I am interested in trying to use category to perform some mathematical computations. It seems natural to want to talk about categories of mathematical objects e.g. Groups,Rings,Algebras ... etc. But I have found some difficulty doing this with Haskell's standard Category libraries. class Category m where id :: m a a comp :: m a b -> m b c -> m a c The main issue is that the objects of the category (represented by the id morphism) are in bijection to some set of haskell types. The problem (I think) this leads too is that 1. All the objects that may appear in a program must be known to the compiler. Which leads to problems (for me at least) as the application I want to study I don't know all the (Groups,Rings,... etc) that I will need. 2. This requires representing (for example) all groups as types. In general I have found while possible it leads to pretty horrendous types. Now this all makes sense Haskell's categories are for reasoning about programs while I want to use it more for pure maths. Has anyone else had a similar problem with categories in haskell? Or am I missing a way of implementing such structures within the standard Category framework for haskell. For a (toy) example take trying to model the category of CyclicGroups --Cyclic n is the object representing the cyclic group Z_n newtype Cyclic = Cyclic Int --CyclicMor n m p is represents a group morphism from Z_n to Z_m taking 1 -> p newtype CyclicMor = CyclicMor Int Int Int This now gives the definition of id :: Cyclic -> CyclicMor id (Cyclic n) = CyclicMor n n 1 comp :: CyclicMor -> CyclicMor -> Maybe CyclicMor comp (CyclicMor n m p) (CyclicMor n' m' p') | m==n' = Just (CyclicMor n m' (p+p')) | otherwise = Nothing The main loss in this approach is compiler can no longer determine if morphisms are compatible. In my project I use a slightly more generic variant : class OrdinaryCategory m where id :: a -> m a a comp :: m a b -> m b c -> Maybe (m a c) I haven't seen a similar construction in haskell libraries does anyone know if this is because 1. It is already there and I just haven't come across it. 2. It solves a problems that no-one else has had. 3. My idea is fundamentally broken / useless in some key way. Thanks for any comments anyone has. Robert