\documentclass[12pt]{amsproc}

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\geometry{twoside=false}

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

\input{../thesis/preamble}

\title{Dold-Kan Correspondence}
\author{Joshua Moerman}

\begin{document}
\maketitle

\begin{definition}
	We define a category $\DELTA$, where the objects are the finite ordinals $[n] = \{0, \dots, n\}$ and maps are monotone increasing functions.
\end{definition}

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

$$ A_0 \to A_1 \to A_2 \to A_3 $$

\end{document}
