# Representing the deformation $\infty$-groupoid

## TL;DR

This paper introduces a simplified yet equivalent model of the Deligne-Hinich-Getzler $
abla$-groupoid for homotopy Lie algebras, using a universal cosimplicial object for differential graded Lie algebras.

## Contribution

It presents a smaller, equivalent representation of the $
abla$-groupoid, enhancing understanding and computational efficiency in the context of homotopy Lie algebras.

## Key findings

- Constructs a universal cosimplicial object for DGLAs
- Provides an explicit smaller model of the $
abla$-groupoid
- Shows equivalence to the original $
abla$-groupoid

## Abstract

The goal of the present paper is to introduce a smaller, but equivalent version of the Deligne-Hinich-Getzler $\infty$-groupoid associated to a homotopy Lie algebra. In the case of differential graded Lie algebras, we represent it by a universal cosimplicial object.