Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T12:58:23.829Z Has data issue: false hasContentIssue false
  • English
  • Français

On dualizability of braided tensor categories

Part of: Lie algebras and Lie superalgebras Modules, bimodules and ideals

Published online by Cambridge University Press:  09 March 2021

Adrien Brochier ,
David Jordan  and
Noah Snyder
Adrien Brochier
Affiliation:
Université de Paris, Sorbonne Université, CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, F-75013Paris, Franceadrien.brochier@imj-prg.fr
David Jordan
Affiliation:
School of Mathematics, University of Edinburgh, Edinburgh, UKdjordan@ed.ac.uk
Noah Snyder
Affiliation:
Indiana University Bloomington, Bloomington, IN, USAnsnyder1@indiana.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

We study the question of dualizability in higher Morita categories of locally presentable tensor categories and braided tensor categories. Our main results are that the 3-category of rigid tensor categories with enough compact projectives is 2-dualizable, that the 4-category of rigid braided tensor categories with enough compact projectives is 3-dualizable, and that (in characteristic zero) the 4-category of braided multi-fusion categories is 4-dualizable. Via the cobordism hypothesis, this produces respectively two-, three- and four-dimensional framed local topological field theories. In particular, we produce a framed three-dimensional local topological field theory attached to the category of representations of a quantum group at any value of $q$.

Keywords

quantum groupsbraided tensor categorieshigher categoriestopological field theories

MSC classification

Primary: 17B37: Quantum groups (quantized enveloping algebras) and related deformations 16D90: Module categories ; module theory in a category-theoretic context; Morita equivalence and duality
Type
Research Article
Information
Compositio Mathematica , Volume 157 , Issue 3 , March 2021 , pp. 435 - 483
Check for updates
Copyright
© The Author(s) 2021

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.)

Footnotes

AB was supported during this work by RTG 1670 ‘Mathematics Inspired by String Theory and Quantum Field Theory’. DJ is supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement no. 637618). NS is supported by NSF grant DMS-1454767.

