  This article develops a theory of cell combinatorics and cell
2-representations for differential graded 2-categories. We introduce two types
of partial preorders, called the strong and weak preorder, and analyse and
compare them. The weak preorder is more easily tractable, while the strong
preorder is more closely related to the combinatorics of the associated
homotopy 2-representations. To each left cell, we associate a maximal ideal
spectrum, and each maximal ideal gives rise to a differential graded cell
2-representation. Any strong cell is contained in a weak cell, and there is a
bijection between the corresponding maximal ideal spectra. We classify weak and
strong cell 2-representation for a class of examples associated to
finite-dimensional differential graded algebras.
