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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 ..."
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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...
Paths in double categories
 Theory Appl. Categ
"... Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, w ..."
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Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinsterâ€™s fcmulticategories, with representable identities in the second case.
A Representation Result for Free Cocompletions
, 1999
"... Given a class F of weights, one can consider the construction that takes a small category C to the free cocompletion of C under weighted colimits, for which the weight lies in F . Provided these free F cocompletions are small, this construction generates a 2monad on Cat, or more generally on VCa ..."
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Given a class F of weights, one can consider the construction that takes a small category C to the free cocompletion of C under weighted colimits, for which the weight lies in F . Provided these free F cocompletions are small, this construction generates a 2monad on Cat, or more generally on VCat for monoidal biclosed complete and cocomplete V. We develop the notion of a dense 2monad on VCat and characterise free F cocompletions by dense KZmonads on VCat. We prove various corollaries about the structure of such 2monads and their Kleisli 2categories, as needed for the use of open maps in giving an axiomatic study of bisimulation in concurrency. This requires the introduction of the concept of a pseudocommutativity for a strong 2monad on a symmetric monoidal 2category, and a characterisation of it in terms of structure on the Kleisli 2category. 1 Introduction Given a class of small categories S, one can consider the construction that takes a small category C to the free co...