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Monads for which structures are adjoint to units
 Aarhus Preprint Series 1972/73 No.35
"... We present here the equational twodimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that alread ..."
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Cited by 33 (1 self)
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We present here the equational twodimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that already
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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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.
Quantum and classical structures in nondeterministic computation
 Proceedings of Quanum Interaction 2009, Lecture
"... Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspon ..."
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Abstract. In categorical quantum mechanics, classical structures characterize the classical interfaces of quantum resources on one hand, while on the other hand giving rise to some quantum phenomena. In the standard Hilbert space model of quantum theories, classical structures over a space correspond to its orthonormal bases. In the present paper, we show that classical structures in the category of relations correspond to direct sums of abelian groups. Although relations are, of course, not an interesting model of quantum computation, this result has some interesting computational interpretations. If relations are viewed as denotations of nondeterministic programs, it uncovers a wide variety of nonstandard quantum structures in this familiar area of classical computation. Ironically, it also opens up a version of what in philosophy of quantum mechanics would be called an onticepistemic gap, as it provides no interface to these nonstandard quantum structures. 1
2008) Classical and quantum structures
"... In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to ..."
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In recent work, symmetric daggermonoidal (SDM) categories have emerged as a convenient categorical formalization of quantum mechanics. The objects represent physical systems, the morphisms physical operations, whereas the tensors describe composite systems. Classical data turn out to correspond to Frobenius algebras with some additional properties. They express the distinguishing capabilities of classical data: in contrast with quantum data, classical data can be copied and deleted. The algebraic approach thus shifts the paradigm of ”quantization ” of a classical theory to ”classicization ” of a quantum theory. Remarkably, the simple SDM framework suffices not only for this conceptual shift, but even allows us to distinguish the deterministic classical operations (i.e. functions) from the nondeterministic classical operations (i.e. relations), and the probabilistic classical operations (stochastic maps). Moreover, a combination of some basic categorical constructions (due to Kleisli, resp. Grothendieck) with the categorical presentations of quantum states, provides a resource sensitive account of various quantumclassical interactions: of classical control of quantum data, of classical data arising from quantum measurements, as well as of the classical data processing inbetween controls and measurements. A salient feature here is the graphical calculus for categorical quantum mechanics, which allows a purely diagrammatic representation of classicalquantum interaction. 1
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional 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 2categories and bicategories, including notions of limit and colimit, 2dimensional 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, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories 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 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
Involutive monoidal categories
, 2010
"... Abstract. In this paper, we consider a nonposetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets ..."
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Abstract. In this paper, we consider a nonposetal analogue of the notion of involutive quantale [MP92]; specifically, a (planar) monoidal category equipped with a covariant involution that reverses the order of tensoring. We study the coherence issues that inevitably result when passing from posets to categories; we also link our subject with other notions already in the literature, such as balanced monoidal categories [JS91] and dagger pivotal categories [Sel09]. 1.
Geometry of abstraction in quantum computation
"... Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction i ..."
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Quantum algorithms are sequences of abstract operations, performed on nonexistent computers. They are in obvious need of categorical semantics. We present some steps in this direction, following earlier contributions of Abramsky, Coecke and Selinger. In particular, we analyze function abstraction in quantum computation, which turns out to characterize its classical interfaces. Some quantum algorithms provide feasible solutions of important hard problems, such as factoring and discrete log (which are the building blocks of modern cryptography). It is of a great practical interest to precisely characterize the computational resources needed to execute such quantum algorithms. There are many ideas how to build a quantum computer. Can we prove some necessary conditions? Categorical semantics help with such questions. We show how to implement an important family of quantum algorithms using just abelian groups and relations.
How Algebraic Is Algebra?
, 2001
"... . The 2category VAR of finitary varieties is not varietal over CAT . We introduce the concept of an algebraically exact category and prove that the 2category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations with rank. Every algebraically exact category ..."
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. The 2category VAR of finitary varieties is not varietal over CAT . We introduce the concept of an algebraically exact category and prove that the 2category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations with rank. Every algebraically exact category K is complete, exact, and has filtered colimits which (a) commute with finite limits and (b) distribute over products; besides (c) regular epimorphisms in K are productstable. It is not known whether (a)  (c) characterize algebraic exactness. An equational hull of VAR w.r.t. all operations is also discussed. 1.
LINEARUSE CPS TRANSLATIONS IN THE ENRICHED EFFECT CALCULUS
"... Abstract. The enriched effect calculus (EEC) is an extension of Moggi’s computational metalanguage with a selection of primitives from linear logic. This paper explores the enriched effect calculus as a target language for continuationpassingstyle (CPS) translations in which the typing of the tran ..."
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Abstract. The enriched effect calculus (EEC) is an extension of Moggi’s computational metalanguage with a selection of primitives from linear logic. This paper explores the enriched effect calculus as a target language for continuationpassingstyle (CPS) translations in which the typing of the translations enforces the linear usage of continuations. We first observe that established callbyvalue and callby name linearuse CPS translations of simplytyped lambdacalculus into intuitionistic linear logic (ILL) land in the fragment of ILL given by EEC. These two translations are uniformly generalised by a single generic translation of the enriched effect calculus into itself. As our main theorem, we prove that the generic selftranslation of EEC is involutive up to isomorphism. As corollaries, we obtain full completeness results, both for the generic translation, and for the original callbyvalue and callbyname translations. 1.
ON FROBENIUS ALGEBRAS IN RIGID MONOIDAL CATEGORIES
, 2008
"... We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal catego ..."
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We show that the equivalence between several possible characterizations of Frobenius algebras, and of symmetric Frobenius algebras, carries over from the category of vector spaces to more general monoidal categories. For Frobenius algebras, the appropriate setting is the one of rigid monoidal categories, and for symmetric Frobenius algebras it is the one of sovereign monoidal categories. We also discuss some properties of Nakayama automorphisms.