# The general linear 2-groupoid

## TL;DR

This paper introduces a strict Lie 2-groupoid structure for symmetries of graded vector spaces, constructs nerves for Lie 2-categories, and classifies 2-term representations up to homotopy of Lie groupoids using smooth pseudofunctors.

## Contribution

It establishes a natural Lie 2-groupoid structure for symmetries, constructs nerves yielding simplicial manifolds, and classifies 2-term representations up to homotopy.

## Key findings

- Symmetries define a strict Lie 2-groupoid.
- Nerves of Lie 2-categories are simplicial manifolds when 2-cells are invertible.
- Smooth pseudofunctors classify 2-term representations up to homotopy.

## Abstract

We deal with the symmetries of a (2-term) graded vector space or bundle. Our first theorem shows that they define a (strict) Lie 2-groupoid in a natural way. Our second theorem explores the construction of nerves for Lie 2-categories, showing that it yields simplicial manifolds if the 2-cells are invertible. Finally, our third and main theorem shows that smooth pseudofunctors into our general linear 2-groupoid classify 2-term representations up to homotopy of Lie groupoids.