We give a definition of weak n-categories based on the theory of operads. We work with operads having an arbitrary set S of types, or ‘S-operads’, and given such an operad O, we denote its set of operations by elt(O). Then for any S-operad O there is an elt(O)-operad O + whose algebras are S-operads over O. Letting I be the initial operad with a one-element set of types, and defining I 0+ = I, I (i+1)+ = (I i+) +, we call the operations of I (n−1)+ the ‘n-dimensional opetopes’. Opetopes form a category, and presheaves on this category are called ‘opetopic sets’. A weak n-category is defined as an opetopic set with certain properties, in a manner reminiscent of Street’s simplicial approach to weak ω-categories. In a similar manner, starting from an arbitrary operad O instead of I, we define ‘n-coherent O-algebras’, which are n times categorified analogs of algebras of O. Examples include ‘monoidal n-categories’, ‘stable n-categories’, ‘virtual n-functors ’ and ‘representable n-prestacks’. We also describe how n-coherent O-algebra objects may be defined in any (n + 1)-coherent O-algebra. 1

We introduce a notion of equipment which generalizes the earlier notion of pro-arrow equipment and includes such familiar constructs as relK, spnK, parK, and proK for a suitable category K, along with related constructs such as the V-pro arising from a suitable monoidal category V. We further exhibit the equipments as the objects of a 2-category, in such a way that arbitrary functors F: L ✲ K induce equipment arrows relF: relL ✲ relK, spnF: spnL ✲ spnK, and so on, and similarly for arbitrary monoidal functors V ✲ W. The article I with the title above dealt with those equipments M having each M(A, B) only an ordered set, and contained a detailed analysis of the case M = relK; in the present article we allow the M(A, B) to be general categories, and illustrate our results by a detailed study of the case M = spnK. We show in particular that spn is a locally-fully-faithful 2-functor to the 2-category of equipments, and determine its image on arrows. After analyzing the nature of adjunctions in the 2-category of equipments, we are able to give a simple characterization of those spnG which arise from a geometric morphism G.

In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for concurrency with a built-in notion of bisimulation. We show how

To a bicategory B (in the sense of Benabou) we assign a simplicial set Ner(B), the (geometric) nerve of B, which completely encodes the structure of B as a bicategory. As a simplicial set Ner(B) is a subcomplex of its 2-Coskeleton and itself isomorphic to its 3-Coskeleton, what we call a 2-dimensional Postnikov complex. We then give, somewhat more delicately, a complete characterization of those simplicial sets which are the nerves of bicategories as certain 2-dimensional Postnikov complexes which satisfy certain restricted "exact horn-lifting" conditions whose satisfaction is controlled by (and here defines) subsets of (abstractly) invertible 2 and 1-simplices. Those complexes which have, at minimum, their degenerate 2-simplices always invertible and have an invertible 2-simplex # 1 2 (x 12 , x 01 ) present for each "composable pair" (x 12 , , x 01 ) # # 1 2 are exactly the nerves of bicategories. At the other extreme, where all 2 and 1-simplices are invertible, are those Kan complexes in which the Kan conditions are satisfied exactly in all dimensions > 2. These are exactly the nerves of bigroupoids -- all 2-cells are isomorphisms and all 1-cells are equivalences. Contents

. In the literature there are several kinds of concrete and abstract cell complexes representing composition in n-categories, !-categories or 1-categories, and the slightly more general partial !-categories. Some examples are parity complexes, pasting schemes and directed complexes. In this paper we give an axiomatic treatment: that is to say, we study the class of `!-complexes' which consists of all complexes representing partial !-categories. We show that !-complexes can be given geometric structures and that in most important examples they become well-behaved CW complexes; we characterise !-complexes by conditions on their cells; we show that a product of !-complexes is again an !-complex; and we describe some products in detail. 1. Introduction In this paper we consider pasting diagrams representing compositions in multiple categories. To be specific, the multiple categories concerned are n-categories and their infinite-dimensional analogues, which are called !-categories or 1-cat...

Abstract. In this paper we introduce a notion of generalized operad containing as special cases various kinds of operad–like objects: ordinary, cyclic, modular, properads etc. We then construct inner cohomomorphism objects in their categories (and categories of algebras over them). We argue that they provide an approach to symmetry and moduli objects in non-commutative geometries based upon these “ring–like ” structures. We give a unified axiomatic treatment of generalized operads as functors on categories of abstract labeled graphs. Finally, we extend inner cohomomorphism constructions to more general categorical contexts. This version differs from the previous ones by several local changes (including the title) and two extra references. 0.1. Inner cohomomorphisms of associative algebras. Let k be a field. Consider pairs A = (A, A1) consisting of an associative k–algebra A and a finite dimensional subspace A1 generating A. For two such pairs A = (A, A1) and B =

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Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higher-dimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of n-categories. Here, we are interested in proving confluence for polygraphs presenting 2-categories, which can be seen as a generalization of term-rewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2-categories. Term rewriting systems have proven very useful to reason about terms modulo equations. In some cases, the equations can be oriented and completed in a way giving rise to a normalizing (i.e. confluent and terminating) rewriting system, thus providing a notion of canonical representative of equivalence classes of terms. Usually, terms are freely generated by a signature (Σn)n∈N, which consists of a family of sets Σn of generators of arity n, and one considers equational theories on such a signature, which are formalized by sets of pairs of terms called equations. For example, the equational theory of monoids contains two generators m and e, whose arities are respectively 2 and 0, and three equations

In this easy introduction to higher gauge theory, we describe parallel transport for particles and strings in terms of 2-connections on 2-bundles. Just as ordinary gauge theory involves a gauge group, this generalization involves a gauge ‘2-group’. We focus on 6 examples. First, every abelian Lie group gives a Lie 2-group; the case of U(1) yields the theory of U(1) gerbes, which play an important role in string theory and multisymplectic geometry. Second, every group representation gives a Lie 2-group; the representation of the Lorentz group on 4d Minkowski spacetime gives the Poincaré 2-group, which leads to a spin foam model for Minkowski spacetime. Third, taking the adjoint representation of any Lie group on its own Lie algebra gives a ‘tangent 2-group’, which serves as a gauge 2-group in 4d BF theory, which has topological gravity as a special case. Fourth, every Lie group has an ‘inner automorphism 2-group’, which serves as the gauge group in 4d BF theory with cosmological constant term. Fifth, every Lie group has an ‘automorphism 2-group’, which plays an important role in the theory of nonabelian gerbes. And sixth, every compact simple Lie group gives a ‘string 2-group’. We also touch upon higher structures such as the ‘gravity 3-group’, and the Lie 3-superalgebra that governs 11-dimensional supergravity. 1

The theory of directed complexes is extended from free !-categories to arbitrary !- categories by defining presentations in which the generators are atoms and the relations are equations between molecules. Our main result relates these presentations to the more standard algebraic presentations; we also show that every !-category has a presentation by directed complexes. The approach is similar to that used by Crans for pasting presentations. 1991 Mathematics Subject Classification: 18D05. 1 Introduction There are at present three combinatorial structures for constructing !-categories: pasting schemes, defined in 1988 by Johnson [8], parity complexes, introduced in 1991 by Street [16, 17] and directed complexes, given by Steiner in 1993 [15]. These structures each consist of cells which have collections of lower dimensional cells as domain and codomain; see for example Definition 2.2 below. They also have `local' conditions on the cells, ensuring that a cell together with its boundin...