A groupoidification of the fermion algebra

TL;DR

This paper develops a categorical framework for the fermion algebra by constructing a groupoid-based analog of the fermionic Fock space and representing creation and annihilation operators as spans of groupoids.

Contribution

It introduces a novel groupoidification approach to the fermion algebra, including a 2-category of spans that aligns with graphical string diagram relations.

Findings

Constructed a groupoid model of fermionic Fock space

Represented fermionic operators as spans of groupoids

Established a 2-category consistent with graphical relations

Abstract

In this paper, we consider the groupoidification of the fermion algebra. We construct a groupoid as the categorical analogues of the fermionic Fock space, and the creation and annihilation operators correspond to spans of groupoids. The categorical fermionic Fock states have some extra structures comparing with the normal forms. We also construct a 2-category of spans of groupoids corresponding to the fermion algebra. The relations of the morphisms in this 2-category are consistent with those in the graphical category which is represented by string diagrams.