapsto\cat{Set}$ + \begin{itemize} + \item $Y\mapsto\mor{\mathscr{C}}{X}{Y}$\\ + Ein $Y\in\mathscr{C}$ wird also auf die Morphismenmenge $\mor{\mathscr{C}}{X}{Y}$, die Morphismen, die in $\mathscr{C}$ von X auf Y existieren, abgebildet. + \item $[Y\xrightarrow{f}Z]\mapsto\mor{\mathscr{C}}{X}{f}:= + \left[ + \begin{aligned} + \mor{\mathscr{C}}{X}{Y}\mapsto\mor{\mathscr{C}}{X}{Z} \\ + [g:X\mapsto Y]\mapsto[f\circ g:X\mapsto Z] + \end{aligned} + \right] + $\\ + Der Funktor bildet also jeden Morphismus $f$ von $Y$ nach $Z$ auf die oben definierte Funktion ab. + \end{itemize} +\end{example} +\begin{example}{$\cat{R1ng}\mapsto\cat{Grp}$\\} + $R\mapsto R^\times=\{r\in R|\exists s\in R:rs=1_R\}$\\ + $ + [R\xrightarrow{}S]\mapsto + \left[ + \begin{aligned} + R^\times\xrightarrow{f^\times}S^\times \\ + f^\times = f + \end{aligned} + \right] + $\\ + Man schränkt den Ring also auf alle invertierbaren Ringelemente ein und + lässt nur noch die Abbildungen übrig, die ohnehin schon zwischen invertierbaren Elementen abgebildet haben. \end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 988a989c047744b1f571af95abd832dfbc1caaf3..b7d2499305c42d9ab0d9b547811d0ab36b57775b 100644 GIT binary patch delta 150571 zcmca}Twwb)!3~~_vd<#+${b2Px%laqiDo}}GIgUE>+N!H=k8;e8&~{wdf*#o;mwtd zzN#$7CI)8H4J#Nm>fgrX=HEFc_-|eO!7cHA4f(s@>udg5xb86vXVamL7jJboJ8jeW zne=kXiziD?tbc!I--J6)KUM51PFl&E5NUI-V8`b(=iV6qR=PFEe9O;29e#h(KG^3* z9NwvV>Ey8j*Et){&OW_(zSM+Ydp<65pTdzI`q{gA`{DgxUI@)wur87-X-9qi1i>Fl z|9IBlk1^YM$&dN@9F2heCItsYwJN{-d7+HI?ehd4ywK7O!7P>{Us?U zKyPZaNYrifZE2IXwMc#8H&60!o}=bffBJ#Ev%0Lv(MeOxU(C-C@aS5olH6j#b|tw&d9od>A{wSNud|^YLs{s$TcX)TCLyD zId7rVoLxtoMIZjIf9#gs8T!e4;8QJceqD|F__^}f!1J%{iAHD5jH)eoO8koo4;vBvR( 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