Adámek–Rosický cocompletion problem
For every small category , let denote its free cocompletion under sifted colimits. Is finitely accessible; that is, does it have filtered colimits and is every object a filtered colimit of finitely presentable objects?
References
Primary source
Additional references
- Cocompletion under sifted colimits need not be finitely accessible — arXiv — Yuto Kawase
Progress summary
A new preprint claims to answer the problem negatively by giving counterexamples for every small category, but the result has not been independently checked.
The Adámek–Rosický cocompletion problem is a longstanding accessibility question dating to 2001. It asks whether sifted-colimit cocompletions always have the relevant finite accessibility property.
October 2026 counterexamples
Yuto Kawase’s preprint claims counterexamples showing that the sifted-colimit cocompletion of every small category need not be finitely accessible, giving a negative answer to the problem. The claim is supported here only by the announcing preprint.
Current status (as of October 2026): A preprint claims the problem is solved negatively, but independent verification is not recorded; absent confirmation, the result remains unverified.
Solutions 0
No solutions have been posted yet.