Bertram Felgenhauer wrote:
This is a rather finely grained hierarchy, but it's still too coarse to define a category of invertible functions. That category has pairs: it supports first, second and (***), but not fst and snd.
You are right, That would suggest class Category cat => CategoryPair cat where swap :: cat (a,b) (b,a) first :: cat a b -> cat (a,c) (b,c) second :: cat a b -> cat (c,a) (c,b) (***) :: cat a b -> cat c d -> cat (a,c) (b,d) class CategoryPair cat => CategorySelect cat where fst :: cat (a,b) a snd :: cat (a,b) b And dually class Category cat => CategoryChoice cat where mirror :: cat (Either a b) (Either b a) left :: cat a b -> cat (Either a c) (Either b c) right :: cat a b -> cat (Either c a) (Either c b) (+++) :: cat a b -> cat c d -> cat (Either a c) (Either b d) class CategoryChoice cat => CategoryInject cat where inl :: cat a (Either a b) inr :: cat b (Either a b) Twan