We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2-category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2-equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...

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

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Results enabling one to exhibit objects as cocomplete in the sense definable within a yoneda structure are presented. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.

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.

d manifolds of a certain dimension and the maps between them are equivalence classes of cobordisms between them, which are manifolds with boundary in the next higher dimension. However, it is in many respects far more natural to deal with an ncobordism "category" constructed from points, edges, surfaces, and so on through n-manifolds that have boundaries with corners. The structure encodes cobordisms between cobordisms between cobordisms. This is an n-category with additional structure, and one needs analogously structured linear categories as targets for the appropriate "functors" that define the relevant TQFT's. One could equally well introduce the basic idea in terms of formulations of programming languages that describe processes between processes between processes. A closely analogous idea has long been used in the study of homotopies between homotopies between homotopies in algebraic topology. Analogous structures appear throughout mathematics. In contrast to the original Eilenb

1. My approach is "foundational". On the one hand, I am motivated by the problem of the foundations of mathematics (an unsolved problem as far as I am concerned). On the other hand-- and this is more relevant here--, I start "from scratch", and thus what I say can be understood with little technical knowledge. I only assume a modest amount of category theory as background. I will talk informally about technical matters that are written down formally elsewhere, where they can be studied further. [The text in square brackets [-] is either some technical explanation, or a digression.] 2. Terminology First, some terminological conventions. I will use the word "category " in its most general sense: weak ω-category. This is completely inclusive: all sorts of "categories " are categories ow. here are two extensions of the original meaning: "weak", and "omega-dimensional". Weak " signifies an indeterminate notion; there are several different specific versions of weak ategory. It can also be used as a vague notion, when one is merely looking at what one would

Abstract. A 2-categorical generalisation of the notion of elementary topos is provided, and some of the properties of the yoneda structure [SW78] it generates are explored. Examples relevant to the globular approach to higher dimensional category theory are discussed. This paper also contains some expository material on the theory of fibrations internal to a finitely complete 2-category [Str74b] and provides a self-contained development of the necessary background material on yoneda structures.

For a given ω-operad A on globular sets we introduce a sequence of symmetric operads on Set called slices of A and show how the connected limit preserving properties of slices are related to the property of the