We define and study a lift of the Boardman-Vogt tensor product of operads to bimodules over operads. Introduction Let Op denote the category of symmetric operads in the monoidal category S of simplicial sets. The Boardman-Vogt tensor product [3] −⊗− : Op× Op→ Op, which endows the category Op with a symmetric monoidal structure, codifies interchanging algebraic structures. For all P,Q ∈ Op, a (P...

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 pseudo-distributive law of the free symmetric strict monoidal category pseudocomonad onMod over itself qua pseudomonad, and show that monads in the ‘two-sided Kleisli bicategory’ ...

We show that Turaev’s group-coalgebras and Hopf group-coalgebras are coalgebras and Hopf algebras in a symmetric monoidal category, which we call the Turaev category. A similar result holds for group-algebras and Hopf group-algebras. As an application, we give an alternative approach to Virelizier’s version of the Fundamental Theorem for Hopf algebras. We introduce Yetter-Drinfeld modules over ...

Within the context of an involutive monoidal category the notion of a comparison relation cp : X ⊗X → Ω is identified. Instances are equality = on sets, inequality ≤ on posets, orthogonality ⊥ on orthomodular lattices, non-empty intersection on powersets, and inner product 〈− |−〉 on vector or Hilbert spaces. Associated with a collection of such (symmetric) comparison relations a dagger category...

Abstract We introduce a general definition for coloured cyclic operads over symmetric monoidal ground category, which has several appealing features. The forgetful functor from to both adjoints, each of is relatively simple. Explicit formulae these adjoints allow us lift the Cisinski–Moerdijk model structure on category enriched in simplicial sets sets.

A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid Q. We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and ‘order-cocomplete’. In fact, tensors and cotensors in a Q-category determine, and are determin...

The monoidal category of Soergel bimodules categorifies the Hecke algebra of a finite Weyl group. In the case of the symmetric group, morphisms in this category can be drawn as graphs in the plane. We define a quotient category, also given in terms of planar graphs, which categorifies the Temperley-Lieb algebra. Certain ideals appearing in this quotient are related both to the 1skeleton of the ...

The advent of fast-casual Mexican-style dining establishments, such as Chipotle and Qdoba, has greatly improved the productivity of research mathematicians and theoretical computer scientists in recent years. Still, many experience confusion upon encountering burritos for the first time. Numerous burrito tutorials (of varying quality) are to be found on the Internet. Some describe a burrito as ...

A new, self-contained, proof of a coherence result for categories equipped with two symmetric monoidal structures bridged by a natural transformation is given. It is shown that this coherence result is sufficient for ω×ω-indexed family of iterated reduced bar constructions based on such a category. Mathematics Subject Classification (2000): 18D10, 55P47

Recently we have reformulated the octonions as quasissociative algebras (quasialgebras) living in a symmetric monoidal category. In this note we provide further examples of quasialgebras, namely ones where the nonassociativity is induced by a Z Z n-grading and a nontrivial 3-cocycle.