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From Coherent Structures to Universal Properties
 J. Pure Appl. Algebra
, 1999
"... Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coh ..."
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Cited by 13 (2 self)
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Given a 2category K admitting a calculus of bimodules, and a 2monad T on it compatible with such calculus, we construct a 2category L with a 2monad S on it such that: • S has the adjointpseudoalgebra property. • The 2categories of pseudoalgebras of S and T are equivalent. Thus, coherent structures (pseudoTalgebras) are transformed into universally characterised ones (adjointpseudoSalgebras). The 2category L consists of lax algebras for the pseudomonad induced by T on the bicategory of bimodules of K. We give an intrinsic characterisation of pseudoSalgebras in terms of representability. Two major consequences of the above transformation are the classifications of lax and strong morphisms, with the attendant coherence result for pseudoalgebras. We apply the theory in the context of internal categories and examine monoidal and monoidal globular categories (including their monoid classifiers) as well as pseudofunctors into Cat.
Pseudodistributive laws
, 2004
"... We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and ..."
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Cited by 11 (0 self)
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We address the question of how elegantly to combine a number of different structures, such as finite product structure, monoidal structure, and colimiting structure, on a category. Extending work of Marmolejo and Lack, we develop the definition of a pseudodistributive law between pseudomonads, and we show how the definition and the main theorems about it may be used to model several such structures simultaneously. Specifically, we address the relationship between pseudodistributive laws and the lifting of one pseudomonad to the 2category of algebras and to the Kleisli bicategory of another. This, for instance, sheds light on the preservation of some structures but not others along the Yoneda embedding. Our leading examples are given by the use of open maps to model bisimulation and by the logic of bunched implications.
KAN EXTENSIONS AND LAX IDEMPOTENT PSEUDOMONADS
"... Abstract. We show that colax idempotent pseudomonads and their algebras can be presented in terms of right Kan extensions. Dually, lax idempotent pseudomonads and their algebras can be presented in terms of left Kan extensions. We also show that a distributive law of a colax idempotent pseudomonad o ..."
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Cited by 2 (1 self)
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Abstract. We show that colax idempotent pseudomonads and their algebras can be presented in terms of right Kan extensions. Dually, lax idempotent pseudomonads and their algebras can be presented in terms of left Kan extensions. We also show that a distributive law of a colax idempotent pseudomonad over a lax idempotent pseudomonad has a presentation in terms of Kan extensions. 1.
How Algebraic Is Algebra?
, 2001
"... . The 2category VAR of finitary varieties is not varietal over CAT . We introduce the concept of an algebraically exact category and prove that the 2category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations with rank. Every algebraically exact category ..."
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. The 2category VAR of finitary varieties is not varietal over CAT . We introduce the concept of an algebraically exact category and prove that the 2category ALG of all algebraically exact categories is an equational hull of VAR w.r.t. all operations with rank. Every algebraically exact category K is complete, exact, and has filtered colimits which (a) commute with finite limits and (b) distribute over products; besides (c) regular epimorphisms in K are productstable. It is not known whether (a)  (c) characterize algebraic exactness. An equational hull of VAR w.r.t. all operations is also discussed. 1.
Continuous Categories Revisited
, 2003
"... Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of ..."
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Generalizing the fact that Scott's continuous lattices form the equational hull of the class of all algebraic lattices, we describe an equational hull of LFP, the category of locally finitely presentable categories, over CAT. Up to a settheoretical hypothesis this hull is formed by the category of all precontinuous categories, i.e., categories in which limits and filtered colimits distribute. This concept is closely related to the continuous categories of P. T. Johnstone and A. Joyal. 1.
THE CORE OF ADJOINT FUNCTORS
"... Abstract. There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the homenriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categ ..."
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Abstract. There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the homenriched context. Then we do it in the cocompletion of a bicategory with respect to Kleisli objects, which we then apply to internal categories. Finally, we describe a doctrinal setting. 1.
NOITERATION PSEUDOMONADS
"... Abstract. We present the noiteration version of the coherence conditions necessary to define a pseudomonad, and a description of the algebras for it in a similar fashion. We show that every noiteration pseudomonad induces a pseudomonad, and that the corresponding algebras are equivalent. We also s ..."
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Abstract. We present the noiteration version of the coherence conditions necessary to define a pseudomonad, and a description of the algebras for it in a similar fashion. We show that every noiteration pseudomonad induces a pseudomonad, and that the corresponding algebras are equivalent. We also show that every pseudomonad induces a noiteration pseudomonad, and again, that the corresponding algebras are equivalent. We conclude with an analysis of the algebras for the 2monad (−) 2 on Cat in the light of the noiteration description of the algebras. 1.
China (2010)" Operads, clones, and distributive laws
, 2012
"... We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day’s tensor pro ..."
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We show how nonsymmetric operads (or multicategories), symmetric operads, and clones, arise from three suitable monads on Cat, each extending to a (pseudo)monad on the bicategory of categories and profunctors. We also explain how other previous categorical analyses of operads (via Day’s tensor products, or via analytical functors) fit with the profunctor approach. 1