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20
Combining effects: sum and tensor
"... We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations ..."
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Cited by 29 (4 self)
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We seek a unified account of modularity for computational effects. We begin by reformulating Moggi’s monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular yields Moggi’s exceptions monad transformer and an interactive input/output monad transformer. We further give a theory of the commutative combination of effects, their tensor, which yields Moggi’s sideeffects monad transformer. Finally we give a theory of operation transformers, for redefining operations when adding new effects; we derive explicit forms for the operation transformers associated to the above monad transformers.
Presheaf models of constructive set theories
, 2004
"... Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic S ..."
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Cited by 18 (5 self)
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Abstract. We introduce a new kind of models for constructive set theories based on categories of presheaves. These models are a counterpart of the presheaf models for intuitionistic set theories defined by Dana Scott in the ’80s. We also show how presheaf models fit into the framework of Algebraic Set Theory and sketch an application to an independence result. 1. Variable sets in foundations and practice Presheaves are of central importance both for the foundations and the practice of mathematics. The notion of a presheaf formalizes well the idea of a variable set, that is relevant in all the areas of mathematics concerned with the study of indexed families of objects [19]. One may then readily see how presheaves are of interest also in foundations: both Cohen’s forcing models for classical set theories and Kripke models for intuitionistic logic involve the idea of sets indexed by stages. Constructive aspects start to emerge when one considers the internal logic of categories of presheaves. This logic, which does not include classical principles such as the law of the excluded middle, provides a useful language to manipulate objects
Constructing strictly positive families
 In The Australasian Theory Symposium (CATS2007
, 2007
"... We present an inductive definition of a universe containing codes for strictly positive families (SPFs) such as vectors or simply typed lambda terms. This construction extends the usual definition of inductive strictly positive types as given in previous joint work with McBride. We relate this to In ..."
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Cited by 9 (3 self)
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We present an inductive definition of a universe containing codes for strictly positive families (SPFs) such as vectors or simply typed lambda terms. This construction extends the usual definition of inductive strictly positive types as given in previous joint work with McBride. We relate this to Indexed Containers, which were recently proposed in joint work with Ghani, Hancock and McBride. We demonstrate by example how dependent types can be encoded in this universe and give examples for generic programs.
General structural operational semantics through categorical logic (Extended Abstract)
, 2008
"... Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rulebased definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formul ..."
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Cited by 7 (6 self)
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Certain principles are fundamental to operational semantics, regardless of the languages or idioms involved. Such principles include rulebased definitions and proof techniques for congruence results. We formulate these principles in the general context of categorical logic. From this general formulation we recover precise results for particular language idioms by interpreting the logic in particular categories. For instance, results for firstorder calculi, such as CCS, arise from considering the general results in the category of sets. Results for languages involving substitution and name generation, such as the πcalculus, arise from considering the general results in categories of sheaves and group actions. As an extended example, we develop a tyft/tyxtlike rule format for open bisimulation in the πcalculus.
Polynomial functors and opetopes
 In preparation
"... We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and exam ..."
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Cited by 4 (0 self)
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We give an elementary and direct combinatorial definition of opetopes in terms of trees, wellsuited for graphical manipulation (e.g. drawings of opetopes of any dimension and basic operations like sources, target, and composition); a substantial part of the paper is constituted by drawings and example computations. To relate our definition to the classical definition, we recast the BaezDolan slice construction for operads in terms of polynomial monads: our opetopes appear naturally as types for polynomial monads obtained by iterating the BaezDolan construction, starting with the trivial monad. Finally we observe a suspension operation for opetopes, and define a notion of stable opetopes. Stable opetopes form a least fixpoint for the BaezDolan construction. The calculus of opetopes is also wellsuited for machine implementation: in an appendix we show how to represent opetopes in XML, and manipulate them with simple Tcl scripts.
Sheaves for predicative toposes
 ArXiv:math.LO/0507480v1
"... Abstract: In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal systems. Among our technical results, we prove that ..."
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Cited by 3 (0 self)
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Abstract: In this paper, we identify some categorical structures in which one can model predicative formal systems: in other words, predicative analogues of the notion of a topos, with the aim of using sheaf models to interprete predicative formal systems. Among our technical results, we prove that all the notions of a “predicative topos ” that we consider, are stable under presheaves, while most are stable under sheaves. 1
When Is a Container a Comonad?
"... Abstract. Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry addi ..."
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Cited by 2 (2 self)
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Abstract. Abbott, Altenkirch, Ghani and others have taught us that many parameterized datatypes (set functors) can be usefully analyzed via container representations in terms of a set of shapes and a set of positions in each shape. This paper builds on the observation that datatypes often carry additional structure that containers alone do not account for. We introduce directed containers to capture the common situation where every position in a datastructure determines another datastructure, informally, the subdatastructure rooted by that position. Some natural examples are nonempty lists and nodelabelled trees, and datastructures with a designated position (zippers). While containers denote set functors via a fullyfaithful functor, directed containers interpret fullyfaithfully into comonads. But more is true: every comonad whose underlying functor is a container is represented by a directed container. In fact, directed containers are the same as containers that are comonads. We also describe some constructions of directed containers. We have formalized our development in the dependently typed programming language Agda. 1