Two Hopf algebras are called monoidally Morita equivalent if module categories over them are equivalent as linear monoidal categories. We introduce monoidal Morita invariants for finite-dimensional Hopf algebras based on certain braid group representations arising from the Drinfeld double construction. As an application, we show, for any integer n, the number of elements of order n is a monoida...

We prove that the monoidal 2-category of cospans of ordinals and surjections is the universal monoidal category with an object X with a semigroup and a cosemigroup structures, where the two structures satisfy a certain 2-dimensional separable algebra condition.

Guided by the idea of letters with finite lists of predecessors and successors, the paper develops the Chomsky-Schützenberger theorem for languages of arrows in any free prop with a finite set of generators on the positive integers. The setting of monoidal categories is essential to obtain a well-behaved generalization of rational set, from monoids to monoidal categories.

We develop the Tannaka-Krein duality for monoidal functors with target in the categories of bimodules over a ring. The Coend of such a functor turns out to be a Hopf algebroid over this ring. Using a result of [4] we characterize a small abelian, locally finite rigid monoidal category as the category of rigid comodules over a transitive Hopf algebroid.

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice.

We show that the monoidal product on the stable homotopy category of spectra is essentially unique. This strengthens work of this author with Schwede on the uniqueness of models of the stable homotopy theory of spectra. Also, the equivalences constructed here give a unified construction of the known equivalences of the various symmetric monoidal categories of spectra (S-modules, W -spaces, orth...

This article investigates cutting planes for mixed-integer disjunctive programs. In the early 1980s, Balas and Jeroslow presented monoidal disjunctive cuts exploiting the integrality of variables. For disjunctions arising from binary variables, it is known that these cutting planes are essentially the same as Gomory mixed-integer and mixed-integer rounding cuts. In this article, we investigate ...

We will construct a monoidal functor (”a monoidal representation”) from the category of framed tangles into the tensor category over a fixed ground vector space which is invariant under Kirby moves and so gives rise to an invariant of 3-manifolds.