Hostname: page-component-848d4c4894-pjpqr Total loading time: 0 Render date: 2024-06-19T04:05:01.040Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  19 February 2021

Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada and Department of Mathematics, Dalhousie University, Halifax, Nova Scotia, Canada e-mail:,
Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada


We show that if ${\mathcal C}$ is a fusion $2$ -category in which the endomorphism category of the unit object is or , then the indecomposable objects of ${\mathcal C}$ form a finite group.

Research Article
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Research at the Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development and Innovation. The Perimeter Institute is in the Haldimand Tract, land promised to the Six Nations. Dalhousie University is in Mi‘kma‘ki, the ancestral and unceded territory of the Mi‘kmaq. We are all Treaty people.


Baez, J. C. and Dolan, J., ‘Higher-dimensional algebra and topological quantum field theory’, J. Math. Phys. 36(11) (1995), 60736105.10.1063/1.531236CrossRefGoogle Scholar
Douglas, C. L. and Reutter, D. J., ‘Fusion 2-categories and a state-sum invariant for 4-manifolds’, Preprint, 2018, arXiv:1812.11933 [math.QA].Google Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V., Tensor Categories, Mathematical Surveys and Monographs, 205 (American Mathematical Society, Providence, RI, 2015).CrossRefGoogle Scholar
Etingof, P., Gelaki, S. and Ostrik, V., ‘Classification of fusion categories of dimension $pq$ ’, Int. Math. Res. Not. 2004(57) (2004), 30413056.10.1155/S1073792804131206CrossRefGoogle Scholar
Freed, D. S. and Teleman, C., ‘Gapped boundary theories in three dimensions’, Preprint, 2020, arXiv:2006.10200 [math.QA].Google Scholar
Gaiotto, D. and Johnson-Freyd, T., ‘Condensations in higher categories’, Preprint, 2019, arXiv:1905.09566 [math.CT].Google Scholar
Johnson-Freyd, T., ‘On the classification of topological orders’, Preprint, 2020, arXiv:2003.06663 [math.CT].Google Scholar
Jordan, D. and Larson, E., ‘On the classification of certain fusion categories’, J. Noncommut. Geom. 3 (2009), 481499.CrossRefGoogle Scholar
Lan, T., Kong, L. and Wen, X.-G., ‘Classification of (2+1)-dimensional topological order and symmetry-protected topological order for bosonic and fermionic systems with on-site symmetries’, Phys. Rev. B 95(23) (2017), 235140.10.1103/PhysRevB.95.235140CrossRefGoogle Scholar
Lan, T., Kong, L. and Wen, X.-G., ‘Classification of (3+1)D bosonic topological orders: the case when pointlike excitations are all bosons’, Phys. Rev. X 8(2) (2018), 021074, 24 pages.Google Scholar
Natale, S., ‘On the classification of fusion categories’, Proc. Int. Congr. Mathematicians, Rio de Janeiro, 2018, Invited Lectures, II (World Scientific, Hackensack, NJ, 2018), 173200.Google Scholar
Wang, Q.-R. and Gu, Z.-C., ‘Towards a complete classification of symmetry-protected topological phases for interacting fermions in three dimensions and a general group supercohomology theory’, Phys. Rev. X 8(1) (2018), 011055, 29 pages.Google Scholar