Distributive laws, pseudodistributive laws and decagons

We give alternative definitions of distributive laws and pseudodistributive laws involving the decagonal coherence conditions which naturally arise when the involved monads and pseudomonads are presented in extensive form. We then use these results to give a number of simplifications in the coherence conditions for distributive laws and pseudodistributive laws. In particular, we show that five coherence axioms suffice in the usual definition of pseudodistributive laws, we give simple descriptions of distributive laws and pseudodistributive laws in terms of (pseudo)algebra structure maps, and we give concise definitions of distributive laws and pseudodistributive laws in no-iteration form.Comment: 27 pages; added the definition of a pseudodistributive law in no-iteration for

On the formal theory of pseudomonads and pseudodistributive laws

We contribute to the formal theory of pseudomonads, i.e. the analogue for pseudomonads of the formal theory of monads. In particular, we solve a problem posed by Steve Lack by proving that, for every Gray-category K, there is a Gray-category Psm(K) of pseudomonads, pseudomonad morphisms, pseudomonad transformations and pseudomonad modifications in K. We then establish a triequivalence between Psm(K) and the Gray-category of pseudomonads introduced by Marmolejo. Finally, these results are applied to give a clear account of the coherence conditions for pseudodistributive laws. 40 pages. Comments welcome.Comment: This submission replaces arXiv:0907:1359v1, titled "On the coherence conditions for pseudo-distributive laws". 40 page

Polycategories via pseudo-distributive laws

AbstractIn 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 pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad on Mod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ of this pseudo-distributive law are precisely symmetric polycategories

Not every pseudoalgebra is equivalent to a strict one

We describe a finitary 2-monad on a locally finitely presentable 2-category for which not every pseudoalgebra is equivalent to a strict one. This shows that having rank is not a sufficient condition on a 2-monad for every pseudoalgebra to be strictifiable. Our counterexample comes from higher category theory: the strict algebras are strict 3-categories, and the pseudoalgebras are a type of semi-strict 3-category lying in between Gray-categories and tricategories. Thus, the result follows from the fact that not every Gray-category is equivalent to a strict 3-category, connecting 2-categorical and higher-categorical coherence theory. In particular, any nontrivially braided monoidal category gives an example of a pseudoalgebra that is not equivalent to a strict one.Comment: 17 pages; added more explanation; final version, to appear in Adv. Mat

Operads within monoidal pseudo algebras

A general notion of operad is given, which includes as instances, the operads originally conceived to study loop spaces, as well as the higher operads that arise in the globular approach to higher dimensional algebra. In the framework of this paper, one can also describe symmetric and braided analogues of higher operads, likely to be important to the study of weakly symmetric, higher dimensional monoidal structures