References

Adámek, J. and Rosicky, J., Locally presentable and accessible categories (Cambridge University Press, Cambridge, 1994).10.1017/CBO9780511600579CrossRefGoogle Scholar
Aganagic, M. and Shakirov, S., Knot homology and refined Chern-Simons index, Comm. Math. Phys. 333 (2015), 187–228; MR 3294947.10.1007/s00220-014-2197-4CrossRefGoogle Scholar
Alekseev, A. Y., Integrability in the Hamiltonian Chern-Simons theory, Algebra i Analiz 6 (1994), 53–66; MR 1290818.Google Scholar
Alekseev, A. Y., Grosse, H. and Schomerus, V., Combinatorial quantization of the Hamiltonian Chern-Simons theory II, Comm. Math. Phys. 174 (1996), 561–604.10.1007/BF02101528CrossRefGoogle Scholar
Andersen, J. E., Mattes, J. and Reshetikhin, N., Quantization of the algebra of chord diagrams, Math. Proc. Cambridge Philos. Soc. 124 (1998), 451–467; MR 1636568 (99m:58040).10.1017/S0305004198002813CrossRefGoogle Scholar
Atiyah, M. F. and Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. A 308 (1983), 523–615; MR 702806 (85k:14006).Google Scholar
Ayala, D. and Francis, J., Factorization homology of topological manifolds, J. Topol. 8 (2015), 1045–1084.10.1112/jtopol/jtv028CrossRefGoogle Scholar
Ayala, D. and Francis, J., The cobordism hypothesis, Preprint (2017), arXiv:1705.02240.Google Scholar
Ayala, D., Francis, J. and Rozenblyum, N., Factorization homology I: Higher categories, Adv. Math. 333 (2018), 1042–1177; MR 3818096.10.1016/j.aim.2018.05.031CrossRefGoogle Scholar
Ayala, D., Francis, J. and Tanaka, H. L., Factorization homology of stratified spaces, Selecta Math. (N.S.) 23 (2017), 293–362.10.1007/s00029-016-0242-1CrossRefGoogle Scholar
Baez, J. C. and Dolan, J., Higher-dimensional algebra and topological quantum field theory, J. Math. Phys. 36 (1995), 6073–6105; MR 1355899 (97f:18003).10.1063/1.531236CrossRefGoogle Scholar
Bakalov, B. and Kirillov, A. Jr., Lectures on tensor categories and modular functors, University Lecture Series, vol. 21 (American Mathematical Society, Providence, RI, 2001); MR 1797619 (2002d:18003).Google Scholar
Barrett, J. W. and Westbury, B. W., Invariants of piecewise-linear $3$-manifolds, Trans. Amer. Math. Soc. 348 (1996), 3997–4022; MR 1357878.10.1090/S0002-9947-96-01660-1CrossRefGoogle Scholar
Bartlett, B., Douglas, C. L., Schommer-Pries, C. J. and Vicary, J., Modular categories as representations of the 3-dimensional bordism 2-category, Preprint (2015), arXiv:1509.06811.Google Scholar
Ben-Zvi, D., Brochier, A. and Jordan, D., Integrating quantum groups over surfaces, J. Topol. 11 (2018), 873–916.10.1112/topo.12072CrossRefGoogle Scholar
Ben-Zvi, D., Brochier, A. and Jordan, D., Quantum character varieties and braided module categories, Selecta Math. (N.S.) 24 (2018), 4711–4748.10.1007/s00029-018-0426-yCrossRefGoogle Scholar
Ben-Zvi, D., Francis, J. and Nadler, D., Integral transforms and Drinfeld centers in derived algebraic geometry, J. Amer. Math. Soc. 23 (2010), 909–966; MR 2669705 (2011j:14023).10.1090/S0894-0347-10-00669-7CrossRefGoogle Scholar
Ben-Zvi, D. and Nadler, D., The character theory of a complex group, Preprint (2009), arXiv:0904.1247.Google Scholar
Beilinson, A. and Drinfeld, V., Chiral algebras, American Mathematical Society Colloquium Publications, vol. 51 (American Mathematical Society, Providence, RI, 2004); MR 2058353 (2005d:17007).10.1090/coll/051CrossRefGoogle Scholar
Beliakova, A., Blanchet, C. and Geer, N., Logarithmic Hennings invariants for restricted quantum $\mathfrak {sl}(2)$, Algebr. Geom. Topol. 18 (2018), 4329–4358; MR 3892247.10.2140/agt.2018.18.4329CrossRefGoogle Scholar
Berest, Y. and Samuelson, P., Double affine Hecke algebras and generalized Jones polynomials, Compos. Math. 152 (2016), 1333–1384.10.1112/S0010437X16007314CrossRefGoogle Scholar
Blanchet, C., Costantino, F., Geer, N. and Patureau-Mirand, B., Non semi-simple TQFTs from unrolled quantum sl(2), in Proceedings of the Gökova Geometry-Topology Conference 2015 (International Press, Somerville, MA, 2016), 218–231; MR 3526845.Google Scholar
Brandenburg, M., Chirvasitu, A. and Johnson-Freyd, T., Reflexivity and dualizability in categorified linear algebra, Theory Appl. Categ. 30 (2015), 808–835.Google Scholar
