From f5ee193e30c0ea0291fbe2b5350c04b04bc8194d Mon Sep 17 00:00:00 2001 From: CDaut Date: Tue, 27 Sep 2022 20:13:43 +0200 Subject: [PATCH] example on functors --- chapters/Grundlagen.tex | 75 ++++++++++++++++++++++++++++++++++++++-- main.pdf | Bin 259180 -> 271324 bytes 2 files changed, 73 insertions(+), 2 deletions(-) diff --git a/chapters/Grundlagen.tex b/chapters/Grundlagen.tex index a19ce57..0d3372c 100644 --- a/chapters/Grundlagen.tex +++ b/chapters/Grundlagen.tex @@ -156,6 +156,77 @@ zu können und diese so in Relation zu setzen definieren wir Funktoren. $(V,+,\cdot)\mapsto V$ bildet einen Vektorraum auf die Menge der Vektoren ab und vergisst dabei alle anderen Informationen wie z.B. über welchem Körper der Vektorraum definiert war und jegliche Operationen. \end{example} -\begin{example} - Sei V ein $\mathbb{K}$-Vektorraum +\begin{example}{Ein Endofunktor\\} + Sei V ein $\mathbb{K}$-Vektorraum. $\mathcal{F}:\cat{K-Vec}\mapsto\cat{K-Vec}$ mit $W\mapsto V\times W$. Funktoren mit gleicher Definitions und Bildkategorie nennt man Endofunktoren. + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(W) at (0,0){$W$}; + \node(X) at (0,-1.5){$X$}; + + \node(VW) at (3,0){$V\times W$}; + \node(VX) at (3,-1.5){$V\times X$}; + + \node(FVW) at (4.5,0){$(v,w)$}; + \node(FVX) at (4.5,-1.5){$(v,f(w))$}; + + \path + (W) edge node[left]{$f$} (X) + (VW) edge (VX) + (FVW) edge (FVX) + (W) edge node[above]{$\mathcal{F}$} (VW) + ; + \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.\\ + 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}] + \node(W1) at (0,0){$W_1$}; + \node(W2) at (2, 0){$W_2$}; + \node(W3) at (4,0){$W_3$}; + + \path + (W1) edge node[above]{$f$} (W2) + (W2) edge node[above]{$g$} (W3) + (W1) edge[bend right=20] node[below]{$g\circ f$} (W3); + \end{tikzpicture} + \end{center} + Also: + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(VW1) at (0,0){$V\times W_1$}; + \node(VW3) at (4, 0){$V\times W_3$}; + \node(vw) at (0,-1){$(v,w)$}; + \node(vw3) at (4,-1){$(v,(g\circ f)(w))$}; + \node(Func) at (6.5,-1){$\mathcal{F}(g\circ f)$}; + + \path + (VW1) edge (VW3) + (vw) edge (vw3) + ; + \end{tikzpicture} + \end{center} + und + \begin{center} + \begin{tikzpicture}[every edge/.style = {draw, -to}] + \node(VW1) at (0,0){$V\times W_1$}; + \node(VW2) at (2,0){$V\times W_2$}; + \node(VW3) at (4, 0){$V\times W_3$}; + \node(vw1) at (0,-1){$(v,w)$}; + \node(vw2) at (2,-1){$(v,f(w)$}; + \node(vw3) at (4,-1){$(v,g(f(w))$}; + \node(Func) at (6.5,-1){$\mathcal{F}(g)\circ \mathcal{F}(f)$}; + + \path + (VW1) edge (VW2) + (VW2) edge (VW3) + + (vw1) edge (vw2) + (vw2) edge (vw3) + ; + \end{tikzpicture} + \end{center} + 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} \ No newline at end of file diff --git a/main.pdf b/main.pdf index 7d491550e1ef601592340ab3ecf8064b002050b3..988a989c047744b1f571af95abd832dfbc1caaf3 100644 GIT binary patch delta 59483 zcmaF!g8$BOfejv#%*N)XlP550)L)H`?3Xqds{6kFMeXGTWxf1O$t>MJPN}tR{BTQ} zdG^WXv@6Qxiw;jTkvqP>ZgoY8x#gRQ&o~m8<;u+?RaSQ$UDcn`8P?quW^c1d*`MwI z<*v{<*N%KR#o?**{OZqF7v)6-{svf{a-Y(1Yt^e&hm!uU&tBOnx@K09(#|7oJr1t* zKcx4~U%zV0u{*o0Es-|lrzkXZPl``?w+daQhi@Y?g ze|~-I-#>5v+sIQK3wK{X{LtpO?BRg#R@1#S5B2PvA0kx}IlDQCb<^RX)RvhcjgO_5 zZfeq4v3GjNR)v^Ts1iv%g4Ol=cM8oB43!ts3Xq+mx%jjT*PDAfLTj8mc>)*p>{{#- z)Z}WhP-Vr9FHRR-Ryc^QoD*_?p3h9Pe+wT@bJ}C3#p}x73F% zD0#8+TFmix51D7E?b&v1`rO_Bmv3B^ZQjef``q%4-#@?d{km&am87Zq6mH2>6Vr!X z+@+g)IIm9EPv$)%$S+a^f|e&!1Ce! zU0%#f*qJXgN)_}utMnaiI(dyja)nw9`~Ht>a_jvRTwa{)|D0BP=xxgVcm4;X-;}RD z`|a@Wvh7Vt?gcyNJ?`$(uln?Us`cO&J(WzlWf 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