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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 precis ..."
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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...
Bireflectivity
, 1999
"... Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempoten ..."
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Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireflectivity. Bireflective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterise them in terms of splitidempotent natural transformations on idA. In the special case that A is a presheaf category, we characterise them in terms of the domain, and prove that any bireflective subcategory of A is itself a presheaf category. We define diagonal structure on a symmetric monoidal category which is still more general than asking the tensor product to be the categorical product. We then obtain a bireflective subcategory of [C op; Set] and deduce results relating its finite product structure with the monoidal structure of [C op; Set] determined by that of C. We also investigate the closed structure. Finally, for completeness, we give results on bireflective subcategories in Rel(A), the category of relations in a topos A, and a characterisation of bireflection functors in terms of modules they define.