Brochier, A., Jordan, D., Safronov, P. and Snyder, N., Invertible braided tensor categories, Preprint (2020), arXiv:2003.13812.Google Scholar
Calaque, D., Lagrangian structures on mapping stacks and semi-classical TFTs, in Stacks and categories in geometry, topology, and algebra, Contemporary Mathematics, vol. 643 (American Mathematical Society, Providence, RI, 2015), 1–23; MR 3381468.Google Scholar
Calaque, D., Pantev, T., Toën, B., Vaquié, M. and Vezzosi, G., Shifted Poisson structures and deformation quantization, J. Topol. 10 (2017), 483–584.10.1112/topo.12012CrossRefGoogle Scholar
Caviglia, G. and Horel, G., Rigidification of higher categorical structures, Algebr. Geom. Topol. 16 (2016), 3533–3562.10.2140/agt.2016.16.3533CrossRefGoogle Scholar
Cherednik, I., Jones polynomials of torus knots via DAHA, Int. Math. Res. Not. IMRN 2013 (2013), 5366–5425.10.1093/imrn/rns202CrossRefGoogle Scholar
Cooke, J., Excision of skein categories and factorisation homology, Preprint (2019), arXiv:1910.02630.Google Scholar
Costello, K. and Gwilliam, O., Factorization algebras in quantum field theory, Vol. 1 (Cambridge University Press, Cambridge, 2016).Google Scholar
Crane, L., Kauffman, L. H. and Yetter, D. N., State-sum invariants of $4$-manifolds, J. Knot Theory Ramifications 6 (1997), 177–234; MR 1452438 (99c:57049).10.1142/S0218216597000145CrossRefGoogle Scholar
Crane, L. and Yetter, D., A categorical construction of 4D topological quantum field theories, in Quantum topology (World Scientific, Singapore, 1993), 120–130.10.1142/9789812796387_0005CrossRefGoogle Scholar
Davydov, A. and Nikshych, D., The Picard crossed module of a braided tensor category, Algebra Number Theory 7 (2013), 1365–1403.10.2140/ant.2013.7.1365CrossRefGoogle Scholar
De Renzi, M., Geer, N. and Patureau-Mirand, B., Renormalized Hennings invariants and 2+1-TQFTs, Comm. Math. Phys. 362 (2018), 855–907.10.1007/s00220-018-3187-8CrossRefGoogle Scholar
Deligne, P., Catégories tannakiennes, in The Grothendieck Festschrift (Birkhäuser, Boston, MA, 2007), 111–195.10.1007/978-0-8176-4575-5_3CrossRefGoogle Scholar
Dimofte, T., Quantum Riemann surfaces in Chern-Simons theory, Adv. Theor. Math. Phys. 17 (2013), 479–599.10.4310/ATMP.2013.v17.n3.a1CrossRefGoogle Scholar
Douglas, C. L., Schommer-Pries, C. and Snyder, N., Dualizable tensor categories II: Homotopy SO(3)-actions, in preparation.Google Scholar
Douglas, C. L., Schommer-Pries, C. and Snyder, N., The balanced tensor product of module categories, Kyoto J. Math. 59 (2019), 167–179; MR 3934626.10.1215/21562261-2018-0006CrossRefGoogle Scholar
Drinfeld, V. G., Almost cocommutative Hopf algebras, Algebra i Analiz 1 (1989), 30–46; MR1025154 (91b:16046).Google Scholar
Drinfeld, V., Gelaki, S., Nikshych, D. and Ostrik, V., On braided fusion categories I, Selecta Math. (N.S.) 16 (2010), 1–119.10.1007/s00029-010-0017-zCrossRefGoogle Scholar
Etingof, P., Gelaki, S., Nikshych, D. and Ostrik, V., Tensor categories, Mathematical Surveys and Monographs, vol. 205 (American Mathematical Society, Providence, RI, 2015), http://www-math.mit.edu/etingof/egnobookfinal.pdf; MR 3242743.10.1090/surv/205CrossRefGoogle Scholar
Etingof, P., Nikshych, D. and Ostrik, V., An analogue of Radford's $S^4$ formula for finite tensor categories, Int. Math. Res. Not. IMRN 2004 (2004), 2915–2933; MR 2097289 (2005m:18007).10.1155/S1073792804141445CrossRefGoogle Scholar
Etingof, P., Nikshych, D. and Ostrik, V., Fusion categories and homotopy theory, Quantum Topol. 1 (2010), 209–273, with an appendix by E. Meir; MR 2677836 (2011h:18007).10.4171/QT/6CrossRefGoogle Scholar
Etingof, P. and Ostrik, V., Finite tensor categories, Mosc. Math. J. 4 (2004), 627–654.10.17323/1609-4514-2004-4-3-627-654CrossRefGoogle Scholar
Fiedorowicz, Z., The symmetric bar construction, Preprint (1992), available for download at https://people.math.osu.edu/fiedorowicz.1/.Google Scholar
Fiorenza, D. and Valentino, A., Boundary conditions for topological quantum field theories, anomalies and projective modular functors, Comm. Math. Phys. 338 (2015), 1043–1074.10.1007/s00220-015-2371-3CrossRefGoogle Scholar
Franco, I. L., Tensor products of finitely cocomplete and abelian categories, J. Algebra 396 (2013), 207–219.10.1016/j.jalgebra.2013.08.015CrossRefGoogle Scholar
