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108
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 23 (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.
On Kontsevich’s Hochschild cohomology conjecture
"... A conjecture of Deligne stated that the Hochschild cohomology complex of an associative algebra has a natural structure of a 2algebra, i.e. an algebra over the chain complex version of the 2cube operad. This indicated a remarkable connection between the deformation theory of associative algebras, ..."
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Cited by 20 (1 self)
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A conjecture of Deligne stated that the Hochschild cohomology complex of an associative algebra has a natural structure of a 2algebra, i.e. an algebra over the chain complex version of the 2cube operad. This indicated a remarkable connection between the deformation theory of associative algebras, and the geometry of
Lax Logical Relations
 In 27th Intl. Colloq. on Automata, Languages and Programming, volume 1853 of LNCS
, 2000
"... Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with th ..."
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Cited by 19 (2 self)
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Lax logical relations are a categorical generalisation of logical relations; though they preserve product types, they need not preserve exponential types. But, like logical relations, they are preserved by the meanings of all lambdacalculus terms. We show that lax logical relations coincide with the correspondences of Schoett, the algebraic relations of Mitchell and the prelogical relations of Honsell and Sannella on Henkin models, but also generalise naturally to models in cartesian closed categories and to richer languages.
Homotopytheoretic aspects of 2–monads
 J. Homotopy Relat. Struct
"... We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the e ..."
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We study 2monads and their algebras using a Catenriched version of Quillen model categories, emphasizing the parallels between the homotopical and 2categorical points of view. Every 2category with finite limits and colimits has a canonical model structure in which the weak equivalences are the equivalences; we use these to construct more interesting model structures on 2categories, including a model structure on the 2category of algebras for a 2monad T, and a model structure on a 2category of 2monads on a fixed 2category K. 1
A Categorical Axiomatics for Bisimulation
 In Proc. of CONCUR’98, LNCS 1466
, 1998
"... We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in ..."
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Cited by 18 (8 self)
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We give an axiomatic category theoretic account of bisimulation in process algebras based on the idea of functional bisimulations as open maps. We work with 2monads, T , on Cat. Operations on processes, such as nondeterministic sum, prefixing and parallel composition are modelled using functors in the Kleisli category for the 2monad T .
Central extensions of smooth 2groups and a finitedimensional string 2group, Geom. Topol. 15 (2011) 609–676
 Department of Mathematics, Massachusetts Institute of Technology
"... ar ..."
Countable Lawvere Theories and Computational Effects
, 2006
"... Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere ..."
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Cited by 17 (3 self)
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Lawvere theories have been one of the two main category theoretic formulations of universal algebra, the other being monads. Monads have appeared extensively over the past fifteen years in the theoretical computer science literature, specifically in connection with computational effects, but Lawvere theories have not. So we define the notion of (countable) Lawvere theory and give a precise statement of its relationship with the notion of monad on the category Set. We illustrate with examples arising from the study of computational effects, explaining how the notion of Lawvere theory keeps one closer to computational practice. We then describe constructions that one can make with Lawvere theories, notably sum, tensor, and distributive tensor, reflecting the ways in which the various computational effects are usually combined, thus giving denotational semantics for the combinations.
Enriched Lawvere Theories
"... We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of mod ..."
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Cited by 17 (0 self)
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We define the notion of enriched Lawvere theory, for enrichment over a monoidal biclosed category V that is locally finitely presentable as a closed category. We prove that the category of enriched Lawvere theories is equivalent to the category of finitary monads on V. Morever, the Vcategory of models of a Lawvere Vtheory is equivalent to the Vcategory of algebras for the corresponding Vmonad. This all extends routinely to local presentability with respect to any regular cardinal. We finally consider the special case where V is Cat, and explain how the correspondence extends to pseudo maps of algebras.
Modelling environments in callbyvalue programming languages
, 2003
"... In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environmen ..."
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Cited by 17 (4 self)
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In categorical semantics, there have traditionally been two approaches to modelling environments, one by use of finite products in cartesian closed categories, the other by use of the base categories of indexed categories with structure. Each requires modifications in order to account for environments in callbyvalue programming languages. There have been two more general definitions along both of these lines: the first generalising from cartesian to symmetric premonoidal categories, the second generalising from indexed categories with specified structure to κcategories. In this paper, we investigate environments in callbyvalue languages by analysing a finegrain variant of Moggi’s computational λcalculus, giving two equivalent sound and complete classes of models: one given by closed Freyd categories, which are based on symmetric premonoidal categories, the other given by closed κcategories.
On PropertyLike Structures
, 1997
"... A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precis ..."
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Cited by 17 (5 self)
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A category may bear many monoidal structures, but (to within a unique isomorphism) only one structure of "category with finite products". To capture such distinctions, we consider on a 2category those 2monads for which algebra structure is essentially unique if it exists, giving a precise mathematical definition of "essentially unique" and investigating its consequences. We call such 2monads propertylike. We further consider the more restricted class of fully propertylike 2monads, consisting of those propertylike 2monads for which all 2cells between (even lax) algebra morphisms are algebra 2cells. The consideration of lax morphisms leads us to a new characterization of those monads, studied by Kock and Zoberlein, for which "structure is adjoint to unit", and which we now call laxidempotent 2monads: both these and their colaxidempotent duals are fully propertylike. We end by showing that (at least for finitary 2monads) the classes of propertylikes, fully propertylike...