2-Vector Spaces and Groupoids

TL;DR

This paper explores the connection between finite groupoids and 2-vector spaces, constructing a functor from the bicategory of groupoids and spans to 2-vector spaces, with detailed proofs and constructions.

Contribution

It introduces a method to construct 2-vector spaces from groupoids and defines 2-linear maps as functors, establishing a weak functor from Span(Gpd) to 2Vect.

Findings

Constructs 2-vector spaces of Vect-valued presheaves on finite groupoids.

Defines 2-linear maps as functors with ambidextrous adjoints.

Establishes a weak functor from Span(Gpd) to 2Vect.

Abstract

This paper describes a relationship between essentially finite groupoids and 2-vector spaces. In particular, we show to construct 2-vector spaces of Vect-valued presheaves on such groupoids. We define 2-linear maps corresponding to functors between groupoids in both a covariant and contravariant way, which are ambidextrous adjoints. This is used to construct a representation--a weak functor--from Span(Gpd) (the bicategory of groupoids and spans of groupoids) into 2Vect. In this paper we prove this and give the construction in detail.