The category theoretic structures of monads and comonads can be used as an abstraction mechanism for simplifying both language semantics and programs. Monads have been used to structure impure computations, whilst comonads have been used to structure context-dependent computations. Interestingly, the class of computations structured by monads and the class of computations structured by comonads...

Abstract Combinatorial games are widely used in finite model theory, constraint satisfaction, modal logic and concurrency theory to characterize logical equivalences between structures. In particular, Ehrenfeucht–Fraïssé games, pebble bisimulation play a central role. We show how each of these types can be described terms an indexed family comonads on the category relational structures homomorp...

We study comodule functors for comonads arising from mixed distributive laws. Their Galois property is reformulated in terms of a (so-called) regular arrow in Street’s bicategory of comonads. Between categories possessing equalizers, we introduce the notion of a regular adjunction. An equivalence is proven between the category of pre-torsors over two regular adjunctions (NA, RA) and (NB, RB) on...

Game comonads have brought forth a new approach to studying finite model theory categorically. By representing comparison games semantically as comonads, they allow important logical and combinatorial properties be exressed in terms of their Eilenberg-Moore coalgebras. As result, number results from theory, such preservation theorems homomorphism counting theorems, been formalised parameterised...

We instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F . Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formul...

We give a coalgebraic formulation of timed processes and their operational semantics. We model time by a monoid called a “time domain”, and we model processes by “timed transition systems”, which amount to partial monoid actions of the time domain or, equivalently, coalgebras for an “evolution comonad” generated by the time domain. All our examples of time domains satisfy a partial closure prop...

We study the notions of relative comonad and comodule over a relative comonad. We use thesenotions to give categorical semantics for the coinductive type families of streams and of infinitetriangular matrices and their respective cosubstitution operations in intensional Martin-Löf typetheory. Our results are mechanized in the proof assistant Coq. 1998 ACM Subject Classification ...

In As94], a correspondence between Lamping-Gonthier's operators for Optimal Reduction of the-calculus Lam90, GAL92a] and the operations associated with the comonad \!" of Linear Logic was established. In this paper, we put this analogy at work, adding new rewriting rules directly suggested by the categorical equations of the comonad. These rules produce an impressive improvement of the performa...