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Cocompactness and Presentability
Abstract
We give a short proof that κ-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable ∞-categories. A consequence is that an ∞-category 𝒞 such that both 𝒞 and 𝒞op are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable ∞-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.
Related
BibTeX
@misc{geiß2026cocompactness,
title = {{Cocompactness and Presentability}},
author = {Thorger Geiß and Phil Pützstück and Maxime Ramzi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20533},
eprint = {2609.20533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20533},
}
RIS
TY - UNPB
TI - Cocompactness and Presentability
AU - Thorger Geiß
AU - Phil Pützstück
AU - Maxime Ramzi
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20533
UR - https://arxiv.org/abs/2609.20533
AB - We give a short proof that κ-cocompact objects in a presentable category are subterminal. As our main result, we extend this to the setting of presentable ∞-categories. A consequence is that an ∞-category 𝒞 such that both 𝒞 and 𝒞^op are presentable is a small complete lattice, extending a classical theorem of Gabriel-Ulmer. Along the way, we prove a nilpotence result for phantom maps in general pointed presentable ∞-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.
ER -