Joshua Moerman
10 years ago
3 changed files with 365 additions and 0 deletions
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.PHONY: thesis fast dirs |
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# We don want to pollute the root dir, so we use a build dir
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# http://tex.stackexchange.com/questions/12686/how-do-i-run-bibtex-after-using-the-output-directory-flag-with-pdflatex-when-f
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thesis: dirs |
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xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex |
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xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex |
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cp build/Rational_Homotopy_Theory.pdf ./ |
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fast: dirs |
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xelatex -file-line-error -output-directory=build Rational_Homotopy_Theory.tex |
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cp build/Rational_Homotopy_Theory.pdf ./ |
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haltfast: dirs |
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xelatex -file-line-error -output-directory=build --halt-on-error Rational_Homotopy_Theory.tex |
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cp build/Rational_Homotopy_Theory.pdf ./ |
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dirs: |
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mkdir -p build |
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\documentclass[14pt]{beamer} |
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\definecolor{todocolor}{rgb}{1, 0.3, 0.2} |
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\newcommand{\td}[1]{\colorbox{todocolor}{*\footnote{TODO: #1}}} |
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\input{preamble} |
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\usepackage{tabularx} |
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\renewcommand{\tabularxcolumn}[1]{p{#1}} |
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\newcommand{\Frame}[2]{ |
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\begin{frame}{#1}#2\end{frame} |
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} |
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\title{Rational Homotopy Theory} |
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\author{Joshua Moerman} |
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\institute[Radboud Universiteit Nijmegen]{Supervisor: Ieke Moerdijk} |
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\date{} |
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\begin{document} |
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\AtBeginSection[]{ |
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\begin{frame}<beamer> |
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\tableofcontents[currentsection] |
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\end{frame} |
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} |
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\Frame{}{ |
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\titlepage |
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} |
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\section{Introduction to homotopy theory} |
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\Frame{Homotopy theory}{ |
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\begin{center} |
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Study of space or shapes \\ |
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with ``weak equivalences'' |
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\bigskip |
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\td{plaatje} |
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\end{center} |
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} |
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\Frame{Important spaces}{ |
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\td{plaatjes} |
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\begin{align*} |
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S^1 &= ... \\[1em] |
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S^2 &= ... \\[1em] |
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S^3 &= \cdots \\[1em] |
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&\cdots |
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\end{align*} |
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} |
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\Frame{Important tool}{ |
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Fundamental group: |
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\[ \pi_1(X) = \text{maps } S^1 \to X \text{ up to homotopy} \] |
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\bigskip |
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\td{plaatje} |
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} |
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\Frame{Important tools}{ |
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Homotopy groups: |
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\begin{align*} |
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\pi_1(X) &= \text{maps } S^1 \to X \text{ up to homotopy} \\[1em] |
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\pi_2(X) &= \text{maps } S^2 \to X \text{ up to homotopy} \\[1em] |
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\pi_3(X) &= \text{maps } S^3 \to X \text{ up to homotopy} \\[1em] |
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&\cdots |
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\end{align*} |
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} |
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\Frame{Torsion-free}{ |
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Serre proved in 1950s: |
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\begin{align*} |
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\text{odd } k: \quad \pi_n(S^k) \tensor \Q &= |
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\begin{cases} |
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\Q &\text{ if } n = k \\ |
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0 &\text{ otherwise } |
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\end{cases} \\[1em] |
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\text{even } k: \quad \pi_n(S^k) \tensor \Q &= |
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\begin{cases} |
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\Q &\text{ if } n = k, 2k-1 \\ |
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0 &\text{ otherwise } |
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\end{cases} \\ |
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\end{align*} |
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} |
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\section{Rational homotopy theory} |
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\Frame{Rational homotopy theory}{ |
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\begin{center} |
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Study of spaces\\ |
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with ``rational equivalences'' \\ |
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and ``rational homotopy groups'' |
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\pause |
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\bigskip |
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or |
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\bigskip |
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Study of \emph{rational} spaces \\ |
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with weak equivalences \\ |
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and ordinary homotopy groups |
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\end{center} |
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} |
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\Frame{Rational spaces}{ |
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$X$ is \emph{rational} if $\pi_n(X)$ is a $\Q$-vector space |
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\bigskip |
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\td{plaatje telescoop} |
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} |
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\section{The main equivalence} |
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\Frame{Main equivalence}{ |
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\begin{theorem} |
