Coalgebras and Coderivations

Abstract

The description of an $L_{\infty }[1]$ structure on a graded vector space V as an element $q\in \prod _{n>0}{\text {Hom}}^1_{\mathbb {K}}(V^{\odot n},V)$ satisfying the equation $[q,q]_{NR}=0$ is somewhat difficult to handle and obscure to understand. One of the goals of this chapter is to reinterpret both the Nijenhuis–Richardson bracket and the category of formal neighbourhoods in the framework of graded coalgebras. This will allow us to give, in Chap. 12, a useful equivalent characterization of $L_{\infty }[1]$ structures which leads naturally to the notion of $L_{\infty }$-morphisms.