From 644cdd5a0287b9dcf65e967108845ea905ffe1fd Mon Sep 17 00:00:00 2001 From: CDaut Date: Sun, 16 Oct 2022 00:19:14 +0200 Subject: [PATCH] =?UTF-8?q?Pullback=20=C3=BCber=20KVR=20und=20Pushout=20de?= =?UTF-8?q?finition?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- chapters/Kegel_und_Ko.tex | 97 ++++++++++++++++++++++++++++++++++++-- main.pdf | Bin 393492 -> 397194 bytes 2 files changed, 94 insertions(+), 3 deletions(-) diff --git a/chapters/Kegel_und_Ko.tex b/chapters/Kegel_und_Ko.tex index 9f52a18..126f515 100644 --- a/chapters/Kegel_und_Ko.tex +++ b/chapters/Kegel_und_Ko.tex @@ -430,7 +430,7 @@ aussehen können. \node (A) at (0,0){$\cdot$}; \node (B) at (2,0){$\cdot$}; \node (C) at (2,2){$\cdot$}; - \node[red] (L) at (0,2){$L$}; + \node[red] (L) at (0,2){$P$}; \node[blue] (T) at (-1,3){$T$}; \draw @@ -448,5 +448,96 @@ aussehen können. \end{figure} \end{definition} \begin{example}{Pullback über der Kategorie \cat{K-VR}\\} - -\end{example} \ No newline at end of file + Damit $P$ ein Pullback ist, muss folgendes Diagramm kommutieren:\\ + \begin{center} + \begin{tikzpicture} + \node (V) at (0,0){$V$}; + \node (X) at (2,0){$X$}; + \node (W) at (2,2){$W$}; + \node[red] (L) at (0,2){$P$}; + + \draw + (V) edge node[below]{$f_V$} (X) + (W) edge node[right]{$f_W$} (X) + (L) edge[red] node[left]{$\Pi_V$} (V) + (L) edge[red] node[above]{$\Pi_V$}(W) + ; + + \end{tikzpicture} + \end{center} + Mit $P:=\{(v,w)\in V\times W|f_V(v)=f_W(w)\}$\\ + Es ist also zu zeigen, dass: + \begin{itemize} + \item P ein $\mathbb{K}$-Vektorraum ist + \item P Limes über dem Diagramm ist + \end{itemize} + $P$ ist $\mathbb{K}$-Vektorraum:\\ + $(v,w)+(\tilde{v},\tilde{w})=(v+\tilde{v},w+\tilde{w})$\\ + $f_V(v+\tilde{v})=f_V(v)+f_V(\tilde{v})=f_W(w)+f_W(\tilde{w})=f_W(w+\tilde{w})$\\ + $\implies P$ ist $\mathbb{K}$-Vektorraum.\\ + P ist ein Limes: + Da $P$ Morphismen auf alle Objekte im Bild des Diagramms hat, ist $P$ ein Kegel.\\ + Betrachte folgendes Diagramm: + \begin{center} + \begin{tikzpicture} + \node (V) at (0,0){$V$}; + \node (X) at (2,0){$X$}; + \node (W) at (2,2){$W$}; + \node[red] (P) at (0,2){$P$}; + \node[blue] (T) at (-1,3){$T$}; + + \draw + (V) edge node[below]{$f_V$}(X) + (W) edge node[right]{$f_W$} (X) + (P) edge[red] node[left]{$\Pi_V$}(V) + (P) edge[red] node[above]{$\Pi_W$} (W) + (T) edge[dotted] node[above]{$\varphi$} (P) + (T) edge[blue, bend right] node[left]{$g_V$} (V) + (T) edge[blue, bend left] node[above]{$g_W$} (W) + ; + \end{tikzpicture} + \end{center} + Wir müssen also ein $\varphi$ definieren, sodass $\varphi\circ\Pi_V=g_V$ und + $\varphi\circ\Pi_W=g_W$ gilt und das Diagramm kommutiert.\\ + Definiere dazu $\varphi:T\mapsto P$ als $t\mapsto (g_V(t),g_W(t))$\\ + $\implies P$ ist Limes.\\ + Da $P$ ein Vektorraum und Limes über dem Pullbackdiagramm ist, ist $P$ Pullback über \cat{K-VR}. + \qed +\end{example} +\begin{definition}{Pushout\\} + Ein Pushout ist der Kolimes des Diagramms + \begin{tikzpicture}[baseline=-5mm] + \node (X) at (0,0){$\cdot$}; + \node (V) at (0,-1){$\cdot$}; + \node (W) at (1,0){$\cdot$}; + + \draw + (X) edge (V) + (X) edge (W) + ; + \end{tikzpicture}. + Also ein Kolimes, sodass folgendes Diagramm kommutiert:\\ + \begin{figure}[h] + \begin{center} + \begin{tikzpicture} + \node (X) at (0,0){$\cdot$}; + \node (V) at (0,-2){$\cdot$}; + \node (W) at (2,0){$\cdot$}; + \node[red] (P) at (2,-2){$P$}; + \node[blue] (T) at (3,-3){$T$}; + + \draw + (X) edge (V) + (X) edge (W) + (V) edge[red] (P) + (W) edge[red] (P) + (V) edge[blue, bend right] (T) + (W) edge[blue, bend left] (T) + (P) edge[dotted] (T) + ; + + \end{tikzpicture} + \end{center} + \caption{Das kommutative Diagramm für einen Pushout.} + \end{figure} +\end{definition} \ No newline at end of file diff --git a/main.pdf b/main.pdf index edc4412c53b9c54ab5f920c4f8d8fa3b9c1991ea..c882bf3c676ebef2f86ad82a7877529fb217ebb6 100644 GIT binary patch delta 27844 zcmbQzB+)frVuJ^xUMYb0Xus9%ds`DGInRshSZ=(ND=Xc&FKJ_h{LN)oBVAYLf6v&p&*s+eSJF?lj&%fgg}X1GTH0bA zC;6^p@ggg$mG|^mCjK@}j?t~#6}kSte*bsY=$c#W>stf6XMdht@%i%EtL@#hU;n+| zx9|0r`menPqKhv)nGx={dRFkFm`lG`Rl2v@#K*4JJGbzN(U&DBBA8{fHq6m@`9VV? 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