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14
Operads In HigherDimensional Category Theory
, 2004
"... The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2 ..."
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Cited by 32 (2 self)
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The purpose of this paper is to set up a theory of generalized operads and multicategories and to use it as a language in which to propose a definition of weak ncategory. Included is a full explanation of why the proposed definition of ncategory is a reasonable one, and of what happens when n <= 2. Generalized operads and multicategories play other parts in higherdimensional algebra too, some of which are outlined here: for instance, they can be used to simplify the opetopic approach to ncategories expounded by Baez, Dolan and others, and are a natural language in which to discuss enrichment of categorical structures.
Categorical structures enriched in a quantaloid: Categories, distributions and functors
 Theory Appl. Categ
"... We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is th ..."
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Cited by 20 (4 self)
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We study the different guises of the projective objects in Cocont(Q): they are the “completely distributive ” cocomplete Qcategories (the left adjoint to the Yoneda embedding admits a further left adjoint); equivalently, they are the “totally continuous ” cocomplete Qcategories (every object is the supremum of the presheaf of objects “totally below ” it); and also are they the Qcategories of regular presheaves on a regular Qsemicategory. As a particular case, the Qcategories of presheaves on a Qcategory are precisely the “totally algebraic” cocomplete Qcategories (every object is the supremum of the “totally compact” objects below it). We think that these results should be part of a yettobeunderstood “quantaloidenriched domain theory”. 1
Monads And Interpolads In Bicategories
, 1997
"... . Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The local ..."
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Cited by 8 (4 self)
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. Given a bicategory, Y , with stable local coequalizers, we construct a bicategory of monads Y mnd by using lax functors from the generic 0cell, 1cell and 2cell, respectively, into Y . Any lax functor into Y factors through Y mnd and the 1cells turn out to be the familiar bimodules. The locally ordered bicategory rel and its bicategory of monads both fail to be Cauchycomplete, but have a wellknown Cauchycompletion in common. This prompts us to formulate a concept of Cauchycompleteness for bicategories that are not locally ordered and suggests a weakening of the notion of monad. For this purpose, we develop a calculus of general modules between unstructured endo1cells. These behave well with respect to composition, but in general fail to have identities. To overcome this problem, we do not need to impose the full structure of a monad on endo1cells. We show that associative coequalizing multiplications suffice and call the resulting structures interpolads. Together with str...
Framed Bicategories and Monoidal Fibrations
, 2007
"... Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such ..."
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Cited by 6 (1 self)
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Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects ’ over the 0cells, such
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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Cited by 6 (0 self)
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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
Coinduction for recursive data types: partial orders, metric spaces and Omegacategories
, 2000
"... In this paper we prove coinduction theorems for nal coalgebras of endofunctors on categories of partial orders and (generalized) metric spaces. These results characterize the order, respectively the metric, on a nal coalgebra as maximum amongst all simulations. As suggested in [15], and motivated by ..."
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Cited by 4 (0 self)
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In this paper we prove coinduction theorems for nal coalgebras of endofunctors on categories of partial orders and (generalized) metric spaces. These results characterize the order, respectively the metric, on a nal coalgebra as maximum amongst all simulations. As suggested in [15], and motivated by the idea that partial orders and metric spaces are types of enriched category, the notion of simulation is based on the enriched categorical counterpart of relations, called bimodules. In fact, the results above arise as instances of a coinduction theorem, parametric in a quantale applying to nal coalgebras of endofunctors on the category of all (small) all) 50147 and 186363 Also, we give a condition under which the operational notion of simulation coincides with the denotational notion of nal semantics. 1 Introduction Coinduction is a principle for reasoning about potentially innite or circular elements of recursive data types, like streams, processes or exact reals [14,5,7]. Typ...
Generalized enrichment of categories
 Also Journal of Pure and Applied Algebra
, 1999
"... We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmultica ..."
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Cited by 3 (1 self)
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We define the phrase ‘category enriched in an fcmulticategory ’ and explore some examples. An fcmulticategory is a very general kind of 2dimensional structure, special cases of which are double categories, bicategories, monoidal categories and ordinary multicategories. Enrichment in an fcmulticategory extends the (more or less wellknown) theories of enrichment in a monoidal category, in a bicategory, and in a multicategory. Moreover, fcmulticategories provide a natural setting for the bimodules construction, traditionally performed on suitably cocomplete bicategories. Although this paper is elementary and selfcontained, we also explain why, from one point of view, fcmulticategories are the natural structures in which to enrich categories.
On Categories of Asynchronous Circuits
, 1994
"... In this paper we describe a general categorical model of asynchronous circuits flexible enough to describe various paradigms of communication between circuit elements  each paradigm gives rise to a specific category of circuits. In each case the operations on circuits and semantics give an algebr ..."
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In this paper we describe a general categorical model of asynchronous circuits flexible enough to describe various paradigms of communication between circuit elements  each paradigm gives rise to a specific category of circuits. In each case the operations on circuits and semantics give an algebra of circuits akin to a process algebra in which series composition is the composition of the category. The essential aspects of the model are ffl a category of data types [Wal89]; ffl a category Circ whose objects are data types, and whose arrows are inputoutput circuits built from data types; ffl a semantic category Sem suitable to contain behaviours of the circuits; ffl operations on Circ and Sem (for example, cases, parallel, and feedback operations); ffl and a behaviour functor bhv : Circ ! Sem, which preserves the operations (compositionality of behaviour). The study of asynchrony in circuit theory is precisely the study of circuit categories and associated behaviour functors ...
FRAMED BICATEGORIES AND MONOIDAL FIBRATIONS MICHAEL SHULMAN
, 706
"... Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects’ over the 0cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, becau ..."
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Abstract. In some bicategories, the 1cells are ‘morphisms ’ between the 0cells, such as functors between categories, but in others they are ‘objects’ over the 0cells, such as bimodules, spans, distributors, or parametrized spectra. Many bicategorical notions do not work well in these cases, because the ‘morphisms between 0cells’, such as ring homomorphisms, are missing. We can include them by using a pseudo double category, but usually these morphisms also induce base change functors acting on the 1cells. We avoid complicated coherence problems by describing base change ‘nonalgebraically’, using categorical fibrations. The resulting ‘framed bicategories ’ assemble into 2categories, with attendant notions of equivalence, adjunction, and so on which are more appropriate for our examples than are the usual bicategorical ones. We then describe two ways to construct framed bicategories. One is an analogue of rings and bimodules which starts from one framed bicategory and builds another. The other starts from a ‘monoidal fibration’, meaning a parametrized family of monoidal categories, and produces an analogue of the framed bicategory of spans. Combining the two, we obtain a construction which includes both enriched and internal categories as special cases.