From 75a6c99d3fd076b00d3eb889791f4136e1439fee Mon Sep 17 00:00:00 2001 From: CDaut Date: Thu, 13 Oct 2022 16:21:42 +0200 Subject: [PATCH] =?UTF-8?q?Beispiele=20f=C3=BCr=20Kegel?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/Kegel_und_Ko.tex | 59 +++++++++++++++++++++++++++++++++++++- main.pdf | Bin 354693 -> 358104 bytes 2 files changed, 58 insertions(+), 1 deletion(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 710db90..a01ade0 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -148,5 +148,62 @@ aussehen können. \end{center} \end{example} \begin{example}{Kategorie mit zwei Objekten\\} - + Es sei I= + \begin{tikzpicture}[baseline=-1mm] + \node(A) at (0,0){$\cdot$}; + \node(B) at (0.5,0){$\cdot$}; + \node (center) at (0.25, 0){}; + + \draw (center) ellipse (0.5 and 0.25); + \end{tikzpicture} + eine kleine Kategorie. Dann suchen wir einen Limes $L$, sodass für alle Kegel + folgendes Diagramm kommutiert:\\ + \begin{center} + + + \begin{tikzpicture} + \node (Fi) at (0,0) {$\cdot$}; + \node (Fj) at (3,0) {$\cdot$}; + \node [red] (L) at (1.5,1.5) {$L$}; + \node [blue] (K) at (1.5,3) {$K$}; + + \draw + (L) edge[red] (Fi) + (L) edge[red] (Fj) + (K) edge[blue, bend right] (Fi) + (K) edge[blue, bend left] (Fj) + (K) edge[dotted] (L) + ; + \end{tikzpicture} + \end{center} + Für \cat{set}:\\ + gegeben $M,N$. Dann ist $\text{lim}\left( + \begin{tikzpicture}[baseline=-1mm] + \node(A) at (0,0){$\cdot$}; + \node(B) at (0.5,0){$\cdot$}; + \node (center) at (0.25, 0){}; + + \draw (center) ellipse (0.5 and 0.25); + \end{tikzpicture} + \right):=$ Produkt.\\ + \begin{center} + + + \begin{tikzpicture} + \node (M) at (0,0) {$M$}; + \node (N) at (3,0) {$N$}; + \node [red] (L) at (1.5,1.5) {$M\times N$}; + \node [blue] (K) at (1.5,3) {$T$}; + + \draw + (L) edge[red] node[left]{$\Pi_M$} (Fi) + (L) edge[red] node[right]{$\Pi_N$} (Fj) + (K) edge[blue, bend right] node[left]{$f_M$} (M) + (K) edge[blue, bend left] node[right]{$f_N$} (N) + (K) edge[dotted] (L) + ; + \end{tikzpicture} + \end{center} + Wobei T hier ein Testobjekt aus \cat{set}, $\Pi_M((m,n))=m$ und der Morphismus $T\mapsto M\times N$ + definiert ist als $t\mapsto (f_M(t),f_N(t))$. Analog funktionieren die Limiten für \cat{Grp} und \cat{K-VR}. \end{example} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 84488313c4092e455d9a3ac2875ffa42672aced4..fcd2dedec4080c41b5f6715e197a0a8d85d5dca4 100644 GIT binary patch delta 77229 zcmZp@EPCUu=mzU}W=kXE>5L~CHR|6+dl$=W7u@$bTw#0aRECFt6SUa1r-f{Lqta6{ zxh7)b;;UY}z1w4^t@`u&Ztrm$!vDcF z)C=1s9Gc0?7c~x=!X}^;wH97Y=SN?t$+{5LVu!||o*5aw> zu8(5Rta6Pu*i2&=T7PKL=0g*WkIU{^@R4oVh206Cmn2^ico9`0?r?McN&Ud$=LR3; zAAV=nie^b(B;#0IKS#ynW5OQhQw*POEy}MrurTF=-PXBB_ix#g_50$bjj4Xz-;F;_ znj%PBS7Ww|eRWn=L zw`RUsJMPV0KKrwal9>G*<;LdNR-5}8Yrluz?miuVfBpJxH|lCx&+3%cS6#ZW=d4I@ zjF6UNR_>)Y-=?27mU&tGgw1E=rm~NjwKMY`>8wj$9j#-2pk#B?$+gAa%)A>;US>YQ ze#p`L-u3?Zl9#XdJ!w0rKGXc!ec_WHjS0y+bX7caX1L0o*V`AbA3pu`TeE03(;MHq zEZ=+IPd7cSP*BNr)HAnFY)!YUZ(+R!&xe9_s)h9{r8Lj>7%MBAP7x_{N?#H&?cI#7 zqe3D(PQQv>YyNvn$*HU#iwfR8_`Uh2&HjwAgxTF9eXPePFKTpam^krFmw`a)3clkl z7nityZIh0h%zo_JC%4(%kG6;$aFYpsS#$JP)QuIl1Q(>mooIe${&iQ((|7;t;_~Ng zn^W)Sv)#Z&N^HrlsW1IZ04{nWHWA|qo`=370xfBp7~ z*GH~3muwM!KV^&FuPx7xDVbgls$V9Z{nT_r!iJZ8CuXl~`*dx|>l?mHvX|ffp7OPy z@tK~ggiwKQw2*vZwDrRi%Zejxezi6pXnocHW^UE#s`^y~K<$Yh%=ccem6Io4jDlFLesukVCxRA7ukU8+ z&Ju87O`Ez)FsmZM;BIk8``=f)jiA%Gl0gvgoS(&qw*@{LOGvPM>db zSxfEb&amY{tmc=2gdv6vNefA8XsT)2yQkzBd#tBLh# zKhq4qo_;v*oxYeCuR~b#d~wzWwm5FZq{~{ItwJsFhAz`q}w}%Gp-UNYj^00)i}z z?m{dI8hexkWHQ}EY%7A+Jw3YP`Qe{U%L+EvTZG;Eyk={Pw1nmQ0|G@XrZ?N_zJGgL zRb00>S~fWSuG72+SCiGOOm)m>hx)ZY{1?1tf#R7=(~iG;-f!~xAR_kup2l+h%d^D~ z@3VUD8)oRrw?gplv)gAsSv{0p8gef2;OhC;Zl939HR08#r1&|HXGr&5Puy{uC-(Z; z(BhdNvb?s`Kh=r8zf3cL^FZ;JH3wJiRu=l}+ToHpHP`G|z^QXL-Z+J=$a7P9RwvN8 zpL1eQdWfUSuWY4BCJlibGM@cY^xUq~m@HD65ovVQJZhTs#`6_@S6dg&RoXB1!;r11 zY`(*i{q-LfTV8u)vu;KMe-Ymwkxfpe>>rHZSuxp1+TG%;P_O^6SoYbg0|o#4o8=ia zPq}Srn(3OFx@Jq`b)#!9*KPd#skT(*uTO`(-o2Mw8qKer?y1<*oRP-&@kHbaWzRR# z|3oIJTj$;Sd`I-iVuQIHY3B>h3OM!)_CIjayW+VgFxK>?*8ZJYW}#n#Hn4>@e&5J% zV>_#hEBtjNEzBaGWvRuje?o$`Hb{XXf zK3~%vrte=W8nb-)jCRv*enZY~Z%dmc7aN}~=g64Ls=0BwaksqIab1JIItL%uKlpm? 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