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Trying to figure out tikz

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Joshua Moerman 2013-05-09 13:21:47 +02:00
parent f050a2e977
commit 439c9f6178
2 changed files with 27 additions and 2 deletions
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16
thesis/3_SimplicialAbelianGroups.tex
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@ -95,7 +95,7 @@ Note that indeed $\Hom{\DELTA}{X}{[n]} \in \Set$, because the collection of morp
\begin{example}
We will compute how $\Delta[0]$ look like. Note that $[0]$ is an one-element set, so for any set $X$, there is only one function $\ast : X \to [0]$. Hence $\Delta[0]_n = \{\ast\}$ for all $n$. The face and degeneracy maps are now functions from $\{\ast\}$ to $\{\ast\}$. Again there is only one, namely $\id : \{\ast\} \to \{\ast\}$. This gives:
\todo{sAb: insert picture}
$$ \Delta[0] = \{\ast\} \to \{\ast\} \to \{\ast\} \to \cdots. $$
\end{example}
\begin{example}
@ -115,6 +115,20 @@ Note that indeed $\Hom{\DELTA}{X}{[n]} \in \Set$, because the collection of morp
\delta^1(\delta_0\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_0 \\
\delta^1(\delta_1\sigma_0) &= \delta_0 \sigma_0 \delta_1 = \delta_1.
\end{align*}
\begin{tikzpicture}
\matrix (m) [matrix of math nodes] {
\Delta[1] = & \{x_0, x_1\} & \{\sigma^0 x_0, \id, \sigma^0 x_1\} & \{ \} & \cdots \\
};
\foreach \r in {-5, 5} \draw [raise line=\r, <-] (m-1-2) -> (m-1-3);
\foreach \r in {0} \draw [raise line=\r, ->] (m-1-2) -> (m-1-3);
\foreach \r in {-10, 0, 10} \draw [raise line=\r, <-] (m-1-3) -> (m-1-4);
\foreach \r in {-5, 5} \draw [raise line=\r, ->] (m-1-3) -> (m-1-4);
\foreach \r in {-15, -5, 5, 15} \draw [raise line=\r, <-] (m-1-4) -> (m-1-5);
\foreach \r in {-10, 0, 10} \draw [raise line=\r, ->] (m-1-4) -> (m-1-5);
\end{tikzpicture}
\end{example}
As we are interested in simplicial abelian group, it would be nice to make these $n$-simplices into simplicial abelian groups. We have seen how to make an abelian group out of any set using the free abelian group. We can use this functor $\Z[-] : \Set \to \Ab$ to induce a functor $\Z^\ast[-] : \sSet \to \sAb$ as shown in the diagram~\ref{fig:diagram_Z}.

13
thesis/preamble.tex
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@ -6,9 +6,20 @@
\usepackage{mathtools}
\usepackage{tikz} % http://pdp7.org/blog/?p=133
\usetikzlibrary{matrix,arrows}
\usetikzlibrary{matrix, arrows, decorations}
\tikzset{node distance=3em, row sep=3em, column sep=3em, auto}
\pgfdeclaredecoration{single line}{initial}{
\state{initial}[width=\pgfdecoratedpathlength-1sp]{\pgfpathmoveto{\pgfpointorigin}}
\state{final}{\pgfpathlineto{\pgfpointorigin}}
}
\tikzset{
raise line/.style={
decoration={single line, raise=#1}, decorate
}
}
\newcommand{\N}{\mathbb{N}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\R}{\mathbb{R}}