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Frobenius objects in the category of relations

Rajan Amit Mehta, Ruoqi Zhang

Abstract

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold.

Frobenius objects in the category of relations

Abstract

We give a characterization, in terms of simplicial sets, of Frobenius objects in the category of relations. This result generalizes a result of Heunen, Contreras, and Cattaneo showing that special dagger Frobenius objects in the category of relations are in correspondence with groupoids. As an additional example, we construct a Frobenius object in the category of relations whose elements are certain cohomology classes in a compact oriented Riemannian manifold.

Paper Structure

This paper contains 19 sections, 22 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Frobenius objects in $\mathbf{Rel}$
  3. Definition
  4. Graphical Calculus
  5. Basic Properties of Frobenius objects in $\mathbf{Rel}$
  6. Nondegeneracy
  7. Relations involving $\hat{\alpha}$
  8. Associativity and coassociativity
  9. Source and target maps
  10. Dual structures
  11. Simplicial Sets
  12. Frobenius objects in $\mathbf{Rel}$ and simplicial sets
  13. From Frobenius objects in $\mathbf{Rel}$ to simplicial sets
  14. From simplicial sets to Frobenius objects in $\mathbf{Rel}$
  15. Groupoids and Frobenius objects in $\mathbf{Rel}$
  16. ...and 4 more sections

Key Result

Lemma 3.2

A binary relation $\alpha$ is nondegenerate if and only if there exists a bijective map $\hat{\alpha}: X \to X$ such that $\alpha = \{(x,\hat{\alpha}(x)) \mid x \in X\}$. In this case, the associated relation $\beta$ is unique and given by $\beta = \{(\hat{\alpha}(x),x) \mid x \in X\}$.

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • ...and 41 more