Motivated by algebraic structures appearing in Rational Conformal Field Theory we study a construction associating to an algebra in a monoidal category a commutative algebra (full centre) in the monoidal centre of the monoidal category. We establish Morita invariance of this construction by extending it to module categories. As an example we treat the case of group-theoretical categories.

We study categorical models for the unitless fragment of multiplicative linear logic. We find that the appropriate notion of model is a special kind of pro-monoidal category. Since the theory of promonoidal categories has not been developed very thoroughly, at least in the published literature, we need to develop it here. The most natural way to do this – and the simplest, once the (substantial...

In this paper we explain the relationship between Frobenius objects in monoidal categories and adjunctions in 2-categories. Specifically, we show that every Frobenius object in a monoidal category M arises from an ambijunction (simultaneous left and right adjoints) in some 2-category D into which M fully and faithfully embeds. Since a 2D topological quantum field theory is equivalent to a commu...

Kornel Szlachányi [28] recently used the term skew-monoidal category for a particular laxi ed version of monoidal category. He showed that bialgebroids H with base ring R could be characterized in terms of skew-monoidal structures on the category of one-sided R-modules for which the lax unit was R itself. We de ne skew monoidales (or skew pseudo-monoids) in any monoidal bicategory M . These are...

It is shown that for every monoidal bi-closed category C left and right (semi)dualization by means of the unit object not only defines a pair of adjoint functors, but that these functors are monoidal as functors from C, the dual monoidal category of C into the transposed monoidal category C. We, thus, generalize the case of a symmetric monoidal category, where this kind of dualization is a spec...

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 study the monoidal structure of the standard strictification functor st : Bicat → 2Cat. In doing so, we construct monoidal structures on the 2-category whose objects are bicategories and on the 2-category whose objects are 2-categories.

If C is a stable model category with a monoidal product then the set of homotopy classes of self-maps of the unit forms a commutative ring, [S, S] . An idempotent e of this ring will split the homotopy category: [X,Y ] ∼= e[X,Y ]⊕(1−e)[X,Y ] . We prove that provided the localised model structures exist, this splitting of the homotopy category comes from a splitting of the model category, that i...

Automata over a symmetric monoidal category M are introduced, and a multi-step simulation is defined among such automata. The collection of M automata is given the structure of a 2-category on the same objects as M , in which the vertical structure is determined by groups of indistinguishable simulations. TwoM -automata are called simulation equivalent if they are connected by an isomorphism of...

This paper introduces axioms for an abstract trace on a monoidal category. This trace can be interpreted in various contexts where it could alternatively be called contraction, feedback, Markov trace or braid closure. Each full submonoidal category of a tortile (or ribbon) monoidal category admits a canonical trace. We prove the structure theorem that every traced monoidal category arises in th...