\[ \newcommand{\blank}{{-}} \newcommand{\cat}[1]{\mathcal{#1}} \newcommand{\comp}{\mathbin{\circ}} \newcommand{\pair}[2]{\mathop{\langle #1, #2 \rangle}} \]
Some interesting degeneracy theorems that I always forget the precise statements of.
Lemma 1 If \(\cat{C}\) is a cartesian closed category with an initial object, then any two maps \(f, g : A \to 0\) are equal. Dually, if \(\cat{C}\) is a cocartesian coclosed category with a terminal object, then any two maps \(f, g : 1 \to A\) are equal.
Proof. \(A \times 0 \cong 0\), since \(A \times (\blank)\) preserves colimits. Hence, \(0 \times 0 \cong 0\), and so \(\pi_1, \pi_2 : 0 \times 0 \to 0\) are equal. If \(f, g : A \to 0\), then \(f = \pi_1 \comp \pair{f}{g} = \pi_2 \comp \pair{f}{g} = g\).
Lemma 2 If \(\cat{C}\) is a cartesian closed category and a cocartesian coclosed category, then \(\cat{C}\) is a poset.
Proof. Suppose \(f, g : A \to B\) are two maps in \(\cat{C}\). Then \(f, g : 1 \to B^A\), and so by Lemma 1, \(f = g\).
Citation
@online{choudhury2026,
author = {Choudhury, Vikraman},
title = {Degeneracy Theorems},
date = {2026-06-20},
url = {https://vikraman.org/notes/degeneracy.html},
langid = {en}
}