Given a horizontal monoid M in a duoidal category F , we examine the relationship between bimonoid structures on M and monoidal structures on the category F ∗M of right M -modules which lift the vertical monoidal structure of F . We obtain our result using a variant of the so-called Tannaka adjunction; that is, an adjunction inducing the equivalence which expresses Tannaka duality. The approach...

We put a monoidal model category structure on the category of chain complexes of quasi-coherent sheaves over a quasi-compact and semiseparated scheme X. The approach generalizes and simplifies the method used by the author in [Gil04] and [Gil06] to build monoidal model structures on the category of chain complexes of modules over a ring and chain complexes of sheaves over a ringed space. Indeed...

At the heart of the axiomatic formulation of 1+1-dimensional topological field theory is the set of all surfaces with boundary assembled into a category. This category of surfaces has compact 1-manifolds as objects and smooth oriented cobordisms as morphisms. Taking disjoint unions gives a monoidal structure and a 1+1-dimensional topological field theory can be defined to be a monoidal functor ...

It is well known that the existence of a braiding in a monoidal category V allows many structures to be built upon that foundation. These include a monoidal 2-category V-Cat of enriched categories and functors over V , a monoidal bicategory V-Mod of enriched categories and modules, a category of operads in V and a 2-fold monoidal category structure on V . We will begin by focusing our expositio...

Symmetric monoidal closed categories may be related to one another not only by the functors between them but also by enrichment of one in another, and it was known to G. M. Kelly in the 1960s that there is a very close connection between these phenomena. In this first part of a two-part series on this subject, we show that the assignment to each symmetric monoidal closed category V its associat...

Given an algebraic theory which can be described by a (possibly symmetric) operad P , we propose a definition of the weakening (or categorification) of the theory, in which equations that hold strictly for P -algebras hold only up to coherent isomorphism. This generalizes the theories of monoidal categories and symmetric monoidal categories, and several related notions defined in the literature...

monoid can be defined. Monoidal categories are ubiquitous: cartesian categories (in particular, the category of sets), free word-category over any category, endofunctors category over any category, category of R-modules over a commutative ring R, etc. The abstract monoids definable in the monoidal category incarnate in the usual monoids, triples (or monads), R-algebras, etc. 2 Given a monoidal ...

We show that a left-rigid monoidal C-category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius--Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S4-invariance of 6j-symbols in such a category.

A monoidal category is a category equipped with extra data, describing how objects and morphisms can be combined ‘in parallel’. This chapter introduces the theory of monoidal categories, and shows how our example categories Hilb, Set and Rel can be given a monoidal structure. We also introduce a visual notation called the graphical calculus, which provides an intuitive and powerful way to work ...