haskell - Typeclass for (what seems to be) a contravariant functor implementing function inversion -

lets have following

import control.category (category, (.), id)  data invertible b = invertible (a -> b) (b -> a)  instance category invertible   id = invertible prelude.id prelude.id   (invertible f f') . (invertible g g') =      invertible (f prelude.. g) (g' prelude.. f')  invert (invertible x y) = invertible y x 

note following true:

invert (g . f) == invert f . invert g 

this structure seems similar contravariant functor (wikipedia), follows same axiom:

 f(g . f) = f(f) . f(g)  

in case, f invert.

i looked @ data.functor.contravariant.contramap, has function of type:

(a -> b) -> f b -> f 

but didn't know how'd i'd implement in situation. example, can't work out sensible choice f, , in situation, there's no function a -> b, invert.

however, invert nevertheless fits mathematical axiom of contravariant functor, i'm thinking can fit existing class, can't find 1 , how it. or pointers appreciated.

a category has 2 collections: objects , morphisms.

the usual haskell prelude, , appears classes in data.functor.contravariant, operate on narrow category, category types objects , functions morphisms, denoted hask. standard functor class narrow: represent endofunctors on hask: must take types types , functions functions.

take example functor maybe. way maybe acts on types takes types a maybe a. maybe maps int maybe int , on (i know sounds bit trivial). morphisms encoded fmap: fmap takes f :: (a -> b), morphism between 2 objects in hask, , maps fmap f :: (maybe -> maybe b), morphism in hask between objects functor maps to. in haskell not define functor takes e.g. int char -- haskell functors have type constructors -- in general category theory could.

control.category generalizes little bit: objects of control.category category c still types[1] in hask, morphisms things of type c b. in example, objects still arbitrary types, morphisms things of type invertible b. since morphisms not functions, not able use standard functor classes.

however, it's fun exercise in building category theory knowhow define functor class operates between category categories rather assuming hask, capture example. remember, functor acts on objects (types) and morphisms.

i'll leave -- feel free comment if more guidance.

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[1] ignoring polykinds, makes bit more general.