Lie coalgebras and rational homotopy theory, I: Graph coalgebras

Abstract

We develop a new, intrinsic, computationally friendly approach to Lie coalgebras through graph coalgebras, which are new and likely to be of independent interest. Our graph coalgebraic approach has advantages both in finding relations between coalgebra elements and in having explicit models for linear dualities. As a result, proofs in the realm of Lie coalgebras are often simpler to give through graph coalgebras than through classical methods, and for some important statements we have only found proofs in the graph coalgebra setting. For applications, we investigate the word problem for Lie coalgebras, we revisit Harrison homology, and we unify the two standard Quillen functors between differential graded commutative algebras and Lie coalgebras.