Document Type
Article
Publication Date
8-29-2022
Publication Title
Reviews in Mathematical Physics
Abstract
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
Spans, category of relations, symplectic category, Frobenius algebra, groupoid, simplicial set, topological quantum field theory
Volume
34
Issue
10
DOI
10.1142/S0129055X22500362
Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Rights
Licensed to Smith College and distributed CC-BY under the Smith College Faculty Open Access Policy.
Recommended Citation
Contreras, Ivan; Keller, Molly; and Mehta, Rajan Amit, "Frobenius Objects in the Category of Spans" (2022). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.
https://scholarworks.smith.edu/mth_facpubs/153
Comments
Peer reviewed accepted manuscript.