Abstract. One of the open problems in higher category theory is the systematic construction of the higher dimensional analogues of the Gray tensor product. In this paper we begin to adapt the machinery of globular operads [1] to this task. We present a general construction of a tensor product on the category of n-globular sets from any normalised (n + 1)-operad A, in such a way that the algebras for A may be recaptured as enriched categories for the induced tensor product. This is an important step in reconciling the globular and simplicial approaches to higher category theory, because in the simplicial approaches one proceeds inductively following the idea that a weak (n + 1)category is something like a category enriched in weak n-categories. In this paper we reveal how such an intuition may be formulated in terms of globular operads.

Generalised Distributive laws in Computer Science are rules governing the transformation of one programming structure into another. In programming, they are programs satisfying certain formal conditions. Their importance has been to date documented in several isolated cases by diverse formal approaches. These applications have always meant leaps in understanding the nature of the subject. However, distributive laws have not yet been given the attention they deserve. One of the reasons for this omission is certainly the lack of a formal notion of distributive laws in their full generality. This hinders the discovery and formal description of occurrences of distributive laws, which is the precursor of any formal manipulation. In this thesis, an approach to formalisation of distributive laws is presented based on the functorial approach to formal Category Theory pioneered by Lawvere and others, notably Gray. The proposed formalism discloses a rather simple nature of distributive laws of the kind found in programming structures based on lax 2-naturality and Gray’s tensor product of 2-categories. It generalises the existing more specific notions of distributive

Based on a study of the 2-category of weak distributive laws, we describe a method of iterating Street’s weak wreath product construction. That is, for any 2-category K and for any non-negative integer n, we introduce 2-categories Wdl (n) (K), of (n + 1)-tuples of monads in K pairwise related by weak distributive laws obeying the Yang-Baxter equation. The first instance Wdl (0) (K) coincides with Mnd(K), the usual 2-category of monads in K, and for other values of n, Wdl (n) (K) contains Mnd n+1 (K) as a full 2-subcategory. For the local idempotent closure K of K, extending the multiplication of the 2-monad Mnd, we equip these 2-categories with n possible ‘weak wreath product ’ 2-functors Wdl (n) (K) → Wdl (n−1) (K), such that all of their possible n-fold composites Wdl (n) (K) → Wdl (0) (K) are equal; that is, such that the weak wreath product is ‘associative’. Whenever idempotent 2-cells in K split, this leads to pseudofunctors Wdl (n) (K) → Wdl (n−1) (K) obeying the associativity property up-to isomorphism. We present a practically important occurrence of an iterated weak wreath product: the algebra of observable quantities in an Ising type quantum spin chain where the spins take their values in a dual pair of finite weak Hopf algebras. We also construct a fully faithful embedding of Wdl (n) (K) into the 2-category of commutative n + 1 dimensional cubes in Mnd(K) (hence into the 2-category of commutative n + 1 dimensional cubes in K whenever K has Eilenberg-Moore objects and its idempotent 2-cells split). Finally we give a sufficient and necessary condition on a monad in K to be isomorphic to an n-ary weak wreath product.