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BASIC CONCEPTS OF ENRICHED CATEGORY THEORY
 REPRINTS IN THEORY AND APPLICATIONS OF CATEGORIES
, 2005
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Structural Induction and Coinduction in a Fibrational Setting
 Information and Computation
, 1997
"... . We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for in ..."
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Cited by 67 (14 self)
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. We present a categorical logic formulation of induction and coinduction principles for reasoning about inductively and coinductively defined types. Our main results provide sufficient criteria for the validity of such principles: in the presence of comprehension, the induction principle for initial algebras is admissible, and dually, in the presence of quotient types, the coinduction principle for terminal coalgebras is admissible. After giving an alternative formulation of induction in terms of binary relations, we combine both principles and obtain a mixed induction/coinduction principle which allows us to reason about minimal solutions X = oe(X) where X may occur both positively and negatively in the type constructor oe. We further strengthen these logical principles to deal with contexts and prove that such strengthening is valid when the (abstract) logic we consider is contextually/functionally complete. All the main results follow from a basic result about adjunc...
Representable Multicategories
 Advances in Mathematics
, 2000
"... We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe ..."
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Cited by 33 (6 self)
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We introduce the notion of representable multicategory , which stands in the same relation to that of monoidal category as bration does to contravariant pseudofunctor (into Cat). We give an abstract reformulation of multicategories as monads in a suitable Kleisli bicategory of spans. We describe representability in elementary terms via universal arrows . We also give a doctrinal characterisation of representability based on a fundamental monadic adjunction between the 2category of multicategories and that of strict monoidal categories. The first main result is the coherence theorem for representable multicategories, asserting their equivalence to strict ones, which we establish via a new technique based on the above doctrinal characterisation. The other main result is a 2equivalence between the 2category of representable multicategories and that of monoidal categories and strong monoidal functors. This correspondence extends smoothly to one between bicategories and a se...
A CategoryTheoretic Account of Program Modules
 Mathematical Structures in Computer Science
, 1994
"... The typetheoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of typeexpressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as inde ..."
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Cited by 23 (6 self)
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The typetheoretic explanation of modules proposed to date (for programming languages like ML) is unsatisfactory, because it does not capture that evaluation of typeexpressions is independent from evaluation of programexpressions. We propose a new explanation based on \programming languages as indexed categories" and illustrates how ML can be extended to support higher order modules, by developing a categorytheoretic semantics for a calculus of modules with dependent types. The paper outlines also a methodology, which may lead to a modular approach in the study of programming languages. Introduction The addition of module facilities to programming languages is motivated by the need to provide a better environment for the development and maintenance of large programs. Nowadays many programming languages include such facilities. Throughout the paper Standard ML (see [Mac85, HMM86, MTH90]) is taken as representative for these languages. The implementation of module facilities has been ...
From Coherent Structures to Universal Properties
 J. Pure Appl. Algebra
, 1999
"... Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coh ..."
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Cited by 13 (2 self)
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Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coherent structures (pseudoTalgebras) are transformed into universally characterised ones (adjointpseudoSalgebras). The 2category L consists of lax algebras for the pseudomonad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudoSalgebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudoalgebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudofunctors into Cat.
Frobenius Algebras and ambidextrous adjunctions
, 2006
"... In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fu ..."
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Cited by 12 (1 self)
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In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2category and utilizing the free completion under EilenbergMoore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2category M, and replacing the 2categoryD into which it embeds by a semistrict 3category. To state this more powerful result, we must first define the notion of a ‘Frobenius pseudomonoid’, which categorifies that of a Frobenius object. We then define the notion of a ‘pseudo ambijunction’, categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2category arises from a pseudo ambijunction in some semistrict 3category.
Lax Factorization Algebras
"... It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facili ..."
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Cited by 9 (4 self)
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It is shown that many weak factorization systems appearing in functorial Quillen model categories, including all those that are cofibrantly generated, come with a rich computational structure, defined by a certain lax algebra with respect to the "squaring monad" on CAT. This structure largely facilitates natural choices for left or right liftings once certain basic natural choices have been made. The use of homomorphisms of such lax algebras is also discussed, with focus on "lax freeness". Mathematics Subject Classification: 18A32, 18C20, 18D05, 55P05. Key words: weak factorization system, cofibrantly generated system, (symmetric) lax factorization algebra, lax homomorphism. Supported by the Ministry of Education of the Czech Republic under project MSM 143100009. y Partial financial assistance by NSERC is acknowledged. 1 1. Introduction Weak factorization systems appear prominently in the definition of Quillen model category: for C, W, F the classes of cofibrations, weak equiva...
The Constructive Lift Monad
 Informix Software, Inc
, 1995
"... ut by applying T to some poset (namely the original poset less the bottom). Both these properties fail to hold constructively, if the lift monad is interpreted as "adding a bottom"; see Remark below. If, on the other hand, we interpret the lift monad as the one which freely provides supremum for ea ..."
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Cited by 5 (0 self)
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ut by applying T to some poset (namely the original poset less the bottom). Both these properties fail to hold constructively, if the lift monad is interpreted as "adding a bottom"; see Remark below. If, on the other hand, we interpret the lift monad as the one which freely provides supremum for each subset with at most one element (which is what we shall do), then the first property holds; and we give a necessary and sufficient condition that the second does. Finally, we shall investigate the lift monad in the context of (constructive) locale theory. I would like to thank Bart Jacobs for guiding me to the litterature on Zsystems; to Gonzalo Reyes for calling my attention to Barr's work on totally connected spaces; to Steve Vickers for some pertinent correspondence. I would like to thank the Netherlands Science Organization (NWO) for supporting my visit to Utrecht, where a part of the present research was carried out, and for various travel support from
Chu I: cofree equivalences, dualities and *autonomous categories
, 1993
"... ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become duali ..."
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Cited by 5 (1 self)
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ing from the technique of dual pairs in functional analysis (Kelley, Nanmioka et al. 1963, ch. 5), they defined the objects of their category to be the triples hA; B; A\Omega B OE !?i, where A and B are arbitrary objects of an autonomous category V, and ? is a fixed object, chosen to become dualizing. A morphism from hA; B; OEi to hC; D; fli was defined as a pair hu : A ! C; B / D : vi of Varrows, making the square A\Omega D A\Omega B C\Omega D ? u\Omega D<Fnan><Fnan> fflffl A\Omega v<Fnan><Fnan> // OE<Fnan><Fnan> fflffl fl<Fnan><Fnan> (1) Cofree equivalences, dualities and autonomous categories 3 commute. This is the setting in which the autonomous structure of a Chu category was originally discovered. The starting point of the present paper is the fact that the category described by Chu is isomorphic to the comma category V=? ? , induced by the homming functor ? ? : V op \Gamma! V : A 7\Gamma! A ? = A \Gammaffi? : (2) By definition, the objects of V=? ? (i.e. Id V =?...