Lyubashenko has described enriched 2–categories as categories enriched over V–Cat, the 2–category of categories enriched over a symmetric monoidal V. Here I generalize this to a k–fold monoidal V. The latter is defined as by Balteanu, Fiedorowicz, Schwänzl and Vogt but with the addition of making visible the coherent associators α. The symmetric case can easily be recovered. The introduction of...

One of the recent advances in Functional Analysis has been the introduction of the notion of an (abstract) operator space. Proponents of this theory have, since its inception, maintained that it provides a much more solid " foundations " for Operator Algebra than the more traditional theory of Banach spaces. In this paper, we show that there exists a (non-trivial) involutive monoidal structure ...

The category of open games, which provides a strongly compositional foundation of economic game theory, is intermediate between symmetric monoidal and compact closed. More precisely it has counits with no corresponding units, and a partially defined duality. There exist open games with the same types as unit maps, given by agents with the strategic goal of predicting a future value. Such agents...

In our paper we follow the development of our approach of regarding programming as logical reasoning in intuitionistic linear logic. We present basic notions of linear logic and its deduction system and we define categorical semantics of linear logic as a symmetric monoidal closed category. Then we construct linear type theory over linear Church’s types involving linear calculus with equational...

By definition, a bialgebra H in a braided monoidal category (C, τ) is an algebra and coalgebra whose multiplication and comultiplication (and unit and counit) are compatible; the compatibility condition involves the braiding τ . The present paper is based upon the following simple observation: If H is a Hopf algebra, that is, if an antipode exists, then the compatibility condition of a bialgebr...

The category of Hopf monoids over an arbitrary symmetric monoidal category as well as its subcategories of commutative and cocommutative objects respectively are studied, where attention is paid in particular to the following questions: (a) When are the canonical forgetful functors of these categories into the categories of monoids and comonoids respectively part of an adjunction? (b) When are ...

After recalling the definition of a bicoalgebroid, we define comodules and modules over a bicoalgebroid. We construct the monoidal category of comodules, and define Yetter–Drinfel’d modules over a bicoalgebroid. It is proved that the Yetter–Drinfel’d category is monoidal and pre–braided just as in the case of bialgebroids, and is embedded into the one–sided center of the comodule category. We p...

We axiomatically define (pre-)Hilbert categories. The axioms resemble those for monoidal Abelian categories with the addition of an involutive functor. We then prove embedding theorems: any locally small pre-Hilbert category whose monoidal unit is a simple generator embeds (weakly) monoidally into the category of pre-Hilbert spaces and adjointable maps, preserving adjoint morphisms and all fini...

Recent work on gravity in two dimensions has a natural generalization to four dimensions. 1. Basic definitions 1.1 The (symmetric monoidal) two-category (Gravity)d+1 has objects: compact oriented d-manifolds, with • morphisms V0 → V1 : (d+ 1)-manifolds W with ∂W ∼= V op 0 t V1, and • diffeomorphisms W̃ →W as two-morphisms. The category Mor(V0, V1) with cobordisms from V0 to V1 as objects and dif...