We consider Frobenius objects in the category Span, where the objects are sets and the morphisms are isomorphism classes of spans of sets. We show that such structures are in correspondence with data that can be characterized in terms of simplicial sets. An interesting class of examples comes from groupoids.

Our primary motivation is that Span can be viewed as a set-theoretic model for the symplectic category, and thus Frobenius objects in Span provide set-theoretic models for classical topological field theories. The paper includes an explanation of this relationship.

Given a finite commutative Frobenius object in Span, one can obtain invariants of closed surfaces with values in the natural numbers. We explicitly compute these invariants in several examples, including examples arising from abelian groups.

Keywords:

AMSC: 18B10, 18B40, 18C40, 18N50, 20L05, 57R56

References

1. L. Abrams , Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory Ramifications 5(5) (1996) 569–587. Link, Web of Science, Google Scholar
2. M. Alexandrov, A. Schwarz, O. Zaboronsky and M. Kontsevich , The geometry of the master equation and topological quantum field theory, Int. J. Mod. Phys. A 12(7) (1997) 1405–1429. Link, Web of Science, ADS, Google Scholar
3. M. Artin and B. Mazur , Etale Homotopy, Lecture Notes in Mathematics, Vol. 100 (Springer-Verlag, 1969). Crossref, Google Scholar
4. J. C. Baez, A. E. Hoffnung and C. D. Walker , Higher dimensional algebra VII: Groupoidification, Theory Appl. Categ. 24(18) (2010) 489–553. Google Scholar
5. J. Bénabou , Introduction to bicategories, in Reports of the Midwest Category Seminar (Springer, 1967), pp. 1–77. Crossref, Google Scholar
6. J. E. Bergner, A. M. Osorno, V. Ozornova, M. Rovelli and C. I. Scheimbauer , 2-Segal sets and the Waldhausen construction, Topol. Appl. 235 (2018) 445–484. Crossref, Web of Science, Google Scholar
7. A. J. Blumberg, R. L. Cohen and C. Teleman , Open-closed field theories, string topology, and Hochschild homology, in Alpine Perspectives on Algebraic Topology, Contemp. Math., Vol. 504 (Amer. Math. Soc., 2009), pp. 53–76. Crossref, Google Scholar
8. D. Calaque , Three lectures on derived symplectic geometry and topological field theories, Indag. Math. (N.S.) 25(5) (2014) 926–947. Crossref, Web of Science, Google Scholar
9. D. Calaque , Lagrangian structures on mapping stacks and semi-classical TFTs, in Stacks and Categories in Geometry, Topology and Algebra (American Mathematical Society, Providence, RI, 2015), pp. 1–23. Crossref, Google Scholar
10. D. Calaque, T. Pantev, B. Toën, M. Vaquié and G. Vezzosi , Shifted Poisson structures and deformation quantization, J. Topol. 10(2) (2017) 483–584. Crossref, Web of Science, Google Scholar
11. D. Calaque, H. Rune and S. Claudia, The AKSZ construction in derived algebraic geometry as an extended topological field theory, preprint (2021), arXiv:2108.02473. Google Scholar
12. A. Carboni, S. Kasangian and R. Street , Bicategories of spans and relations, J. Pure Appl. Algebra 33(3) (1984) 259–267. Crossref, Web of Science, Google Scholar
13. A. S. Cattaneo and G. Felder , Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients (Birkhäuser, Basel, 2001), pp. 61–93. Crossref, Google Scholar
14. A. Coste, P. Dazord and A. Weinstein , Groupoïdes Symplectiques, Publications du Département de Mathématiques, Nouvelle Série. A, Vol. 2 (University Claude 1987), pp. i–ii, 1–62. Google Scholar
15. D. R. J. MacG, R. Paré and D. A. Pronk , Universal properties of Span, Theory Appl. Categ. 13(4) (2004) 61–85. Google Scholar
