Universal algebra is often known within computer science in the guise of algebraic specification or equational logic. In 1963, it was given a category theoretic characterisation in terms of what are now called Lawvere theories. Unlike operations and equations, a Lawvere theory is uniquely determined by its category of models. Except for a caveat about nullary operations, the notion of Lawvere t...

Dynamic evaluation is a paradigm in computer algebra which was introduced for computing with algebraic numbers. In linear algebra, for instance, dynamic evaluation can be used to apply programs which have been written for matrices with coefficients modulo some prime number to matrices with coefficients modulo some composite number. A way to implement dynamic evaluation in modern computing langu...

Girard worked with the category of coherence spaces and continuous stable maps and observed that the functor that forgets the linearity of linear stable maps has a left adjoint. This fundamental observation gave rise to the discovery of Linear Logic. Since then, the category of coherence spaces and linear stable maps, with the comonad induced by the adjunction, has been considered a canonical m...

Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical models of DLL are now commune, and the simplest one is Rel, the category of sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic coun...

Lenses are a heavily studied form of bidirectional transformation, with diverse applications including database view updating, software development and memory management. Previous work has explored lenses category-theoretically, and established that the category of lenses for a fixed ‘view’ V is, up to isomorphism, the category of algebras for a particular monad on set/V . It has recently been ...

Monads are a de facto standard for the type-based analysis of impure aspects of programs, such as runtime cost [9, 5]. Recently, the logical dual of a monad, the comonad, has also been used for the cost analysis of programs, in conjunction with a linear type system [6, 8]. The logical duality of monads and comonads extends to cost analysis: In monadic type systems, costs are (side) effects, whe...

In Martin-Löf type theory, simple inductive types—W-types—are characterized categorically as initial algebras of a polynomial functor. Dually, coinductive types are characterized as terminal coalgebras of polynomial functors. In the case of coinductive types, the meta-theoretic notion of equality given by Martin-Löf’s identity type is too weak: instead, the idea of bisimilarity as equality for ...

Since its discovery, differential linear logic (DLL) inspired numerous domains. In denotational semantics, categorical models of DLL are now commune, and the simplest one is Rel, the category of sets and relations. In proof theory this naturally gave birth to differential proof nets that are full and complete for DLL. In turn, these tools can naturally be translated to their intuitionistic coun...

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a datatypegeneric representation theorem. More precisely, we prove a representation theorem for a wide class of second-order functionals which are polymorphic over a class of functors. Types polymorphic over a class o...

In this paper, we look at two categorical accounts of computational effects (strong monad as a model of the monadic metalanguage, adjunction as a model of call-bypush-value with stacks), and we adapt them to incorporate global exceptions. In each case, we extend the calculus with a construct, due to Benton and Kennedy, that fuses exception handling with sequencing. This immediately gives us an ...