The study of codescent in Gray-categories is the study of certain kinds of colimits. These colimits are a generalization of coequalizers, and we shall see that they naturally appear in the study of algebras for Gray-monads. The Gray-category which represents codescent diagrams is here denoted ΔG for its close connection with the simplicial category. In fact, the underlying category of ΔG is the free category on the subcategory of Δop with objects [0], [1], [2],[3], and morphisms consisting of all face maps together with all degeneracy maps whose source is [0] or [1]. The definition given below actually uses the objects [1], [2], [3], [4], identifying this category with a subcategory of Δ instead (and here one should use the algebraist's Δ which includes the empty ordinal [0]). This presentation is not coincidental, as ΔG should be viewed as a kind of three-dimensional version of Δ which has the universal property of being the free strict monoidal category generated by a monoid. While we do not pursue this perspective any further, computing the higher dimensional analogues of the strict monoidal category Δ is an interesting open problem.

The Gray-category ΔG produces the lax version of the notion of codescent diagram. It is a Gray-computad, meaning that cells are added freely one dimension at a time, but that at each dimension we impose the conditions necessary to ensure that the resulting structure is a Gray-category.