# Weak representations, representations up to homotopy, and VB-groupoids

## TL;DR

This paper introduces weak representations of Lie groupoids, establishes their equivalence with 2-term representations up to homotopy, and shows their correspondence with VB-groupoids, unifying these concepts in a categorical framework.

## Contribution

It defines weak representations of Lie groupoids and proves their categorical equivalence with 2-term representations up to homotopy and VB-groupoids, providing a unified perspective.

## Key findings

- Weak representations are introduced for Lie groupoids.
- An equivalence of categories between weak representations and 2-term representations up to homotopy is established.
- VB-groupoids are shown to correspond to weak representations via action groupoids.

## Abstract

In this paper, I introduce weak representations of a Lie groupoid $G$. I also show that there is an equivalence of categories between the categories of 2-term representations up to homotopy and weak representations of $G$. Furthermore, I show that any VB-groupoid is isomorphic to an action groupoid associated to a weak representation on its kernel groupoid; this relationship defines an equivalence of categories between the categories of weak representations of $G$ and the category of VB-groupoids over $G$.