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Adds stuff to presentation :D:D:D

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Joshua Moerman 2015-01-27 17:49:10 +01:00
parent 2143582542
commit e58eb67f61
2 changed files with 44 additions and 6 deletions
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49
presentation/Rational_Homotopy_Theory.tex
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@ -162,10 +162,10 @@
\Frame{Dictionary}{ \Frame{Dictionary}{
\Dict{ \Dict{
$S^n$ with $n$ odd $S^n_\Q$ with $n$ odd
& $\Lambda(e)$ with $\deg{e} = n$ \\[1em] & $\Lambda(e)$ with $\deg{e} = n$ \\[1em]
$S^n$ with $n$ even $S^n_\Q$ with $n$ even
& $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em] & $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em]
Eilenberg-MacLane space $K(\Q, n)$ Eilenberg-MacLane space $K(\Q, n)$
@ -175,12 +175,49 @@
\Frame{Dictionary}{ \Frame{Dictionary}{
\Dict{ \Dict{
homotopy $$h: X \times I \to Y$$
& homotopy $$h: A \to B \tensor \Lambda(t, dt)$$ \\[1em]
weak equivalence $$\pi_n(f): \pi_n(X) \iso \pi_n(Y)$$ weak equivalence $$\pi_n(f): \pi_n(X) \iso \pi_n(Y)$$
& weak equivalence if $$H(f): H(X) \iso H(Y)$$ & weak equivalence $$H(f): H(X) \iso H(Y)$$ \\[1em]
homotopy $$h: X \times I \to Y$$
& homotopy $$h: A \to B \tensor \Lambda(t, dt)$$
} }
} }
\Frame{Dictionary}{
\Dict{
$$ \pi_n(X) = [S^n, X] $$
& {\begin{align*}
\pi^n(A) &= H(Q(A)) \\
\pi^n(A)^\ast &\iso [A, \Lambda(e)] \\
&\text{or } [A, \Lambda(e, f)]
\end{align*}} \\[1em]
Long exact sequence of a fibration
& Long exact sequence of a cofibration \\[1em]
}
}
\Frame{Dictionary}{
\bf topological $n$-simplex
\[ \Delta^n = \left\{ (x_0, \ldots, x_n) \in \R^{n+1} \,|\, \sum x_i = 1, x_i \geq 0 \right\} \]
\bigskip
\bf cdga $n$-simplex
\[ \Delta_n = \frac{\Lambda(x_0, \ldots x_n, dx_0, \ldots, dx_n)}{\langle \sum x_i - 1, \sum dx_i \rangle} \]
}
\Frame{Construction}{
\begin{center}
\begin{tikzcd}[column sep=huge, row sep=huge, ampersand replacement=\&]
\DELTA \arrow[d, "y"] \arrow[rd, "\Delta_{(-)}"] \& \\
\sSet \arrow[r, dashed, shift left = 1ex, "A"] \& \opCat{\CDGA_\Q} \arrow[l, dashed, shift left = 1ex]
\end{tikzcd}
\end{center}
\bigskip
\pause
\[ A(X) = \Hom_\sSet(X, \Delta_{(-)}) \]
}
\end{document} \end{document}

1
presentation/preamble.tex
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@ -16,6 +16,7 @@
% fancy diagrams % fancy diagrams
\usepackage{tikz} \usepackage{tikz}
\usepackage{tikz-cd}
\usetikzlibrary{matrix, arrows, decorations} \usetikzlibrary{matrix, arrows, decorations}
\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} \tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto}