Parametrized L-infinity Spaces

TL;DR

This paper generalizes L-infinity spaces by incorporating sheaves over nilpotent dg manifolds, linking them to Lie algebroids and their invariants, thus broadening the framework for formal moduli problems.

Contribution

It introduces a new notion of parametrized L-infinity spaces over nilpotent dg manifolds, connecting them to Lie algebroids and their characteristic classes.

Findings

L-infinity spaces can be parametrized by sheaves over nilpotent dg manifolds.

Characteristic classes of these spaces recover Lie algebroid invariants.

Provides examples including those from Lie algebroids.

Abstract

Motivated by families of formal moduli problems, in this note we generalize the notion of L-infinity space by allowing sheaves of L-infinity algebras over any (reasonable) nilpotent dg manifold. We discuss various examples including those coming from Lie algebroids. Given a Lie algebroid, we show that there is an L-infinity space parametrized by (cochains of) the Lie algebroid; further, characteristic classes of this L-infinity space recover the primary invariants of the original algebroid.