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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 21 (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 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 16 (4 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...
A UNIFIED FRAMEWORK FOR GENERALIZED MULTICATEGORIES
"... Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “l ..."
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Cited by 12 (1 self)
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Abstract. Notions of generalized multicategory have been defined in numerous contexts throughout the literature, and include such diverse examples as symmetric multicategories, globular operads, Lawvere theories, and topological spaces. In each case, generalized multicategories are defined as the “lax algebras ” or “Kleisli monoids ” relative to a “monad ” on a bicategory. However, the meanings of these words differ from author to author, as do the specific bicategories considered. We propose a unified framework: by working with monads on double categories and related structures (rather than bicategories), one can define generalized multicategories in a way that unifies all previous
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 12 (3 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.
ON PROPERTYLIKE STRUCTURES G. M. KELLY
"... ABSTRACT. 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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ABSTRACT. 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 Zöberlein, 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 propertylikes, and laxidempotents are each coreflective among all 2monads. 1.
Contents
, 2002
"... We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad cons ..."
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We define the category of tidy symmetric multicategories. We construct for each tidy symmetric multicategory Q a cartesian monad (EQ, TQ) and extend this assignation to a functor. We exhibit a relationship between the slice construction on symmetric multicategories, and the ‘free operad ’ monad construction on suitable monads. We use this to give an explicit description of the relationship between BaezDolan and Leinster opetopes.