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\begin{center} |
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Homotopy theory of rational spaces \\ |
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= \\ |
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Homotopy theory of commutative differential graded algebras |
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\end{center} |
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\end{theorem} |
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} |
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\Frame{Main equivalence (precise version)}{ |
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\begin{theorem} |
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\[ \Ho(\Top_{\Q, 1, f}) \simeq \opCat{\Ho(\CDGA_{\Q, 1, f})} \] |
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\end{theorem} |
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} |
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\Frame{What is a cdga?}{ |
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\begin{definition} |
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a cdga $A$ is |
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\begin{itemize} |
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\item a $\Q$-vector space |
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\item with a multiplication $A \tensor A \tot{\mu} A$ |
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\item with a differential $A \tot{d} A$ such that $d^2 = 0$ |
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\item with a grading $A = \bigoplus_{n \in \N} A^n$ |
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\item it is commutative: $ x y = (-1)^{\deg{x}\cdot\deg{y}} y x $ |
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\end{itemize} |
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\end{definition} |
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} |
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\Frame{Free cdga's}{ |
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As always: there is a free guy: $\Lambda(...)$ |
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For example |
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\[ \Lambda(t, dt) \text{ with } \deg{t} = 0 \] |
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is just polynomials in $t$, with its differential $dt$ |
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} |
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\newcommand{\Dict}[1]{ |
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\noindent |
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\begin{tabularx}{\textwidth}{ X X } |
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{\bf rational spaces} & {\bf cdga's} \\[1em] |
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#1 |
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\end{tabularx} |
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} |
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\Frame{Dictionary}{ |
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\Dict{ |
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$S^n$ with $n$ odd |
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& $\Lambda(e)$ with $\deg{e} = n$ \\[1em] |
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$S^n$ with $n$ even |
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& $\Lambda(e, f)$ with $\deg{e} = n$, $\deg{f} = 2n-1$ and $d f = e^2$ \\[1em] |
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Eilenberg-MacLane space $K(\Q, n)$ |
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& $\Lambda(e)$ with $\deg{e} = n$ |
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} |
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} |
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\Frame{Dictionary}{ |
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\Dict{ |
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homotopy $$h: X \times I \to Y$$ |
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& homotopy $$h: A \to B \tensor \Lambda(t, dt)$$ \\[1em] |
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$f: X \to Y$ weak equivalence if $\pi_n(f): \pi_n(X) \iso \pi_n(Y)$ |
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& $f: A \to B$ weak equivalence if $H(f): H(X) \iso H(Y)$ |
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} |
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} |
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\end{document} |
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@ -0,0 +1,160 @@ |
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% normally included with amsart |
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% \usepackage{amsmath, amsthm} |
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% font with unicode support, does not work with classicthesis |
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% \usepackage{fontspec} |
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% clickable tocs |
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\usepackage{hyperref} |
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% floating figures |
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\usepackage{float} |
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% for multiple cites |
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\usepackage{cite} |
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% fancy diagrams |
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\usepackage{tikz} |
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\usetikzlibrary{matrix, arrows, decorations} |
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\tikzset{node distance=2.5em, row sep=2.2em, column sep=2.7em, auto} |
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% simple diagrams |
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% \usepackage[all,cmtip]{xy} |
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\usepackage{graphicx} |
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% \graphicspath{ {./images/} } |
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\usepackage{caption} |
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\usepackage{subcaption} |
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% for the fib arrow |
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\usepackage{amssymb} |
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% Some basic objects |
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\newcommand{\N}{\mathbb{N}} % natural numbers |
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\newcommand{\Np}{{\mathbb{N}^{>0}}} % positive numbers |
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\newcommand{\Z}{\mathbb{Z}} % integers |
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\newcommand{\R}{\mathbb{R}} % reals |
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\DeclareRobustCommand{\Q}{\mathbb{Q}} % rationals |
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\renewcommand{\k}{\mathrm{I\!k}} % default ground ring |
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% Basic category stuff |
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\newcommand{\cat}[1]{\mathbf{#1}} % the category of ... |
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\newcommand{\opCat}[1]{{#1}^{\text{op}}}% opposite category |
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\newcommand{\Hom}{\mathbf{Hom}} |
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\newcommand{\id}{\mathbf{id}} |
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\newcommand{\Ho}{\cat{Ho}} |
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% Categories |
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\newcommand{\Set}{\cat{Set}} % sets |
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\newcommand{\Top}{\cat{Top}} % topological spaces |
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\newcommand{\Grp}{\cat{Grp}} % groups |
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\newcommand{\Ab}{\cat{Ab}} % abelian groups |
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\newcommand{\DELTA}{\boldsymbol{\Delta}}% the simplicial cat |
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\newcommand{\simplicial}[1]{\cat{s{#1}}}% simplicial objects |
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\newcommand{\sSet}{\simplicial{\Set}} % simplicial sets |
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\newcommand{\Mod}[1]{\cat{{#1}Mod}} % modules over a ring |
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\newcommand{\Alg}[1]{\cat{{#1}Alg}} % algebras over a ring |
