The essential interaction between classical and intuitionistic features in the system of linear logic is best described in the language of category theory. Given a symmetric monoidal closed category with products, the category can be given the structure of a *-autonomous category by a special case of the Chu construction. The main result of the paper is to show that the intuitionistic translati...

Before one can attach a meaning to a sentence, one must distinguish different ways of parsing it. When analyzing a language with pregroup grammars, we are thus led to replace the free pregroup by a free compact strict monoidal category. Since a strict monoidal category is a 2-category with one 0-cell, we investigate the free compact 2-category generated by a given category, and we describe its ...

The Reshetikhin-Turaev invariant, Turaev’s TQFT, and many related constructions rely on the encoding of certain tangles (n-string links, or ribbon n-handles) as n-forms on the coend of a ribbon category. We introduce the monoidal category of Hopf diagrams, and describe a universal encoding of ribbon string links as Hopf diagrams. This universal encoding is an injective monoidal functor and admi...

The definition and calculus of extraordinary natural transformations [EK] is extended to a context internal to any autonomous monoidal bicategory [DyS]. The original calculus is recaptured from the geometry [SV], [MT] of the monoidal bicategory V-Mod whose objects are categories enriched in a cocomplete symmetric monoidal category V and whose morphisms are modules. Mathematics Subject Classific...

Motivated by a model for syntactic control of interference, we introduce a general categorical concept of bireeectivity. Bireeective subcategories of a category A are subcategories with left and right adjoint equal, subject to a coherence condition. We characterize them in terms of split-idempotent natural transformations on id A. In the special case that A is a presheaf category, we characteri...

We show that the nerve of a braided monoidal category carries a natural action of a simplicial E2-operad and is thus up to group completion a double loop space. Shifting up dimension twice associates to each braided monoidal category a 1-reduced lax 3-category whose nerve realizes an explicit double delooping whenever all cells are invertible. We deduce that lax 3-groupoids are algebraic models...

Given a categorical crossed module H→ G, where G is a group, we show that the category of derivations, Der(G,H), from G into H has a natural monoidal structure. We introduce the Whitehead categorical group of derivations as the Picard category of Der(G,H) and then we characterize the invertible derivations, with respect to the tensor product, in this monoidal category. 2000 Mathematics Subject ...

We prove that the monoidal 2-category of cospans of finite linear orders 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.

We consider the composition product of symmetric sequences in the case where the underlying symmetric monoidal structure does not commute with coproducts. Even though this composition product is not a monoidal structure on symmetric sequences, it has enough properties to be able to define monoids (which are then operads on the underlying category) and make a bar construction. The main benefit o...

Hopf algebras are closely related to monoidal categories. More precise, k-Hopf algebras can be characterized as those algebras whose category of finite dimensional representations is an autonomous monoidal category such that the forgetful functor to k-vectorspaces is a strict monoidal functor. This result is known as the Tannaka reconstruction theorem (for Hopf algebras). Because of the importa...