We describe the categorical semantics for a simply typed variant and simplified dependently of Cocon , contextual modal type theory where box modality mediates between weak function space that is used to represent higher-order abstract syntax (HOAS) trees strong describes (recursive) computations about them. What makes different from standard theories presence first-class contexts objects are c...

The category-theoretic concept of a monad occurs widely as a design pattern for functional programming with effects. The utility and ubiquity of monads is such that some languages provide syntactic sugar for this pattern, further encouraging its use. We argue that comonads, the dual of monads, similarly provide a useful design pattern, capturing notions of context dependence. However, comonads ...

Abstract We exhibit the cartesian differential categories of Blute, Cockett and Seely as a particular kind enriched category. The base for enrichment is category commutative monoids—or in straightforward generalisation, modules over rig k . However, tensor product on this not usual one, but rather warping it by certain monoidal comonad Q Thus sense, skew sense Szlachányi. Our first main result ...

Linear logic is well known for its resource-awareness, which hasvinspired the design of several resource management mechanisms in programming language design. Its resource-awareness arises from the distinction between linear, single-use data and non-linear, reusable data. The latter is marked by the so-called exponential modality, which, from the categorical viewpoint, is a (monoidal) comonad. ...

The exception monad, while an adequate mechanism for providing the denotational semantics of exceptions, is somewhat awkward to program with. Just as any other monad, it forces a programming style in which exceptional computations are explicitly sequentialized in the program text. In addition, values of computation types must usually be tested before use, in order to determine if they correspon...

Reddy 13] introduced an extended intuitionistic linear calculus to model some features of state-manipulation. His calculus LLMS { for Linear Logic Model of State { includes the connective \before" and an its associated modality y. De Paiva 5] presents a (collection of) dialectica categorical models for Classical Linear Logic, the categories GC. These categories contain an extra tensor product f...

Whilst the relationship between initial algebras and monads is well understood, the relationship between final coalgebras and comonads is less well explored. This paper shows that the problem is more subtle than might appear at first glance: final coalgebras can form monads just as easily as comonads, and, dually, initial algebras form both monads and comonads. In developing these theories we s...

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...

Various typing system have been recently introduced giving a parametric version of the exponential modality of linear logic, e.g. [6, 2]. The parameters are taken from a semi-ring, and allow to express coeffects – i.e. specific requirements of a program with respect to the environment (availability of a resource, some prerequisite of the input, etc.). We show that all these systems can be inter...