{"ID":23577921,"CreatedAt":"2026-09-18T05:07:30.921092366Z","UpdatedAt":"2026-09-18T05:07:30.921092366Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20533","arxiv_id":"2609.20533","title":"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 $\\infty$-categories. A consequence is that an $\\infty$-category $\\mathcal{C}$ such that both $\\mathcal{C}$ and $\\mathcal{C}^\\mathsf{op}$ 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 $\\infty$-categories. Additionally, we show that a strengthening of our main result is equivalent to the existence of a proper class of measurable cardinals.","short_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 $\\infty$-categories. A consequence is that an $\\infty$-category $\\mathcal{C}$ such that both $\\mathcal{C}$ and $\\mathcal{C}^\\mathsf{op}$ are presentable is a small...","url_abs":"https://arxiv.org/abs/2609.20533","url_pdf":"https://arxiv.org/pdf/2609.20533v1","authors":"[\"Thorger Geiß\",\"Phil Pützstück\",\"Maxime Ramzi\"]","published":"2026-09-17T15:06:51Z","proceeding":"math.CT","tasks":"[\"math.CT\",\"math.AT\",\"math.GN\",\"math.LO\"]","methods":"[]","has_code":false}