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11
Relations in Concurrency
"... The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the seman ..."
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Cited by 273 (34 self)
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The theme of this paper is profunctors, and their centrality and ubiquity in understanding concurrent computation. Profunctors (a.k.a. distributors, or bimodules) are a generalisation of relations to categories. Here they are first presented and motivated via spans of event structures, and the semantics of nondeterministic dataflow. Profunctors are shown to play a key role in relating models for concurrency and to support an interpretation as higherorder processes (where input and output may be processes). Two recent directions of research are described. One is concerned with a language and computational interpretation for profunctors. This addresses the duality between input and output in profunctors. The other is to investigate general spans of event structures (the spans can be viewed as special profunctors) to give causal semantics to higherorder processes. For this it is useful to generalise event structures to allow events which “persist.”
The cartesian closed bicategory of generalised species of structures
, 2006
"... Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised ..."
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Cited by 9 (3 self)
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Abstract. The concept of generalised species of structures between small categories and, correspondingly, that of generalised analytic functor between presheaf categories are introduced. An operation of substitution for generalised species, which is the counterpart to the composition of generalised analytic functors, is also put forward. These definitions encompass most notions of combinatorial species considered in the literature—including of course Joyal’s original notion—together with their associated substitution operation. Our first main result exhibits the substitution calculus of generalised species as arising from a Kleisli bicategory for a pseudocomonad on profunctors. Our second main result establishes that the bicategory of generalised species of structures is cartesian closed. 1.
Differential Structure in Models of Multiplicative Biadditive Intuitionistic Linear Logic (Extended Abstract)
"... Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigati ..."
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Cited by 6 (0 self)
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Abstract. In the first part of the paper I investigate categorical models of multiplicative biadditive intuitionistic linear logic, and note that in them some surprising coherence laws arise. The thesis for the second part of the paper is that these models provide the right framework for investigating differential structure in the context of linear logic. Consequently, within this setting, I introduce a notion of creation operator (as considered by physicists for bosonic Fock space in the context of quantum field theory), provide an equivalent description of creation operators in terms of creation maps, and show that they induce a differential operator satisfying all the basic laws of differentiation (the product and chain rules, the commutation relations, etc.). 1
Operads, clones, and distributive laws
, 2008
"... Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory ..."
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Cited by 4 (0 self)
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Abstract We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a monad on profunctors thanks to a distributivelaw. The presentation builds upon recent work by Fiore, Gambino, Hyland, and Winskel on a theory of generalized species of structures,but, for the multicategory case, the general idea goes back to Burroni's Tcategories (1971). We show how other previous categorical analysesof operad (via Day's tensor products, or via analytical functor) fit with the profunctor approach.
REFLECTIVE KLEISLI SUBCATEGORIES OF THE CATEGORY OF EILENBERGMOORE ALGEBRAS FOR FACTORIZATION MONADS
"... It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left ..."
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Cited by 2 (1 self)
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It is well known that for any monad, the associated Kleisli category is embedded in the category of EilenbergMoore algebras as the free ones. We discovered some interesting examples in which this embedding is reflective; that is, it has a left
Monoidal indeterminates and categories of possible worlds
 In Proc. of MFPS XXV
, 2009
"... Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffin ..."
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Given any symmetric monoidal category C, a small symmetric monoidal category Σ and a strong monoidal functor j:Σ C, we construct C[x: jΣ], the polynomial category with a system of (freely adjoined) monoidal indeterminates x: I j(w), natural in w ∈ Σ. As a special case, we construct the free coaffine category (symmetric monoidal category with initial unit) on a given small symmetric monoidal category. We then exhibit all the known categories of “possible worlds ” used to treat languages that allow for dynamic creation of “new ” variables, locations, or names as instances of this construction and explicate their associated universality properties. As an application of the resulting characterisation of O(W), Oles’s category of possible worlds, we present an O(W)indexed Lawvere theory of manysorted storage, generalizing the singlesorted one introduced by J. Power, and we describe explicitly an associated
A foundation for GADTs and inductive families: dependent polynomial functor approach
 In WGP’11
, 2011
"... Every Algebraic Datatype (ADT) is characterised as the initial algebra of a polynomial functor on sets. This paper extends the characterisation to the case of more advanced datatypes: Generalised Algebraic Datatypes (GADTs) and Inductive Families. Specifically, we show that GADTs and Inductive Fa ..."
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Every Algebraic Datatype (ADT) is characterised as the initial algebra of a polynomial functor on sets. This paper extends the characterisation to the case of more advanced datatypes: Generalised Algebraic Datatypes (GADTs) and Inductive Families. Specifically, we show that GADTs and Inductive Families are characterised as initial algebras of dependent polynomial functors. The theoretical tool we use throughout is an abstract notion of polynomial between sets together with its associated general form of polynomial functor between categories of indexed sets introduced by Gambino and Hyland. In the context of ADTs, this fundamental result is the basis for various generic functional programming techniques. To establish the usefulness of our approach for such developments in the broader context of inductively defined dependent types, we apply the theory to construct zippers for Inductive Families.
China (2010)" Operads, clones, and distributive laws
, 2012
"... We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day’s tensor pro ..."
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We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day’s tensor products, or via analytical functors) fit with the profunctor approach. 1