Abstract
Groupoids graded by the groupoid of bijections between finite sets admit generating functions which encode the groupoid cardinalities of their graded components. As suggested in the work of Baez and Dolan, we use analytic continuation of such generating functions to define a complex-valued cardinality for groupoids whose usual groupoid cardinality diverges. The complex nature of such a cardinality invariant is shown to reflect a recursion of structure which we refer to as ‘nested equivalence’.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Baez, J.C., Dolan, J.: From finite sets to Feynman diagrams. In: Engquist, B., Schmid, W. (eds.) Mathematics Unlimited–2001 and Beyond, pp. 29–50. Springer, Berlin (2001)
Baez, J.C., Hoffnung, A.E., Walker, C.D.: Higher dimensional algebra VII: Groupoidification. Theory Appl. Categ. 24(18), 489–553 (2010)
Behrend, K.A.: The Lefschetz trace formula for algebraic stacks. Invent. Math. 112(1), 127–149 (1993)
Berger, C., Leinster, T.: The Euler characteristic of a category as the sum of a divergent series. Homology Homotopy Appl. 10(1), 41–51 (2008)
Blass, A.: Seven trees in one. J. Pure Appl. Algebra 103(1), 1–21 (1995)
Fiore, M.: Isomorphisms of Generic Recursive Polynomial Types. In: POP’04: Proceedings of the 31st ACM SIGPLAN-SIGACT Symposium on Principles of Programming Languages pp. 77–88 (2004)
Fiore, M., Leinster, T.: Objects of categories as complex numbers. Adv. Math. 190(2), 264–277 (2005)
Gates, R.: On the generic solution to \(P(X)\cong X\) in distributive categories. J. Pure Appl. Algebra 125(1–3), 191–212 (1998)
Joyal, A.: Une théorie combinatoire des séries formelles. Adv. Math. 42(1), 1–82 (1981)
Lawvere, F.W.: Some thoughts on the future of category theory. In: Carboni, A., et al. (eds.) Category Theory (Como, 1990). Lecture Notes in Mathematics, vol. 1488, pp. 1–13. Springer, Berlin (1991)
Leinster, T.: The Euler characteristic of a category. Doc. Math. 13, 21–49 (2008)
Morton, J.: Categorified algebra and quantum mechanics. Theory Appl. Categ. 16(29), 785–854 (2006)
Weinstein, A.: The volume of a differentiable stack. Lett. Math. Phys. 90(1–3), 353–371 (2009)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Fullwood, J. On analytic groupoid cardinality. European Journal of Mathematics 8 (Suppl 1), 274–289 (2022). https://doi.org/10.1007/s40879-022-00554-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40879-022-00554-4