Comonadic Notions of Computation

A Notation for Comonads

“…for a comonad c. Uustalu and Vene gave the semantics of a context-dependent calculus in this form [8].…”

“…Monoidal operation The czip :: (c a, c b) → c (a, b) operation introduced in Section 1 corresponds to that of a (semi)-monoidal functor which may satisfy various laws with respect to the comonad (see the discussion of (semi)-monoidal comonads in [8]). The following property, which we call idempotency of a semimonoidal functor, frequently holds of comonad/czip implementations:…”

“…The desugaring of codo is based on Uustalu and Vene's semantics for a contextdependent λ-calculus [8]. It has two parts: translation of statements into composition via extend , and management of the environment for variables bound in a codo-block.…”

“…For labelled graphs, the costate comonad is combined with product comonad (see [8]) pairing the label of a node with the list of its successors, thus the type is isomorphic to C a = (s → (a ×…”

Monads Need Not Be Endofunctors

“…Apart from their use to model a generic notion of effect they also serve as a convenient interface to generalized notions of substitution. Research in the area on the border between category theory and functional programming focusses on unveiling new programming and reasoning constructions similar to monads, such as comonads [36], arrows [20] and idioms (closed functors) [25]. Indeed, especially when working in an expressive and total language with dependent types, such as Agda [27], we can exploit monads as a way to structure not only our programs but also their verification.…”