In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 1-1 correspondence between commutative monads and symmetric monoidal monads (Theorem 2.3 below). The main computational work needed consists in constructing an equivalenc...

The category of flows is not cartesian closed. We construct a closed symmetric monoidal structure which has moreover a satisfactory behavior from the computer scientific viewpoint.

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...

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...

A relevant category is a symmetric monoidal closed category with a diagonal natural transformation that satisfies some coherence conditions. Every cartesian closed category is a relevant category in this sense. The denomination relevant comes from the connection with relevant logic. It is shown that the category of sets with partial functions, which is isomorphic to the category of pointed sets...

We introduce a notion of category with feedback-withdelay, closely related to the notion of traced monoidal category, and show that the Circ construction of [15] is the free category with feedback on a symmetric monoidal category. Combining with the Int construction of Joyal et al. [12] we obtain a description of the free compact closed category on a symmetric monoidal category. We thus obtain ...

We give a new formulation of attribute grammars (AG for short) called monoidal AGs in traced symmetric monoidal categories. Monoidal AGs subsume existing domain-theoretic, graph-theoretic and relational formulations of AGs. Using a 2-categorical aspect of monoidal AGs, we also show that every monoidal AG is equivalent to a synthesised one when the underlying category is closed, and that there i...

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...

We introduce two-dimensional linear algebra, by which we do not mean two-dimensional vector spaces but rather the systematic replacement in linear algebra of sets by categories. This entails the study of categories that are simultaneously categories of algebras for a monad and categories of coalgebras for comonad on a category such as SymMons, the category of small symmetric monoidal categories...