Monads with arities and their associated theories

After a review of the concept of "monad with arities" we show that the category of algebras for such a monad has a canonical dense generator. This is used to extend the correspondence between finitary monads on sets and Lawvere's algebraic theories to a general correspondence between monads and theories for a given category with arities. As application we determine arities for the free groupoid monad on involutive graphs and recover the symmetric simplicial nerve characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section 2; Subsections on symmetric operads (3.14) and symmetric simplicial sets (4.17) added; Bibliography complete

Monads in Double Categories

We extend the basic concepts of Street's formal theory of monads from the setting of 2-categories to that of double categories. In particular, we introduce the double category Mnd(C) of monads in a double category C and define what it means for a double category to admit the construction of free monads. Our main theorem shows that, under some mild conditions, a double category that is a framed bicategory admits the construction of free monads if its horizontal 2-category does. We apply this result to obtain double adjunctions which extend the adjunction between graphs and categories and the adjunction between polynomial endofunctors and polynomial monads.Comment: 30 pages; v2: accepted for publication in the Journal of Pure and Applied Algebra; added hypothesis in Theorem 3.7 that source and target functors preserve equalizers; on page 18, bottom, in the statement concerning the existence of a left adjoint, "if and only if" was replaced by "a sufficient condition"; acknowledgements expande

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