Recursive specifications of domains plays a crucial role in denotational semantics as developed by Scott and Strachey and their followers. The purpose of the present paper is to set up a categorical framework in which the known techniques for solving these equations find a natural place. The idea is to follow the well-known analogy between partial orders and categories, generalizing from least fixed-points of continuous functions over cpos to initial ones of continuous functors over $\omega $-categories. To apply these general ideas we introduce Wand’s ${\bf O}$-categories where the morphism-sets have a partial order structure and which include almost all the categories occurring in semantics. The idea is to find solutions in a derived category of embeddings and we give order-theoretic conditions which are easy to verify and which imply the needed categorical ones. The main tool is a very general form of the limit-colimit coincidence remarked by Scott. In the concluding section we outline how compatibility considerations are to be included in the framework. A future paper will show how Scott’s universal domain method can be included too.

The solution of a recursive domain equation, of the form D ~= F(D) may be viewed as the finding of a fixpoint (up to isomorphism) of the functor F. This has led to the idea of formulating a category-theoretic analogue of Tarski's fixpoint theorem for lattices, as a basis for a general method of solution for this kind of equation; see especially Reynolds [1], Wand [2], Plotkin [3].