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An Australian conspectus of higher categories

, 2004
"... 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 higherdimensional wo ..."
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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 higherdimensional 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
Weak Bisimulation and Open Maps (Extended Abstract)
, 1999
"... A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (rough ..."
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A systematic treatment of weak bisimulation and observational congruence on presheaf models is presented. The theory is developed with respect to a "hiding" functor from a category of paths to observable paths. Via a view of processes as bundles , we are able to account for weak morphisms (roughly only required to preserve observable paths) and to derive a saturation monad (on the category of presheaves over the category of paths). Weak morphisms may be encoded as strong ones via the Kleisli construction associated to the saturation monad. A general
THE COMPREHENSIVE FACTORIZATION AND TORSORS
, 2010
"... This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the tors ..."
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This is an expanded, revised and corrected version of the first author's preprint [1]. The discussion of onedimensional cohomology H1 in a fairly general category E involves passing to the (2)category Cat(E) of categories in E. In particular, the coe cient object is a category B in E and the torsors that H1 classifies are particular functors in E. We only impose conditions on E that are satisfied also by Cat(E) and argue that H1 for Cat(E) is a kind of H2 for E, and so on recursively. For us, it is too much to ask E to be a topos (or even internally complete) since, even if E is, Cat(E) is not. With this motivation, we are led to examine morphisms in E which act as internal families and to internalize the comprehensive factorization of functors into a nal functor followed by a discrete bration. We de ne Btorsors for a category B in E and prove clutching and classification theorems. The former theorem clutches ƒech cocycles to construct torsors while the latter constructs a coefficient category to classify structures locally isomorphic to members of a given internal family of structures. We conclude with applications to examples.
Reprints in Theory and Applications of Categories, No. 4, 2004, pp. 1–16. CAUCHY CHARACTERIZATION OF ENRICHED CATEGORIES
"... Preface to the reprinted edition Soon after the appearance of enriched category theory in the sense of EilenbergKelly1, I wondered whether Vcategories could be the same as Wcategories for nonequivalent monoidal categories V and W. It was not until my fourmonth sabbatical in Milan at the end of ..."
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Preface to the reprinted edition Soon after the appearance of enriched category theory in the sense of EilenbergKelly1, I wondered whether Vcategories could be the same as Wcategories for nonequivalent monoidal categories V and W. It was not until my fourmonth sabbatical in Milan at the end of 1981 that I made a serious attempt to properly formulate this question and try to solve it. By this time I was very impressed by the work of Bob Walters [28] showing that sheaves on a site were enriched categories. On sabbatical at Wesleyan University (Middletown) in 197677, I had looked at a preprint of Denis Higgs showing that sheaves on a Heyting algebra H couldbeviewedassomekindofHvalued sets. The latter seemed to be understandable as enriched categories without identities. Walters ’ deeper explanation was that they were enriched categories (with identities) except that the base was not H but rather a bicategory built from H. A stream of research was initiated in which the base monoidal category for enrichment was replaced, more generally, by a base bicategory. In analysis, Cauchy complete metric spaces are often studied as completions of more readily defined metric spaces. Bill Lawvere [15] had found that Cauchy completeness could be expressed for general enriched categories with metric spaces as a special case. Cauchy sequences became left adjoint modules2 and convergence became representability. In Walters ’ work it was the Cauchy complete enriched categories that were the sheaves. It was natural then to ask, rather than my original question, whether Cauchy complete Vcategories were the same as Cauchy complete Wcategories for appropriate base bicategories V and W. I knew already [20] that the bicategory VMod whose morphisms were modules between Vcategories could be constructed from the bicategory whose morphisms were Vfunctors. So the question became: given a base bicategory V, for which
Paracategories II: Adjunctions, fibrations and examples from probabilistic automata theory
, 2002
"... In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction betwee ..."
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In this sequel to [HM02], we explore some of the global aspects of the category of paracategories. We establish its (co)completeness and cartesian closure. From the closed structure we derive the relevant notion of transformation for paracategories. We setup the relevant notion of adjunction between paracategories and apply it to define (co)completeness and cartesian closure, exemplified by the paracategory of bivariant functors and dinatural transformations. We introduce partial multicategories to account for partial tensor products. We also consider fibrations for paracategories and their indexedparacategory version. Finally, we instantiate all these concepts in the context of probabilistic automata.
CATEGORICAL GEOMETRY AND THE MATHEMATICAL FOUN DATIONS OF QUANTUM GRAVITY
, 2006
"... ABSTRACT: We consider two related approaches to quantizing general relativity which involve replacing point set topology with category theory as the foundation for the theory. The ideas of categorical topology are introduced in a way we hope is physicist friendly. I. ..."
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ABSTRACT: We consider two related approaches to quantizing general relativity which involve replacing point set topology with category theory as the foundation for the theory. The ideas of categorical topology are introduced in a way we hope is physicist friendly. I.
CATEGORICAL GEOMETRY AND THE MATHEMATICAL FOUN DATIONS OF QUANTUM GRAVITY
, 2006
"... ABSTRACT: We consider two related approaches to quantizing general relativity which involve replacing point set topology with category theory as the foundation for the theory. The ideas of categorical topology are introduced in a way we hope is physicist friendly. I. ..."
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ABSTRACT: We consider two related approaches to quantizing general relativity which involve replacing point set topology with category theory as the foundation for the theory. The ideas of categorical topology are introduced in a way we hope is physicist friendly. I.