category-theory

来源： https://stackoverflow.com/questions/59869399/why-does-mutual-yielding-make-arrowapply-and-monads-equivalent-unlike-arrow-and

来源： https://stackoverflow.com/questions/59869399/why-does-mutual-yielding-make-arrowapply-and-monads-equivalent-unlike-arrow-and

来源： https://stackoverflow.com/questions/64380092/what-type-corresponds-to-a-xor-b-in-type-theory

来源： https://stackoverflow.com/questions/3273373/are-all-haskell-functors-endofunctors

来源： https://stackoverflow.com/questions/62949200/why-are-monoidal-and-applicative-laws-telling-us-the-same-thing

问题 I tried writing down joinArr :: ??? a => a r (a r b) -> a r b . I came up with a solution which uses app , therefore narrowing the a down to ArrowApply 's: joinArr :: ArrowApply a => a r (a r b) -> a r b joinArr g = g &&& Control.Category.id >>> app Is it possible to have this function written down for arrows? My guess is no. Control.Monad.join could have been a good stand-in for >>= in the definition of the Monad type class: m >>= k = join $ k <$> m . Having joinArr :: Arrow a => a r (a r b)

问题 It is widely known that Applicative generalises Arrows . In the Idioms are oblivious, arrows are meticulous, monads are promiscuous paper by Sam Lindley, Philip Wadler and Jeremy Yallop it is said that Applicative is equivalent to Static Arrows, that is arrows for which the following isomorphism holds: arr a b :<->: arr () (a -> b) As far as I can understand, it could be illustrated the following way: Note: newtype Identity a = Id { runId :: a } . Klesli Identity is a Static Arrow as it wraps

问题 The category of sets is both cartesian monoidal and cocartesian monoidal. The types of the canonical isomorphisms witnessing these two monoidal structures are listed below: type x + y = Either x y type x × y = (x, y) data Iso a b = Iso { fwd :: a -> b, bwd :: b -> a } eassoc :: Iso ((x + y) + z) (x + (y + z)) elunit :: Iso (Void + x) x erunit :: Iso (x + Void) x tassoc :: Iso ((x × y) × z) (x × (y × z)) tlunit :: Iso (() × x) x trunit :: Iso (x × ()) x For the purposes of this question I

问题 The category of sets is both cartesian monoidal and cocartesian monoidal. The types of the canonical isomorphisms witnessing these two monoidal structures are listed below: type x + y = Either x y type x × y = (x, y) data Iso a b = Iso { fwd :: a -> b, bwd :: b -> a } eassoc :: Iso ((x + y) + z) (x + (y + z)) elunit :: Iso (Void + x) x erunit :: Iso (x + Void) x tassoc :: Iso ((x × y) × z) (x × (y × z)) tlunit :: Iso (() × x) x trunit :: Iso (x × ()) x For the purposes of this question I

问题 The category of sets is both cartesian monoidal and cocartesian monoidal. The types of the canonical isomorphisms witnessing these two monoidal structures are listed below: type x + y = Either x y type x × y = (x, y) data Iso a b = Iso { fwd :: a -> b, bwd :: b -> a } eassoc :: Iso ((x + y) + z) (x + (y + z)) elunit :: Iso (Void + x) x erunit :: Iso (x + Void) x tassoc :: Iso ((x × y) × z) (x × (y × z)) tlunit :: Iso (() × x) x trunit :: Iso (x × ()) x For the purposes of this question I