homotopy hypothesis-theorem
delooping hypothesis-theorem
stabilization hypothesis-theorem
This page is meant to list problems in (higher) category theory that have been stated in the literature (here meant in a broad sense including papers, preprints, blog posts, and so on). A problem is considered closed when it is solved by a published paper, after which it should be removed from this page (with a link to the corresponding paper in the comment accompanying the change).
Open problems in topos theory:
Lawvere 2009 states a list of Open problems in topos theory
(some of which have by now been resolved)
Open problems in higher category theory:
an extensive proof sketch was given (and published as such) by Lurie 2009; a proof along a new strategy is claimed by Grady & Pavlov 2021, but not yet published
(closely related to the cobordism hypothesis)
On open problems in topos theory:
(original statements, some by now solved)
(original statement)
On the cobordism hypothesis/tangle hypothesis:
John Baez, James Dolan, p. 28 of: Higher dimensional algebra and Topological Quantum Field Theory, J. Math. Phys. 36 (1995) 6073-6105 [doi:10.1063/1.531236, arXiv:q-alg/9503002]
(original statement)
Jacob Lurie, On the Classification of Topological Field Theories, Current Developments in Mathematics 2008 (2009) 129-280 [doi:10.4310/CDM.2008.v2008.n1.a, arXiv:0905.0465]
(extensive proof sketch)
Daniel Grady, Dmitri Pavlov, The geometric cobordism hypothesis [arXiv:2111.01095]
(claim of new proof)
Last revised on September 10, 2025 at 18:13:50. See the history of this page for a list of all contributions to it.