On the Breen-Baez-Dolan stabilization hypothesis for Tamsamani’s weak n-categories

In [2] Baez and Dolan established their stabilization hypothesis as one of a list of the key properties that a good theory of higher categories should have. It is the analogue for n-categories of the well-known stabilization theorems in homotopy theory. To explain the statement, recall that Baez-Dolan introduce the notion of k-uply monoidal n-category which is an n+ k-category having only one i-morphism for all i < k. This includes the notions previously defined and examined by many authors, of monoidal (resp. braided monoidal, symmetric monoidal) category (resp. 2-category) and so forth, as is explained in [2] [4]. See the bibliographies of those preprints as well as that of the the recent preprint [9] for many references concerning these types of objects. In the case where the n-category in question is an n-groupoid, this notion is—except for truncation at n—the same thing as the notion of k-fold iterated loop space, or “Ek-space” which appears in Dunn [10] (see also some anterior references from there). The fully stabilized notion of k-uply monoidal n-categories for k ≫ n is what Grothendieck calls Picard n-categories in [12]. The stabilization hypothesis [2] states that for n + 2 ≤ k ≤ k, the k-uply monoidal n-categories are the same thing as the k-uply monoidal n-categories. This statement first appeared in a preliminary way in Breen [8]; also there is some related correspondence between Breen and Grothendieck in [12]. We will consider this hypothesis for Tamsamani’s theory of (“weak”) n-categories [23], and show one of the main parts of the statement, namely that a k-uply monoidal n-category can be “delooped” to a k + 1-uply monoidal n-category, when k ≥ n + 2. Before giving the precise statement, we make a change of indexing. A k-connected n-category is an n-category which has up to equivalence only one i-morphism for each i ≤ n. More precisely this means that the truncation τ≤k(A) is trivial, equivalent to ∗. Note that a k− 1-connected n+ k-category is equivalent to a k-uply monoidal n-category (see 2.2.5 below). We prove the following theorem (Corollary 2.4.8) for the theory of [23]: