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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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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.
Pseudodistributive laws
, 2004
"... We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and ..."
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Cited by 11 (0 self)
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We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and we show how the definition and the main theorems about it may be used to model several such structures simultaneously. Specifically, we address the relationship between pseudodistributive laws and the lifting of one pseudomonad to the 2category of algebras and to the Kleisli bicategory of another. This, for instance, sheds light on the preservation of some structures but not others along the Yoneda embedding. Our leading examples are given by the use of open maps to model bisimulation and by the logic of bunched implications.
Combining algebraic effects with continuations
, 2007
"... We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor ext ..."
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Cited by 8 (3 self)
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We consider the natural combinations of algebraic computational effects such as sideeffects, exceptions, interactive input/output, and nondeterminism with continuations. Continuations are not an algebraic effect, but previously developed combinations of algebraic effects given by sum and tensor extend, with effort, to include commonly used combinations of the various algebraic effects with continuations. Continuations also give rise to a third sort of combination, that given by applying the continuations monad transformer to an algebraic effect. We investigate the extent to which sum and tensor extend from algebraic effects to arbitrary monads, and the extent to which Felleisen et al.’s C operator extends from continuations to its combination with algebraic effects. To do all this, we use Dubuc’s characterisation of strong monads in terms of enriched large Lawvere theories.
Resource modalities in tensor logic
"... The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more ..."
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Cited by 6 (0 self)
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The description of resources in game semantics has never achieved the simplicity and precision of linear logic, because of the misleading conception that linear logic is more primitive than game semantics. Here, we defend the opposite view, and thus advocate that game semantics is conceptually more primitive than linear logic. This revised point of view leads us to introduce tensor logic, a primitive variant of linear logic where negation is not involutive. After formulating its categorical semantics, we interpret tensor logic in a model based on Conway games equipped with a notion of payoff, in order to reflect the various resource policies of the logic: linear, affine, relevant or exponential.
CLOSED CATEGORIES VS. CLOSED MULTICATEGORIES
"... Abstract. We prove that the 2category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2subcategory of the 2category of closed multicategories. ..."
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Abstract. We prove that the 2category of closed categories of Eilenberg and Kelly is equivalent to a suitable full 2subcategory of the 2category of closed multicategories.
Department of Mathematics,
, 2009
"... Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1 Preliminaries A categorical group structure (A, j) consists of a monoidal category A and an assignment for every object a of ..."
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Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1 Preliminaries A categorical group structure (A, j) consists of a monoidal category A and an assignment for every object a of A of an object a • with an isomorphism ja: I → a • ⊗ a, (a • is an inverse of a). We are concerned in this paper with symmetric Picard categories which are the categorical groups (A, j) for which A has a symmetric monoidal structure and its underlying category is a groupoid. SPC denotes the 2category with objects symmetric Picard categories, arrows symmetric monoidal functors and 2cells monoidal natural transformations. There is a forgetful 2functor SPC → SMC forgetting the group structure where SMC denotes the 2category with objects symmetric monoidal categories, arrows symmetric monoidal functors, and monoidal natural transformations as 2cells. The 2categorical properties of SPC are similar to those of SMC, the latter 2category has been studied in different works in particular in [HyPo02] and in [Sch08]. We refer the reader to this last work for basic notations, and more elaborate results. In this first section,
Enrichments over symmetric Picard categories
, 2009
"... Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1 ..."
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Categorical rings were introduced in [JiPi07], which we call 2rings. In these notes we present basic definitions and results regarding 2modules. This is work in progress. 1
ON THE REGULAR REPRESENTATION OF AN (ESSENTIALLY) FINITE
, 907
"... Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard ..."
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Abstract. The regular representation of an essentially finite 2group G in the 2category 2Vectk of (Kapranov and Voevodsky) 2vector spaces is defined and cohomology invariants classifying it computed. It is next shown that all homcategories in Rep2Vectk G are 2vector spaces under quite standard assumptions on the field k, and a formula giving the corresponding “intertwining numbers ” is obtained which proves they are symmetric. Finally, it is shown that the forgetful 2functor ω: Rep2Vectk G�2Vectk is representable with the regular representation as representing object. As a consequence we obtain a klinear equivalence between the 2vector space Vect G k of functors from the underlying groupoid of G to Vectk, on the one hand, and the klinear category End ω of pseudonatural endomorphisms of ω, on the other hand. We conclude that End ω is a 2vector space, and we (partially) describe a basis of it. 1.
Tensor and unit for symmetric monoidal categories
, 2008
"... Let SMC denote the 2category with objects small symmetric monoidal categories, 1cells symmetric monoidal functors and 2cells monoidal natural transformations. It is shown that the category quotient of SMC by the congruence generated by its 2cells is symmetric monoidal closed. 1 Summary of result ..."
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Let SMC denote the 2category with objects small symmetric monoidal categories, 1cells symmetric monoidal functors and 2cells monoidal natural transformations. It is shown that the category quotient of SMC by the congruence generated by its 2cells is symmetric monoidal closed. 1 Summary of results Thomason’s famous result claims that symmetric monoidal categories model all connective spectra [Tho95]. The discovery of a symmetric monoidal structure on the category of structured spectra [EKMM97] suggests that a similar structure should exist on an adequate category with symmetric monoidal categories as objects. The first aim of this work is to give a reasonable candidate for a tensor of symmetric monoidal categories. Some structure is investigated on the 2category SMC with objects symmetric monoidal categories, with 1cells symmetric monoidal functors and 2cells monoidal natural transformations. It is shown to induce a symmetric monoidal closed structure on the category SMC / ∼ quotient of SMC by the congruence ∼ generated by its 2cells. This is Theorem 22.2.