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

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

A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2-category those 2-monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2-monads property-like. We further consider the more restricted class of fully property-like 2-monads, consisting of those property-like 2-monads for which all 2-cells between (even lax) algebra morphisms are algebra 2-cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which "structure is adjoint to unit", and which we now call lax-idempotent 2-monads: both these and their colax-idempotent duals are fully property-like. We end by showing that (at least for finitary 2-monads) the classes of property-likes, fully property-like...

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present three applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang–Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq. The third application is to Hall algebras. We explain how the standard construction of the Hall algebra from the category of Fq representations of a simply-laced quiver can be seen as an example of degroupoidification. This in turn provides a new way to categorify—or more precisely, groupoidify—the positive part of the quantum group associated to the quiver.

Groupoidification is a form of categorification in which vector spaces are replaced by groupoids, and linear operators are replaced by spans of groupoids. We introduce this idea with a detailed exposition of ‘degroupoidification’: a systematic process that turns groupoids and spans into vector spaces and linear operators. Then we present two applications of groupoidification. The first is to Feynman diagrams. The Hilbert space for the quantum harmonic oscillator arises naturally from degroupoidifying the groupoid of finite sets and bijections. This allows for a purely combinatorial interpretation of creation and annihilation operators, their commutation relations, field operators, their normal-ordered powers, and finally Feynman diagrams. The second application is to Hecke algebras. We explain how to groupoidify the Hecke algebra associated to a Dynkin diagram whenever the deformation parameter q is a prime power. We illustrate this with the simplest nontrivial example, coming from the A2 Dynkin diagram. In this example we show that the solution of the Yang– Baxter equation built into the A2 Hecke algebra arises naturally from the axioms of projective geometry applied to the projective plane over the finite field Fq.

Abstract. This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2-categories and bicategories, including notions of limit and colimit, 2-dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2-categories, and Cat-categories. The latter two are exactly the same (except that strictly speaking a Cat-category should have small hom-categories, but that need not concern us here). The first two are nominally different — the 2-categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2-categories. Nonetheless, the theories of bicategories, 2-categories, and Catcategories have rather different flavours.

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

Let C , D and E be n-dimensional teisi, i.e., higher-dimensional Gray-categorical structures. The following questions can be asked. Does a left q-transfor C ! D , i.e., a functor 2 q\Omega C ! D , induce a right q-transfor C ! D , i.e., a functor C\Omega 2 q ! D ? More generally, does a functor C\Omega D ! E induce a functor D\Omega C ! E? For c; c 0 elements of C whose (k \Gamma 1)-sources and (k \Gamma 1)-targets agree, does a q-transfor C ! D induce a q-transfor C (c;c 0 ) ! D (d;d 0 ), for appropriate d;d 0 2 D ? For c; c 0 2 C and d;d 0 2 D whose (k \Gamma 1)-sources and (k \Gamma 1)-targets agree, does a q-transfor C\Omega D ! E induce a (q+k+1)-transfor C (c;c 0 )\Omega D (d;d 0 ) ! E(e;e 0 ), for appropriate e; e 0 2 E? I give answers to these questions in the cases where n-dimensional teisi and their tensor product have been defined, i.e., for n 3, and in some cases for n up to 5 which do not need all data and axioms...

We study 2-monads and their algebras using a Cat-enriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2-categorical points of view. Every 2-category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2-categories, including a model structure on the 2-category of algebras for a 2-monad T, and a model structure on a 2-category of 2-monads on a fixed 2-category K. 1.

Abstract. We study homotopy limits for 2-categories using the theory of Quillen model categories. In order to do so, we establish the existence of projective and injective model structures on diagram 2-categories. Using this result, we describe the homotopical behaviour not only of conical limits but also of weighted limits for 2-categories. Finally, homotopy limits are related to pseudo-limits. 1. Quillen model structures in 2-category theory The 2-category of groupoids, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the Grothendieck fibrations [1, 5, 13]. Similarly, the 2-category of small categories, functors, and natural transformations admits a model structure in which the weak equivalences are the equivalence of categories and the fibrations are the isofibrations, which are functors satisfying a restricted version of the lifting condition for Grothendieck fibrations which involves only isomorphisms [13, 19]. Steve Lack has vastly