We describe a construction that to each algebraically specified notion of higher-dimensional category associates a notion of homomorphism which preserves the categorical structure only up to weakly invertible higher cells. The construction is such that these homomorphisms admit a strictly associative and unital composition. We give two applications of this construction. The first is to tricategories; and here we do not obtain the trihomomorphisms defined by Gordon, Power and Street, but rather something which is equivalent in a suitable sense. The second application is to Batanin’s weak ω-categories.

A. Joyal [J] has introduced the category D of the so-called finite disks, and used it to define the concept of #-category, a notion of weak #-category. We introduce the notion of an #-graph being composable (meaning roughly that 'it has a unique composite'), and call an #-category simple if it is freely generated by a composable #-graph. The category S of simple #-categories is a full subcategory of the category, with strict #-functors as morphisms, of all #-categories. The category S is a key ingredient in another concept of weak #-category, called protocategory [MM1], [MZ]. We prove that D and S are contravariantly equivalent, by a duality induced by a suitable schizophrenic object living in both categories. In [MZ], this result is one of the tools used to show that the concept of #-category and that of protocategory are equivalent in a suitable sense. We also prove that composable #-graphs coincide with the #-graphs of the form T # considered by M.Batanin [B], which were characterized by R. Street (as announced in [S]) and called `globular cardinals'. Batanin's construction, using globular cardinals, of the free #-category on a globular set plays an important role in our paper. We give a self-contained presentation of Batanin's construction that suits our purposes.

We begin with a chronology tracing the rise of symmetry concepts in physics, starting with groups and their role in relativity, and leading up to more sophisticated concepts from n-category theory, which manifest themselves in Feynman diagrams and their higherdimensional generalizations: strings, membranes and spin foams.

We demonstrate how to organize 1-dimensional cellular automata into an operad of spaces. The nth term C(k) is the space of radius r = k − 1 automata. The operad composition operation involves both automata composition and shifting of domain. Pointwise operations such as addition of automata become important when we look at the structure of the individual terms in the operad, the spaces of automata with a given radius. Having adopted the discrete topology on such a space, we demonstrate an action of the little n-cubes operad on (n − 1)-dimensional radius r automata. There are clear applications of this action to parallel programming issues. Finally we discuss ways of generalizing both the idea of an operad of automata and the n-cubes action to higher dimensional automata, using higher dimensional operads.

that the objects of Ω are trees and a morphism t → t ′ is an operad map from Ω(t) to Ω(t ′), that is, from the operad generated by the vertices of t to that generated by the vertices of t ′. [Globular pasting diagrams are pasting diagrams which correspond to trees with height. Note that the two sorts of trees just mentioned are not directly related. For one thing there are two sorts of tree composition around, one the ordinary grafting, and the other the special composition that reflects composition of pasting diagrams. We won’t draw the trees with height unless they make definitions or proofs more efficient. Sources are Batanin [4] and Leinster [13]. For examples of pasting diagrams together with their trees see page 8 of [6] at

Operads were originally defined as V-operads, that is, enriched in a symmetric or braided monoidal category V. The symmetry or braiding in V is required in order to describe the associativity axiom the operads must obey, as well as the associativity that must be a property of the action of an operad on any of its algebras. A sequence of categorical types that filter the category of monoidal categories and monoidal functors is given by Balteanu, Fiedorowicz, Schwänzl and Vogt in [2]. These subcategories of MonCat have objects that are called n-fold monoidal categories. A k– fold monoidal category is n-fold monoidal for all n ≤ k, and a symmetric monoidal category is n-fold monoidal for all n. After a review of the role of operads in loop space theory and higher categories we go over definitions of iterated monoidal categories and introduce the lower branches of an extended family tree of simple examples. Then we generalize the definition of operad by defining n-fold operads and their algebras in an iterated monoidal category. It is seen that the interchanges in an iterated monoidal category are the natural requirement for expressing operad associativity. The definition is developed from the starting point of iterated monoids in a category of collections. Since monoids are special cases of enriched categories this allows us to describe the iterated monoidal and higher dimensional categorical structure of iterated operads. We show that for V k-fold monoidal the structure of a (k − n)-fold monoidal strict n-category is possessed by the category of n-fold operads in V. We discuss examples of these operads that live in the previously described categories. Finally we describe the algebras of n-fold V-operads and their products.

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In this paper we describe how to give a particular global category of rings and modules the structure of a relaxed multi category, and we describe an algebra in this relaxed multi category such that vertex algebras appear as such algebras. Key words: Multicategory, relaxed multicategory, vertex algebra, ring and module. Our intention for this paper is to describe a method for giving the category of modules for a cocommutative, coassociative Hopf algebra, the structure of a

The theory of operads, defined through categories of labeled graphs, is generalized to suit definitions of higher categories with arbitrary basic shapes. Constructions of cubical, globular and opetopic weak higher categories are obtained as examples. 1

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