The purpose of this paper is to work out the categorical basis for the foundations of Conformal Field Theory. The definition of Conformal Field Theory was outlined in Segal [45] and recently given in [24] and [25]. Concepts of 2-category theory, such as versions of algebra, limit, colimit, and adjunction, are necessary for this

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

The theory of enriched accessible categories over a suitable base category V is developed. It is proved that these enriched accessible categories coincide with the categories of flat functors, but also with the categories of models of enriched sketches. A particular attention is devoted to enriched locally presentable categories and enriched functors.

asked that it be expanded to study the relation of operads to clubs. The author found this too daunting a task at a busy time and the manuscript was never published. Reading through the manuscript now, more than thirty years later, elicits two strong impressions. First, the treatment is very complete: the only item not discussed in detail is the coherence of the monoidal structure given by the functor T ◦ S on [P, V]. Secondly, it was done—for instance in proving the associativity (R ◦ T) ◦ S ∼ = R ◦ (T ◦ S)—with bare hands. Today one could argue as follows, using universal properties; the author learned this approach from Aurelio Carboni. P op, which is in fact isomorphic to P, is the free symmetric monoidal category on 1. So to give an object of [P, V], or a functor T:1 → [P, V], is equally to give a strong monoidal functor P op → [P, V], where the latter has the convolution monoidal structure ⊗; this is the strong monoidal functor sending m to the tensor power T m = T ⊗T ⊗...⊗T. By Theorem 5.1 of [12], this is equally to give a cocontinuous strong monoidal functor T ′:[P, V] → [P, V]; this is the left Kan extension −◦T,andT is recovered from T ′ as T ′ (J) =J ◦ T. Now the desired associativity ( − ◦T) ◦ S ∼ = −◦(T ◦ S) isjustthe associativity of these cocontinuous strong monoidal functors. I am grateful to my colleagues Lack, Street, and Wood for suggesting this article for the TAC Reprint series, and to Flora Armaghanian for producing the LaTeX version. 1.

Abstract not found

Generalized ultrametric spaces are a common generalization of preorders and ordinary ultrametric spaces, as was observed by Lawvere (1973). Guided by his enriched-categorical view on (ultra)metric spaces, we generalize the standard notions of Cauchy sequence and limit in an (ultra)metric space, and of adjoint pair between preorders. This leads to a solution method for recursive domain equations that combines and extends the standard order-theoretic (Smyth and Plotkin, 1982) and metric (America and Rutten, 1989) approaches.