The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal

In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor and par connectives identified. We show that the system allows a fairly simple cut-elimination, and the proofs in the system have a natural interpretation in compact closed categories. However, the soundness of the cut-elimination procedure in terms of the categorical interpretation is by no means evident. We answer to this question affirmatively and establish the soundness by using the coherence result on compact closed categories by Kelly and Laplaza. 1 Introduction In this paper, we introduce the system CMLL of sequent calculus and establish its correspondence with compact closed categories. CMLL is equivalent in provability to the system MLL of classical linear logic with the tensor ffl and par O connectives identified. Compact closed categories are abundant in ...

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

Proofs Without Syntax [37] introduced polynomial-time checkable combinatorial proofs for classical propositional logic. This sequel approaches Hilbert’s 24 th Problem with combinatorial proofs as abstract invariants for sequent calculus proofs, analogous to homotopy groups as abstract invariants for topological spaces. The paper lifts a simple, strongly normalising cut elimination from combinatorial proofs to sequent calculus, factorising away the mechanical commutations of structural rules which litter traditional syntactic cut elimination. Sequent calculus fails to be surjective onto combinatorial proofs: the paper extracts a semantically motivated closure of sequent calculus from which there is a surjection, pointing towards an abstract combinatorial refinement of Herbrand’s theorem.

Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous

POVMs and Naimark’s theorem without sums

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

Abstract not found

Abstract. We propose the categorification of algebraic analysis in terms of a specific 2-category, called here the Leibniz 2-category given by generators and relations which include the Leibniz-like relation (strict 3-cell) among extended 2-cells. The Leibniz 2-category offers the ‘most general ’ notion of a ‘(co)derivation’, as a strict 3-cell, for a general (al- co-)gebra, not necessarily (co-)associative, not necessarily (co-)unital, nor necessarily (co-)commutative. We outline a program in which every 2-cell related to a partial (co-)derivation (called a Leibniz strict 3-cell) is translated into an appropriate 2-cell related to a Cartan’s-like (co-)derivation (called a Cartan strict 3-cell), and vice versa. We found also that a non-Leibniz component (2-cell related to a Leibniz 3-cell), responsible