  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.
