this paper we try to lay some of the foundations of such a theory of categories `up to homotopy' or more exactly `up to coherent homotopies'. The method we use is based on earlier work on:

This document was typeset using L

Much Australian work on categories is part of, or relevant to, the development of higher categories and their theory. In this note, I hope to describe some of the origins and achievements of our efforts that they might perchance serve as a guide to the development of aspects of higher-dimensional work. I trust that the somewhat autobiographical style will add interest rather than be a distraction. For so long I have felt rather apologetic when describing how categories might be helpful to other mathematicians; I have often felt even worse when mentioning enriched and higher categories to category theorists. This is not to say that I have doubted the value of our work, rather that I have felt slowed down by the continual pressure to defend it. At last, at this meeting, I feel justified in speaking freely amongst motivated researchers who know the need for the subject is well established. Australian Category Theory has its roots in homology theory: more precisely, in the treatment of the cohomology ring and the Künneth formulas in the book by Hilton and Wylie [HW]. The first edition of the book had a mistake concerning the cohomology ring of a product. The Künneth formulas arise from splittings of the natural short exact sequences

As an application of Roberts ’ net cohomology theory, we positively answer about the completeness of the known set of DHR sectors of the observables of the model in the title, detailed in [7]. This result is achieved via the triviality of the 1-cohomology of the underlying causal index poset with values in the field net, enhancing this tool for the case of Weyl nets not satisfying the split property and with anyonic commutation rules. We also take advantage of different causal index posets for the nets involved, and obtain the description of the twisted and untwisted sectors of the

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...

The pasting theorem showed that pasting schemes are useful in studying free !-categories. It was thought that their inflexibility with respect to composition and identities prohibited wider use. This is not the case: there is a way of dealing with identities which makes it possible to describe !-categories in terms of generating pasting schemes and relations between generated pastings, i.e., with pasting presentations. In this chapter I develop the necessary machinery for this. The main result, that the !-category generated by a pasting presentation is universal with respect to respectable families of realizations, is a generalization of the pasting theorem. Contents 1 Introduction 3 2 Pasting schemes according to Johnson 4 2.1 Graded sets : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 2.2 !-categories : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 2.3 Pasting schemes : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 2.4 The pasting theorem : : ...

Nonabelian differential n-cocycles provide the data for an assignment of “quantities ” to n-dimensional “spaces ” which is • locally controlled by a given “typical quantity”; • globally compatible with all possible gluings of volumes. For n = 1 this encompasses the notion of parallel transport in fiber bundles with connection. In general we think of it as parallel n-transport. For low n and/or “sufficiently abelian quantities ” this has been modeled by differential characters, (n − 1)-gerbes, (n − 1)-bundle gerbes and n-bundles with connection. We give a general definition for all n in terms of descent data for transport n-functors along the lines of [7, 57, 58, 59]. Concrete realizations, notably Chern-Simons n-cocycles, are obtained by integrating L∞-algebras and their higher Cartan-Ehresmann connections [52]. Here we assume all gluing to happen through equivalences. If one

A “Σ-model ” can be thought of as a quantum field theory (QFT) which is determined by pulling back n-bundles with connection (aka (n−1)-gerbes with connection, aka nonabelian differential cocycles) along all possible maps (the “fields”) from a “parameter space ” to the given base space. If formulated suitably, such Σ-models include gauge theories such as notably (higher) Chern-Simons theory. If the resulting QFT is considered as an “extended ” QFT, it should itself be a nonabelian differential cocycle on parameter space whose parallel transport along pieces of parameter space encodes the QFT propagation and correlators. We are after a conception of nonabelian differential cocycles and their quantization which captures this. Our main motivation is the quantization of differential Chern-Simons cocycles to extended Chern-Simons QFT and its boundary conformal QFT, reproducing the cocycle structure implicit in [23]. • Classical – We conceive nonabelian differential cohomology in terms of cohomology with coefficients in ω-category-valued presheaves [48] of parallel transport ω-functors from ω-paths to a given

Abstract. The primary purpose of this work is to characterise strict ω-categories as simplicial sets with structure. We prove the Street-Roberts conjecture which states that they are exactly the “complicial sets ” defined and named by John Roberts in his handwritten notes of that title [26].2 VERITY