16. R. H. Dijkgraaf, A geometrical approach to two-dimensional conformal field theory, Ph.D. Thesis (1989). Google Scholar
17. T. Dyckerhoff and M. Kapranov , Higher Segal Spaces, Lecture Notes in Mathematics, Vol. 2244 (Springer, Cham, 2019). Crossref, Google Scholar
18. M. Feller, R. Garner, J. Kock, M. U. Proulx and M. Weber , Every 2-Segal space is unital, Commun. Contemp. Math. 23(2) (2021) 2050055. Link, Web of Science, Google Scholar
19. J. Fuchs and C. Stigner , On Frobenius algebras in rigid monoidal categories, Arab. J. Sci. Eng. Sect. C Theme Issues 33(2) (2008) 175–191. Google Scholar
20. M. Fukuma, S. Hosono and H. Kawai , Lattice topological field theory in two dimensions, Comm. Math. Phys. 161(1) (1994) 157–175. Crossref, Web of Science, ADS, Google Scholar
21. I. Gálvez-Carrillo, J. Kock and A. Tonks , Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory, Adv. Math. 331 (2018) 952–1015. Crossref, Web of Science, Google Scholar
22. R. Haugseng , Iterated spans and classical topological field theories, Math. Z. 289(3–4) (2018) 1427–1488. Crossref, Web of Science, Google Scholar
23. C. Heunen, I. Contreras and A. S. Cattaneo , Relative Frobenius algebras are groupoids, J. Pure Appl. Algebra 217(1) (2013) 114–124. Crossref, Web of Science, Google Scholar
24. C. Heunen and J. Vicary , Categories for Quantum Theory : An Introduction, Oxford Graduate Texts in Mathematics, Vol. 28 (Oxford University Press, Oxford, 2019). Crossref, Google Scholar
25. J. Kock , Frobenius Algebras and 2D Topological Quantum Field Theories, London Mathematical Society Student Texts, Vol. 59 (Cambridge University Press, Cambridge, 2004). Google Scholar
26. A. D. Lauda and H. Pfeiffer , State sum construction of two-dimensional open-closed topological quantum field theories, J. Knot Theory Ramifications 16(9) (2007) 1121–1163. Link, Web of Science, Google Scholar
27. A. D. Lauda and H. Pfeiffer , Open-closed strings: Two-dimensional extended TQFTs and Frobenius algebras, Topol. Appl. 155(7) (2008) 623–666. Crossref, Web of Science, Google Scholar
28. D. Li-Bland and A. Weinstein, Selective categories and linear canonical relations, Symmetry Integr. Geom. Methods Appl. 10 (2014), Paper 100, 31 pp. Google Scholar
29. R. A. Mehta and R. Zhang , Frobenius objects in the category of relations, Lett. Math. Phys. 110(7) (2020) 1941–1959. Crossref, Web of Science, ADS, Google Scholar
30. J. C. Morton , Two-vector spaces and groupoids, Appl. Categr. Struct. 19(4) (2011) 659–707. Crossref, Web of Science, Google Scholar
31. M. D. Penney, Simplicial spaces, lax algebras and the 2-Segal condition, preprint (2017), arXiv:1710.02742. Google Scholar
32. P. Schaller and T. Strobl , Poisson structure induced (topological) field theories, Mod. Phys. Lett. A 9(33) (1994) 3129–3136. Link, Web of Science, ADS, Google Scholar
33. C. J. Schommer-Pries, The classification of two-dimensional extended topological field theories, ProQuest LLC, Ann Arbor, MI, Thesis Ph.D., University of California, Berkeley (2009). Google Scholar
34. W. H. Stern , 2-Segal objects and algebras in spans, J. Homotopy Relat. Struct. 16(2) (2021) 297–361. Crossref, Web of Science, Google Scholar
35. C. Walton and H. Yadav, Filtered Frobenius algebras in monoidal categories, preprint (2021), arXiv:2106.01999. Google Scholar
36. K. Wehrheim and C. T. Woodward , Functoriality for Lagrangian correspondences in Floer theory, Quantum Topol. 1(2) (2010) 129–170. Crossref, Google Scholar
37. A. Weinstein , A note on the Wehrheim-Woodward category, J. Geom. Mech. 3(4) (2011) 507–515. Crossref, Web of Science, Google Scholar