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144
Parameterised notions of computation
 In MSFP 2006: Workshop on mathematically structured functional programming, ed. Conor McBride and Tarmo Uustalu. Electronic Workshops in Computing, British Computer Society
, 2006
"... Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, sideeffects, input/output and continuations. We present generalisations of both constructs, which we call para ..."
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Cited by 52 (3 self)
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Moggi’s Computational Monads and Power et al’s equivalent notion of Freyd category have captured a large range of computational effects present in programming languages such as exceptions, sideeffects, input/output and continuations. We present generalisations of both constructs, which we call parameterised monads and parameterised Freyd categories, that also capture computational effects with parameters. Examples of such are composable continuations, sideeffects where the type of the state varies and input/output where the range of inputs and outputs varies. By also considering monoidal parameterisation, we extend the range of effects to cover separated sideeffects and multiple independent streams of I/O. We also present two typed λcalculi that soundly and completely model our categorical definitions — with and without monoidal parameterisation — and act as prototypical languages with parameterised effects.
Towards an algebraic theory of Boolean circuits
 Journal of Pure and Applied Algebra
, 2003
"... Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to ..."
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Cited by 47 (5 self)
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Boolean circuits are used to represent programs on finite data. Reversible Boolean circuits and quantum Boolean circuits have been introduced to modelize some physical aspects of computation. Those notions are essential in complexity theory, but we claim that a deep mathematical theory is needed to make progress in this area. For that purpose, the recent developments of knot theory is a major source of inspiration. Following the ideas of Burroni, we consider logical gates as generators for some algebraic structure with two compositions, and we are interested in the relations satisfied by those generators. For that purpose, we introduce canonical forms and rewriting systems. Up to now, we have mainly studied the basic case and the linear case, but we hope that our methods can be used to get presentations by generators and relations for the (reversible) classical case and for the (unitary) quantum case.
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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Cited by 38 (4 self)
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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 &quot;strongly separable &quot; Frobenius algebras and &quot;weak monoidal Morita equivalence&quot;. Wreath products of Frobenius algebras are discussed.
Replacing model categories with simplicial ones
 Trans. Amer. Math. Soc
"... Abstract. In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories. 1. ..."
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Cited by 36 (2 self)
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Abstract. In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories. 1.
Equational Reasoning With 2Dimensional Diagrams (preliminary Version)
 Term Rewriting, volume 909 of LNCS
, 1992
"... The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special ca ..."
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Cited by 27 (1 self)
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The significance of the 2dimensional calculus, which goes back to Penrose, has already been pointed out by Joyal and Street in [JoS91]. Independently, Burroni has introduced a general notion of ndimensional presentation in [Bur91] and he has shown that the equational logic of terms is a special case of 2dimensional calculus. Here, we propose a combinatorial definition of 2dimensional diagrams and a simple method for proving that certain monoidal categories are finitely 2presentable. We illustrate the translation of terms into diagrams and we explain the change from groups to quantum groups in a purely syntactical way. This paper should serve as a reference for our future work on symbolic computation, including a theory of 2dimensional rewriting and the design of software for interactive diagrammatic reasoning. New address: CNRS  Laboratoire de math'ematiques discr`etes, 163 avenue de Luminy  Case 930, 13288 Marseille Cedex 9, France. Email: lafont@lmd.univmrs.fr 2 1 FROM T...
On the homotopy of simplicial algebras over an operad
 Trans. Amer. Math. Soc
"... According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work o ..."
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Cited by 21 (5 self)
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According to a result of H. Cartan (cf. [5]), the homotopy of a simplicial commutative algebra is equipped with divided power operations. In this paper, we provide a general approach to the construction of such operations in the context of simplicial algebras over an operad. To be precise, we work over a xed eld F, and we consider operads in the
Additivity for derivator Ktheory
, 2008
"... We prove the additivity theorem for the Ktheory of triangulated derivators. This solves one of the conjec ..."
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Cited by 18 (0 self)
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We prove the additivity theorem for the Ktheory of triangulated derivators. This solves one of the conjec