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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 precise mathemat ..."
Abstract

Cited by 9 (3 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...
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.