Freed, D., 3-dimensional TQFTs through the lens of the cobordism hypothesis, slides available at https://www.ma.utexas.edu/users/dafr/StanfordLecture.pdf (2012).Google Scholar
Freed, D. S., 4-3-2-8-7-6, Aspects of Topology Conference talk slides (2012), slides available at https://www.ma.utexas.edu/users/dafr/Aspects.pdf.Google Scholar
Freed, D. S. and Teleman, C., Relative quantum field theory, Comm. Math. Phys. 326 (2014), 459–476; MR 3165462.10.1007/s00220-013-1880-1CrossRefGoogle Scholar
Fresse, B., Homotopy of operads and Grothendieck-Teichmüller groups. Part 1: The algebraic theory and its topological background, Mathematical Surveys and Monographs, vol. 217 (American Mathematical Society, Providence, RI, 2017); MR 3643404.Google Scholar
Frohman, C. and Gelca, R., Skein modules and the noncommutative torus, Trans. Amer. Math. Soc. 352 (2000), 4877–4888.10.1090/S0002-9947-00-02512-5CrossRefGoogle Scholar
Fuchs, J., Schweigert, C. and Valentino, A., Bicategories for boundary conditions and for surface defects in 3-d TFT, Comm. Math. Phys. 321 (2013), 543–575.10.1007/s00220-013-1723-0CrossRefGoogle Scholar
Gaitsgory, D., Sheaves of categories and the notion of 1-affineness, in Stacks and categories in geometry, topology, and algebra, Contemporary Mathematics, vol. 643 (American Mathematical Society, Providence, RI, 2015), 127–225; MR 3381473.10.1090/conm/643/12899CrossRefGoogle Scholar
Garner, R. and Shulman, M., Enriched categories as a free cocompletion, Adv. Math. 289 (2016), 1–94.10.1016/j.aim.2015.11.012CrossRefGoogle Scholar
Garoufalidis, S., On the characteristic and deformation varieties of a knot, Geom. Topol. Monogr. 7 (2004), 291–309.10.2140/gtm.2004.7.291CrossRefGoogle Scholar
Ginot, G., Notes on factorization algebras, factorization homology and applications, in Mathematical aspects of quantum field theories (Springer, Cham, 2015), 429–552.Google Scholar
Goldman, W., The symplectic nature of fundamental groups of surfaces, Adv. Math. 54 (1984), 200–225.10.1016/0001-8708(84)90040-9CrossRefGoogle Scholar
Greenough, J., Monoidal 2-structure of bimodule categories, J. Algebra 324 (2010), 1818–1859.10.1016/j.jalgebra.2010.06.018CrossRefGoogle Scholar
Gukov, S., Three-dimensional quantum gravity, Chern-Simons theory, and the A-polynomial, Comm. Math. Phys. 255 (2005), 577–627.10.1007/s00220-005-1312-yCrossRefGoogle Scholar
Gukov, S. and Sulkowski, P., A-polynomial, B-model, and quantization, J. High Energy Phys. 2012 (2012), 70.10.1007/JHEP02(2012)070CrossRefGoogle Scholar
Gwilliam, O. and Scheimbauer, C., Duals and adjoints in the factorization higher Morita category, Preprint (2018), arXiv:1804.10924.Google Scholar
Haugseng, R., The higher Morita category of $E_n$–algebras, Geom. Topol. 21 (2017), 1631–1730.10.2140/gt.2017.21.1631CrossRefGoogle Scholar
Hinich, V., Enriched Yoneda lemma, Theory Appl. Categ. 31 (2016), 833–838; MR 3551499.Google Scholar
Idrissi, N., Swiss-Cheese operad and Drinfeld center, Israel J. Math. 221 (2017), 941–972.10.1007/s11856-017-1579-7CrossRefGoogle Scholar
Johnson-Freyd, T., Heisenberg-picture quantum field theory, Preprint (2015), arXiv:1508.05908.Google Scholar
Johnson-Freyd, T. and Scheimbauer, C., (Op) lax natural transformations, twisted quantum field theories, and ‘even higher’ Morita categories, Adv. Math. 307 (2017), 147–223.10.1016/j.aim.2016.11.014CrossRefGoogle Scholar
Kassel, C., Quantum groups, Graduate Texts in Mathematics, vol. 155 (Springer, New York, 1995); MR1321145 (96e:17041).10.1007/978-1-4612-0783-2CrossRefGoogle Scholar
Kauffman, L. H., Knots and physics, vol. 1 (World Scientific, Singapore, 2001).10.1142/4256CrossRefGoogle Scholar
Kelly, M., Basic concepts of enriched category theory, vol. 64 (Cambridge University Press, Cambridge, 1982).Google Scholar
Lickorish, W. B. R., What is … a skein module?, Notices Amer. Math. Soc. 56 (2009), 240–242. Available at https://www.ams.org/notices/200902/rtx090200240p.pdf.Google Scholar
Lurie, J., Higher algebra, Preprint, available at https://www.math.ias.edu/lurie/papers/HA.pdf.Google Scholar
