With the very recent work of Hopkins and Lurie, the Baez–Dolan conjectures have come to the forefront of research on n-categories. Baez and Dolan [8] proposed a whole series of definitions and properties to be expected of a good theory of n-categories. The main elements were their conjectures on the universal property of certain n-categories defined by looking at cobordisms. These generalize the known ideas of topological and conformal field theories associated to knot invariants. It now appears that Hopkins and Lurie have proven a good portion of these conjectures, see Lurie's expository account [193].

In the present chapter, we will consider a small and beginning piece of the picture, which Baez and Dolan called the stabilization hypothesis. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. This chapter constitutes a revised version of my preprint [237], which had been occasioned by Baez and Dolan's paper “Categorification” [12].

We work in the context of weak n-categories obtained by iterating our weak enrichment construction starting from the model category Set. This is the first place where we really consider the higher iterates all together, rather than just using one step of the iteration procedure. It will therefore be convenient to stick to the case PCn(Set) although parts of the discussion could undoubtedly be done for iterations of the form PCn(M) starting from a different model category.