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A Basic Distributive Law
 JOURNAL OF PURE AND APPLIED ALGEBRA
, 2002
"... We pursue distributive laws between monads, particularly in the context of KZdoctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect ..."
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Cited by 14 (3 self)
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We pursue distributive laws between monads, particularly in the context of KZdoctrines, and show that a very basic distributive law has (constructively) completely distributive lattices for its algebras. Moreover, the resulting monad is shown to be also the double dualization monad (with respect to the subobject classifier) on ordered sets.
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.
Frobenius Algebras and ambidextrous adjunctions
, 2006
"... In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fu ..."
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Cited by 12 (1 self)
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In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2categoryDinto which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commutative Frobenius algebra, this result also shows that every 2D TQFT is obtained from an ambijunction in some 2category. Our theorem is proved by extending the theory of adjoint monads to the context of an arbitrary 2category and utilizing the free completion under EilenbergMoore objects. We then categorify this theorem by replacing the monoidal category M with a semistrict monoidal 2category M, and replacing the 2categoryD into which it embeds by a semistrict 3category. To state this more powerful result, we must first define the notion of a ‘Frobenius pseudomonoid’, which categorifies that of a Frobenius object. We then define the notion of a ‘pseudo ambijunction’, categorifying that of an ambijunction. In each case, the idea is that all the usual axioms now hold only up to coherent isomorphism. Finally, we show that every Frobenius pseudomonoid in a semistrict monoidal 2category arises from a pseudo ambijunction in some semistrict 3category.
A Classification of Accessible Categories
 Max Kelly volume, J. Pure Appl. Alg
, 1996
"... For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the ..."
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Cited by 11 (0 self)
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For a suitable collection D of small categories, we define the Daccessible categories, generalizing the #accessible categories of Lair, Makkai, and Pare; here the
Polycategories via pseudodistributive laws
"... In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomo ..."
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Cited by 7 (1 self)
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In this paper, we give a novel abstract description of Szabo’s polycategories. We use the theory of double clubs – a generalisation of Kelly’s theory of clubs to ‘pseudo ’ (or ‘weak’) double categories – to construct a pseudodistributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘twosided Kleisli bicategory’ of this pseudodistributive law are precisely symmetric polycategories. 1
A 2categories companion
"... Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal gu ..."
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Cited by 4 (0 self)
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Abstract. This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. 1. Overview and basic examples This paper is a rather informal guide to some of the basic theory of 2categories and bicategories, including notions of limit and colimit, 2dimensional universal algebra, formal category theory, and nerves of bicategories. As is the way of these things, the choice of topics is somewhat personal. No attempt is made at either rigour or completeness. Nor is it completely introductory: you will not find a definition of bicategory; but then nor will you really need one to read it. In keeping with the philosophy of category theory, the morphisms between bicategories play more of a role than the bicategories themselves. 1.1. The key players. There are bicategories, 2categories, and Catcategories. The latter two are exactly the same (except that strictly speaking a Catcategory should have small homcategories, but that need not concern us here). The first two are nominally different — the 2categories are the strict bicategories, and not every bicategory is strict — but every bicategory is biequivalent to a strict one, and biequivalence is the right general notion of equivalence for bicategories and for 2categories. Nonetheless, the theories of bicategories, 2categories, and Catcategories have rather different flavours.
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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Cited by 1 (0 self)
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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.
ON THE COHERENCE CONDITIONS FOR
, 907
"... Abstract. We survey the development of the formal theory of pseudomonads, the analogue for pseudomonads of the formal theory of monads. One of the main achievements of the theory is a satisfactory axiomatisation of the notion of a pseudodistributive law between pseudomonads. ..."
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Abstract. We survey the development of the formal theory of pseudomonads, the analogue for pseudomonads of the formal theory of monads. One of the main achievements of the theory is a satisfactory axiomatisation of the notion of a pseudodistributive law between pseudomonads.