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\newcommand{\grMod}[1]{\cat{gr\mbox{-}{#1}Mod}} % graded modules over a ring |
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\newcommand{\grAlg}[1]{\cat{gr\mbox{-}{#1}Alg}} % graded algebras over a ring |
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\newcommand{\dgMod}[1]{\cat{dg\mbox{-}{#1}Mod}} % differential graded modules over a ring |
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\newcommand{\dgAlg}[1]{\cat{dg\mbox{-}{#1}Alg}} % differential graded algebras over a ring |
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\newcommand{\Ch}[1]{\cat{Ch_{n\geq0}({#1})}} % chain complexes |
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\newcommand{\CoCh}[1]{\cat{Ch^{n\geq0}({#1})}} % cochain complexes |
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\DeclareRobustCommand{\DGA}{\cat{DGA}} % cochain algebras |
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\DeclareRobustCommand{\CDGA}{\cat{CDGA}} % commutative cochain algebras |
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\DeclareRobustCommand{\AugCDGA}{\cat{CDGA^\ast}}% augmentedcommutative cochain algebras |
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\newcommand{\cof}{\hookrightarrow} % cofibration |
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\newcommand{\fib}{\twoheadrightarrow} % fibration |
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\newcommand{\we}{\tot{\simeq}} % weak equivalence |
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% for use in xy diagrams |
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\newcommand{\arcof}{\ar@{^{(}->}} |
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\newcommand{\artcof}{\ar@{^{(}->}|\simeq} |
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\newcommand{\arfib}{\ar@{->>}} |
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\newcommand{\artfib}{\ar@{->>}|\simeq} |
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\newcommand{\arwe}{\ar|-\simeq} |
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\newcommand{\ariso}{\ar|-\iso} |
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% adjunction symbol for xymatrices |
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\newcommand{\xyadj}{\raisebox{0.2\height}{\scalebox{0.5}{$\perp$}}} |
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% pushout and pullback for xymatrices (makes empty arrow with text) |
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\newcommand{\xypo}{\ar@{}[dr]|(.75){\scalebox{1.2}{$\ulcorner$}}} |
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\newcommand{\xypb}{\ar@{}[dr]|(.25){\scalebox{1.2}{$\lrcorner$}}} |
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%\newcommand{\leftadj}{\ooalign{\hss\rightleftarrows\hss\cr\bot}} |
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\newcommand{\leftadj}{\rightleftarrows} |
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% Notation and operators |
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\newcommand{\I}{\,\mid\,} % seperator in set notation |
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\newcommand{\del}{\partial} % boundary |
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\newcommand{\iso}{\cong} % isomorphic |
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\newcommand{\eq}{\sim} % homotopic |
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\newcommand{\ison}[1]{\stackrel{(#1)}{\iso}} % isos to refer to |
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\newcommand{\refison}[1]{{\small $(#1)$}} % ref |
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\newcommand{\tot}[1]{\xrightarrow{\,\,{#1}\,\,}} % arrow with name |
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\newcommand{\toti}[1]{\xleftarrow{\,\,{#1}\,\,}} % left arrow with name |
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\newcommand{\mapstot}[1]{\xmapsto{\,\,{#1}\,\,}} % mapsto with name |
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\newcommand{\unit}{\eta} |
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\newcommand{\counit}{\epsilon} |
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\DeclareMathOperator*{\im}{im} |
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\DeclareMathOperator*{\coker}{coker} |
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\DeclareMathOperator*{\colim}{colim} |
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\DeclareMathOperator*{\Tor}{Tor} |
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\DeclareMathOperator*{\Ext}{Ext} |
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\DeclareMathOperator*{\tensor}{\otimes} |
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\DeclareMathOperator*{\bigtensor}{\bigotimes} |
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\renewcommand{\deg}[1]{{|{#1}|}} |
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\newcommand{\Char}[1]{char({#1})} |
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\newcommand{\RH}{\widetilde{H}} % reduced homology |
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\DeclareRobustCommand{\C}{\mathcal{C}} % Serre mod C class |
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\newcommand{\Apl}[0]{{A_{PL}}} % Apl simplicial set of polynomials |
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\newcommand{\titleCDGA}{\texorpdfstring{$\CDGA$}{CDGA}} |
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% restriction of a function |
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\newcommand\restr[2]{{% we make the whole thing an ordinary symbol |
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\left.\kern-\nulldelimiterspace % automatically resize the bar with \right |
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#1 % the function |
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\vphantom{\big|} % pretend it's a little taller at normal size |
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\right|_{#2} % this is the delimiter |
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}} |
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% Todos in the margin |
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\newcommand{\todo}[1]{ |
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\addcontentsline{tdo}{todo}{\protect{#1}} |
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$\ast$ \marginpar{\tiny $\ast$ #1} |
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} |
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% Big todos in text |
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\newcommand{\TODO}[1]{ |
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\addcontentsline{tdo}{todo}{\protect{#1}} |
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{\tiny $\ast$ #1} |
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} |
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% TODO item, as itemize does not work |
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\newcommand{\titem}[0]{\\-\qquad} |
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% List of todos |
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\makeatletter |
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\newcommand \listoftodos{\section*{Todo list} \@starttoc{tdo}} |
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\newcommand\l@todo[2]{ |
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\par\noindent \textit{#2}, \parbox{10cm}{#1}\par |
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} |
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\makeatother |
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% simple way to center an image |
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\newcommand{\cimage}[2][]{ |
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\begin{center} |
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\includegraphics[#1]{#2} |
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\end{center} |
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} |
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% simple way to center a diagram |
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\newcommand{\cdiagrambase}[1]{ |
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\begin{displaymath} |
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\input{#1} |
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\end{displaymath} |
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} |
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\newcommand{\cdiagram}[1]{ |
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\cdiagrambase{diagrams/#1} |
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} |
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