Lurie, J., On the classification of topological field theories, Curr. Dev. Math. 2008 (2009), 129–280.10.4310/CDM.2008.v2008.n1.a3CrossRefGoogle Scholar
Makkai, M. and Paré, R., Accessible categories: the foundations of categorical model theory, Contemporary Mathematics, vol. 104 (American Mathematical Society, Providence, RI, 1989).10.1090/conm/104CrossRefGoogle Scholar
Morrison, S. and Walker, K., Higher categories, colimits, and the blob complex, Proc. Nat. Acad. Sci. 108 (2011), 8139–8145.10.1073/pnas.1018168108CrossRefGoogle ScholarPubMed
Morrison, S. and Walker, K., Blob homology, Geom. Topol. 16 (2012), 1481–1607; MR 2978449.10.2140/gt.2012.16.1481CrossRefGoogle Scholar
Ostrik, V., Module categories, weak Hopf algebras and modular invariants, Transform. Groups 8 (2003), 177–206.10.1007/s00031-003-0515-6CrossRefGoogle Scholar
Pantev, T., Toën, B., Vaquié, M. and Vezzosi, G., Shifted symplectic structures, Publ. Math. IHÉS 117 (2013), 271–328.10.1007/s10240-013-0054-1CrossRefGoogle Scholar
Przytycki, J. H., Skein modules of $3$-manifolds, Bull. Pol. Acad. Sci. Math. 39 (1991), 91–100; MR 1194712.Google Scholar
Przytycki, J. H., Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999), 45–66; MR 1723531.Google Scholar
Przytyckia, J. H. and Sikora, A. S., On skein algebras and ${Sl}_2 (\mathbb {C})$-character varieties, Topology 39 (2000), 115–148.10.1016/S0040-9383(98)00062-7CrossRefGoogle Scholar
Radford, D. E., The order of the antipode of a finite dimensional Hopf algebra is finite, Amer. J. Math. 98 (1976), 333–355; MR 0407069 (53 $\#$10852).10.2307/2373888CrossRefGoogle Scholar
Reshetikhin, N. and Turaev, V. G., Invariants of $3$-manifolds via link polynomials and quantum groups, Invent. Math. 103 (1991), 547–597; MR 1091619 (92b:57024).10.1007/BF01239527CrossRefGoogle Scholar
Roche, P. and Szenes, A., Trace functionals on noncommutative deformations of moduli spaces of flat connections, Adv. Math. 168 (2002), 133–192.10.1006/aima.2001.2045CrossRefGoogle Scholar
Samuelson, P., Iterated torus knots and double affine Hecke algebras, Int. Math. Res. Not. IMRN (2017), rnx198.Google Scholar
Scheimbauer, C. I., Factorization homology as a fully extended topological field theory, PhD thesis, ETH Zurich (2014), https://www.research-collection.ethz.ch/handle/20.500.11850/154981.Google Scholar
Shulman, M., Enriched indexed categories, Theory Appl. Categ. 28 (2013), 616–696; MR 3094435.Google Scholar
Stay, M., Compact closed bicategories, Theory Appl. Categ. 31 (2016), 755–798; MR 3542382.Google Scholar
Street, R., Enriched categories and cohomology, Quaest. Math. 6 (1983), 265–283.10.1080/16073606.1983.9632304CrossRefGoogle Scholar
Tillmann, U., $S$-structures for $k$-linear categories and the definition of a modular functor, J. Lond. Math. Soc. (2) 58 (1998), 208–228; MR 1670122.10.1112/S0024610798006383CrossRefGoogle Scholar
Turaev, V. G., Quantum invariants of knots and 3-manifolds, Revised edition, de Gruyter Studies in Mathematics, vol. 18 (de Gruyter, Berlin, 2010); MR 2654259 (2011f:57023).10.1515/9783110221848CrossRefGoogle Scholar
Turaev, V. G. and Viro, O. Y., State sum invariants of $3$-manifolds and quantum $6j$-symbols, Topology 31 (1992), 865–902; MR 1191386.10.1016/0040-9383(92)90015-ACrossRefGoogle Scholar
Voronov, A. A., The Swiss-Cheese operad, in Homotopy invariant algebraic structures (Baltimore, MD, 1998), Contemporary Mathematics, vol. 239 (American Mathematical Society, Providence, RI, 1999), 365–373; MR 1718089 (2000i:55032).10.1090/conm/239/03610CrossRefGoogle Scholar
Walker, K., Note on topological field theories, available at http://canyon23.net/math/tc.pdf (2006).Google Scholar
Witten, E., Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399; MR 990772 (90h:57009).10.1007/BF01217730CrossRefGoogle Scholar
Yetter, D., On right adjoints to exponential functors, J. Pure Appl. Algebra 45 (1987), 287–304; MR 890028.10.1016/0022-4049(87)90077-6CrossRefGoogle Scholar
Yuan, Q., Projective objects, Annoying Precision (Blog) (March 28, 2015), available at https://qchu.wordpress.com/2015/03/28/projective-objects/.Google Scholar
Yuan, Q., Tiny objects, Annoying Precision (Blog) (May 7, 2015), available at https://qchu.wordpress.com/2015/05/07/tiny-objects/.Google Scholar