On monads and warpings

We explain the sense in which a warping on a monoidal category is the same as a pseudomonad on the corresponding one-object bicategory, and we describe extensions of this to the setting of skew monoidal categories: these are a generalization of monoidal categories in which the associativity and unit maps are not required to be invertible. Our analysis leads us to describe a normalization process for skew monoidal categories, which produces a universal skew monoidal category for which the right unit map is invertible.Comment: 15 pages. Version 2: revised based on a very helpful report from the referee. To appear in the Cahiers de Topologie and Geometrie Differentielle Categorique

Triangulations, orientals, and skew monoidal categories

A concrete model of the free skew-monoidal category Fsk on a single generating object is obtained. The situation is clubbable in the sense of G.M. Kelly, so this allows a description of the free skew-monoidal category on any category. As the objects of Fsk are meaningfully bracketed words in the skew unit I and the generating object X, it is necessary to examine bracketings and to find the appropriate kinds of morphisms between them. This leads us to relationships between triangulations of polygons, the Tamari lattice, left and right bracketing functions, and the orientals. A consequence of our description of Fsk is a coherence theorem asserting the existence of a strictly structure-preserving faithful functor from Fsk to the skew-monoidal category of finite non-empty ordinals and first-element-and-order-preserving functions. This in turn provides a complete solution to the word problem for skew monoidal categories.Comment: 48 page

A skew-duoidal Eckmann-Hilton argument and quantum categories

A general result relating skew monoidal structures and monads is proved. This is applied to quantum categories and bialgebroids. Ordinary categories are monads in the bicategory whose morphisms are spans between sets. Quantum categories were originally defined as monoidal comonads on endomorphism objects in a particular monoidal bicategory M. Then they were shown also to be skew monoidal structures (with an appropriate unit) on objects in M. Now we see in what kind of M quantum categories are merely monads.Comment: 14 pages, dedicated to George Janelidze on the occasion of his 60th birthday; v2 final version, 15 pages, to appear in Applied Categorical Structure

Sounding the past: three silent films

The project was an experiment in linking music and poetry to archive films, not only to provide an enhancing accompaniment but, in some cases, with the aim of making something new which would quite profoundly change the way that these films were perceived by audiences

The Catalan simplicial set

The Catalan numbers are well-known to be the answer to many different counting problems, and so there are many different families of sets whose cardinalities are the Catalan numbers. We show how such a family can be given the structure of a simplicial set. We show how the low-dimensional parts of this simplicial set classify, in a precise sense, the structures of monoid and of monoidal category. This involves aspects of combinatorics, algebraic topology, quantum groups, logic, and category theory.Comment: 15 pages. Replaces and expands upon parts of arXiv:1307.0265; remaining parts of arXiv:1307.0265 will be incorporated into a sequel. Version 2: minor revision; to appear in Math. Proc. Camb. Phil. So

The effects of Ancymidol on Petunia hybrida Hort. \u27Orchid Bouquet\u27

Differences in the response of Petunia hybrida Hort. \u27Orchid Bouquet* to varying rates of ancymidol were found. Plant size varied according to time of year grown; however, there was no impairment in the activity of ancymidol on plant retarda-tion. Soil drench applications were more consistent in overall plant retardation than foliar spray applications. Height and foliar spread were reduced in proportion to the amount of active ingredient applied, up to a maximum of 66 ppm. Stem diameter, leaf and stem dry weight, bloom size, root fresh weight, and root dry weight were all signifi-cantly reduced by ancymidol treatments. The number of blooms and fresh weights of leaves and stems were not affected by ancymidol treatments

On the 2-categories of weak distributive laws

A weak mixed distributive law (also called weak entwining structure) in a 2-category consists of a monad and a comonad, together with a 2-cell relating them in a way which generalizes a mixed distributive law due to Beck. We show that a weak mixed distributive law can be described as a compatible pair of a monad and a comonad, in 2-categories extending, respectively, the 2-category of comonads and the 2-category of monads. Based on this observation, we define a 2-category whose 0-cells are weak mixed distributive laws. In a 2-category K which admits Eilenberg-Moore constructions both for monads and comonads, and in which idempotent 2-cells split, we construct a fully faithful 2-functor from this 2-category of weak mixed distributive laws to K^{2 x 2}.Comment: 15 pages LaTeX source, final version to appear in Comm. Algebr

Weak bimonads and weak Hopf monads

We define a weak bimonad as a monad T on a monoidal category M with the property that the Eilenberg-Moore category M^T is monoidal and the forgetful functor from M^T to M is separable Frobenius. Whenever M is also Cauchy complete, a simple set of axioms is provided, that characterizes the monoidal structure of M^T as a weak lifting of the monoidal structure of M . The relation to bimonads, and the relation to weak bimonoids in a braided monoidal category are revealed. We also discuss antipodes, obtaining the notion of weak Hopf monad.Comment: 29 pages; version 2 minor corrections and added references, also added remark 4.3; title changed from "Weak bimonads" to "Weak bimonads and weak Hopf monads"; to appear in Journal of Algebr

Idempotent splittings, colimit completion, and weak aspects of the theory of monads

We show that some recent constructions in the literature, named `weak' generalizations, can be systematically treated by passing from 2-categories to categories enriched in the Cartesian monoidal category of Cauchy complete categories.Comment: LaTeX source 23 pages; v3 final version, minor corrections only, to appear in Journal of Pure and Applied Algebr