From e26185ed08c61e7b42bfc02a1526718402f12617 Mon Sep 17 00:00:00 2001 From: CDaut Date: Tue, 11 Oct 2022 10:45:11 +0200 Subject: [PATCH] Hom Funktor --- chapters/Grundlagen.tex | 12 ++++++++---- main.pdf | Bin 300764 -> 301423 bytes 2 files changed, 8 insertions(+), 4 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index 8bfe5f5..974d144 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -237,7 +237,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \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}:= + \item $[Y\xrightarrow{f}Z]\mapsto f^*=\mor{\mathscr{C}}{X}{f}:= \left[ \begin{aligned} \mor{\mathscr{C}}{X}{Y}\mapsto\mor{\mathscr{C}}{X}{Z} \\ @@ -245,7 +245,11 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{aligned} \right] $\\ - Der Funktor bildet also jeden Morphismus $f$ von $Y$ nach $Z$ auf die oben definierte Funktion ab. + Gehen wir also von beliebigen $X,Y\in\mathscr{C}$ aus, zwischen denen ein Morphismus $f$ existiert, so werden diese vom Hom-Funktor auf die + Morphismenmengen $\mor{\mathscr{C}}{X}{Y}$ und $\mor{\mathscr{C}}{X}{Z}$ abgebildet. Das heißt, dass wir jetzt + noch definieren müssen, worauf die Morphismen $\mor{\mathscr{C}}{Y}{Z}$ abgebildet werden. Wir müssen also jeden Morphismus aus $\mor{\mathscr{C}}{Y}{Z}$ auf einen Morphismus $\mor{\cat{set}}{\mor{\mathscr{C}}{X}{Y}}{\mor{\mathscr + {C}}{X}{Z}}$ abbilden. Dies geschieht, indem wir ihn einfach mit einem entsprechenden passenden Morphismus verknüpfen. Wir identifizieren also jeden Morphismus $g: X\mapsto Y$ mit seiner Verknüpfung mit $f: Y\mapsto Z$ + und erhalten so einen neuen Morphismus $f^*$, der die geforderten Bedingungen erfüllt. \end{itemize} \end{example} \begin{example}{$\cat{R1ng}\mapsto\cat{Grp}$\\} @@ -336,7 +340,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \begin{itemize} \item $Y\mapsto\mor{\mathscr{C}}{Y}{X}$\\ Ein $Y\in\mathscr{C}$ wird also auf die Morphismenmenge $\mor{\mathscr{C}}{Y}{X}$, die Morphismen, die in $\mathscr{C}$ von Y auf X existieren, abgebildet. - \item $[Y\xrightarrow{f}Z]\mapsto\mor{\mathscr{C}}{X}{f}:= + \item $[Y\xrightarrow{f}Z]\mapsto f^* =\mor{\mathscr{C}}{X}{f}:= \left[ \begin{aligned} \mor{\mathscr{C}}{Z}{X}\mapsto\mor{\mathscr{C}}{Y}{X} \\ @@ -344,7 +348,7 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. \end{aligned} \right] $\\ - Der Funktor bildet also jeden Morphismus $g$ von $Z$ nach $X$ auf die oben definierte Funktion ab. + Der Funktor agiert analog zu seinem Kovarianten Gegenstück. \end{itemize} \end{example} \begin{example}{Punktierung von Mengen\\} diff --git a/main.pdf b/main.pdf index 9362d0377dfd073db7252c7ae918b64fc745118c..9812067cb52675a87886214106daacfaac6199b1 100644 GIT binary patch delta 77287 zcmcb!R_OgEp$P%?zWzrP1lr#J6)kRI<~ST)ZF=*rORrqco)zp{a&FI9y6pDX@7#wD z9%)vOlXzCyrjWqG*zWdcbKLi{SK~At&c?o0`Qkod?XyFg1uJ$Ki~VApck=P_`oMF6 z%Szn+K8GDzd;idmHN_oYw(j>-;MLf%HN&W5X5Zqx%d0G;LLGKnwq2>$xizP+bLH}{ zLdR>aOthSK$>6&6ewK`rX03BppKiDkli6+WJbTHDhRB0k4=&pE{r%2tFO8QT9*=@M z{kMpwL~g&DzCP!qxU2iLh*jl@ou30fC4QYg$tHEumAWl4OO|sUvEjP$WZHox4)=Dx zx)*$1u-Eohx*Vo_JqA!x*_q@(?Li@75U=@xo1vg=HKCW9)Cwo*+iLTu7 zVf}l{;)n0foGa1{>Rh_5A@|+6dA8Qit?tk4&Mdxo;n`|ot52sN#NXP`R_wiGa__?~ zi{#nY!`1(1HN0AVZ1TJSr8%Kzf3wX#JpaopnR^S?&E!tnQ9rR>utMPfxApPyrk0m= z+0R)99;nNE(4?L6>FDRl_O_EhOi?PoW@Dc(6sl5HckPeK?}uNWKaZbZ^K*~7ha1bB zeXs3As&>8Au=$y%(5vDn_2Xj8;rES?I2KLlH0zT%==PK|rSlZ`m&XAM=4egV__FPF 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