\documentclass[12pt]{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
% $$ [0] \to [1] \to [2] \to [3] \to \ldots $$
% $$\delta_i: [n] \to [n+1], k \mapsto \begin{cases} k & \text{if } k < i;\\ k+1 & \text{if } k \geq i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n+1$$
% $$\sigma_i: [n+1] \to [n], k \mapsto \begin{cases} k & \text{if } k \leq i;\\ k-1 & \text{if } k > i. \end{cases} \hspace{0.5cm} 0 \leq i \leq n$$
% $$ A_0 \to A_1 \to A_2 \to A_3 $$

% 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 $=$}}
$$ + - \mapsto $$
$$ \{ \} $$

\end{document}
