\documentclass[11pt]{amsproc}

% a la fullpage
\usepackage{geometry}
\geometry{a4paper}
\geometry{twoside=false}

% Activate to begin paragraphs with an empty line rather than an indent
\usepackage[parfill]{parskip}
\setlength{\marginparwidth}{2cm}

\newtheorem{theorem}{Theorem}[section]
\newtheorem{definition}[theorem]{Definition}
\newtheorem{lemma}[theorem]{Lemma}

\input{../thesis/preamble}

\begin{document}

% For basic categorical picture of simplicial objects
% $$ [0] \to [1] \to [2] \to [3] \to \ldots $$
% $$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
% $$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
% $$ X_0 \to X_1 \to X_2 \to X_3 $$

% For geometric picture of simplicial objects
% $$ 0 \tot{\delta_0} 1 \tot{\delta_1} 2 \tot{\delta_2} 3 \tot{\delta_3} \cdots $$

% For the pictures in the presentation (singular chain complex)
% $$ \cdots \tot{\del_2} C_2 \tot{\del_1} C_1 \tot{\del_0} C_0 $$
% \reflectbox{\rotatebox[origin=c]{90}{\large $=$}}
% $$ + - \mapsto $$
% $$ \{ \} $$

% For singular chain complex, face maps
% $$ C_n(X) = \Z[\Hom{\cat{Top}}{\Delta^n}{X}] $$
% $$ \Delta^2 \to X \sigma \circ \delta^1$$
% $$ \Delta^1 \mono $$
% $$ \delta^0 - \delta^1 + \delta^2 $$

% For singular homology
% $$ X' \, X \subseteq \R^2 $$
% $$ \sigma_1 \sigma_2 \sigma_3 \tau $$

% For simplicial htp (degree 1)
$$ 0, x, x', y $$
$$ x_0, x_1, x_2 $$

\end{document}
