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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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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
Frobenius monads and pseudomonoids
 2CATEGORIES COMPANION 73
, 2004
"... Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalenc ..."
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Six equivalent definitions of Frobenius algebra in a monoidal category are provided. In a monoidal bicategory, a pseudoalgebra is Frobenius if and only i f it is star autonomous. Autonomous pseudoalgebras are also Frobenius. What i t means for a morphism of a bicategory to be a projective equivalence is defined; this concept is related to "strongly separable " Frobenius algebras and "weak monoidal Morita equivalence". Wreath products of Frobenius algebras are discussed.
THE CHU CONSTRUCTION
, 1996
"... We take another look at the Chu construction and show how to simplify it by looking at ..."
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We take another look at the Chu construction and show how to simplify it by looking at
First Order Linear Logic in Symmetric Monoidal Closed Categories
, 1991
"... There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encodin ..."
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There has recently been considerable interest in the development of `logical frameworks ' which can represent many of the logics arising in computer science in a uniform way. Within the Edinburgh LF project, this concept is split into two components; the first being a general proof theoretic encoding of logics, and the second a uniform treatment of their model theory. This thesis forms a case study for the work on model theory. The models of many first and higher order logics can be represented as fibred or indexed categories with certain extra structure, and this has been suggested as a general paradigm. The aim of the thesis is to test the strength and flexibility of this paradigm by studying the specific case of Girard's linear logic. It should be noted that the exact form of this logic in the first order case is not entirely certain, and the system treated here is significantly different to that considered by Girard.
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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. 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...
Lax Naturality Through Enrichment
, 1995
"... We develop the relationship between algebraic structure and monads enriched over the monoidal biclosed category LocOrd l of small locally ordered categories, with closed structure given by Lax(A; B). We state the theorem, give a series of examples, and incorporate an account of sketches and cont ..."
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We develop the relationship between algebraic structure and monads enriched over the monoidal biclosed category LocOrd l of small locally ordered categories, with closed structure given by Lax(A; B). We state the theorem, give a series of examples, and incorporate an account of sketches and contravariance into the theory. This was motivated by C.A.R. Hoare's use of category theoretic structures to model data refinement. 1 Introduction In 1987, C.A.R. Hoare wrote a draft paper, "Data refinement in a categorical setting" [10] in which he used category theory to provide an abstract formalism for his development of data refinement over the previous twenty years [9]. The notion of data refinement is central to the programming method called stepwise refinement proposed by Wirth [19], and gave rise to work on abstract data types such as the IOTA programming system developed by Nakajima, Honda and Nakahara [16]. As Hoare said in [10], there was evidently a unified body of category theo...
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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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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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
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