agdacomonad

Agda Store Comonad

I'm just starting with Agda but know some Haskell and would like to know how to define the Store Comonad in Agda.

This is what I have until now:

   open import Category.Comonad
   open import Data.Product

   Store : Set → Set → ((Set → Set) × Set)
   Store s a = ((λ s → a) , s)

   StoreComonad : RawComonad (λ s a → (Store s a))
   StoreComonad = record
     { extract (Store s a) = extract s a
     ; extend f (Store s a = Store (extend (λ s' a' →­ f (Store s' a')) s) a
     } where open RawComonad

For now I'm getting the following error:

Parse error
=<ERROR>
 extract s a
  ; extend f (Sto...

I'm not too sure what it is I'm doing wrong. Any help would be appreciated! Thanks!

EDIT

I think I'm getting closer. I replaced the fields in the record using matching lambdas:

Store : Set → Set → ((Set → Set) × Set)
Store s a = ((λ s → a) , s)

StoreComonad : RawComonad (λ s a → (Store s a))
StoreComonad = record
  { extract   = λ st → (proj₁ st) (proj₂ st)
  ; duplicate = λ st → Store (λ s → Store (proj₁ st) s) (proj₂ st)
  ; extend    = λ g st → g (duplicate st)
  } where open RawComonad

RawComonad is from https://github.com/agda/agda-stdlib/blob/master/src/Category/Comonad.agda and has the signature

record RawComonad (W : Set f → Set f) : Set (suc f)

and Store is based on type Store s a = (s -> a, s) in Haskell.

Now the error I'm getting is:

(a : Set) → Σ (Set → Set) (λ x → Set) !=< Set
when checking that the expression λ a → Store s a has type Set

I'm wondering if this error has to do with this line:

StoreComonad : RawComonad (λ s a → (Store s a))

I find that the compilation error messages in Agda don't give many clues or I haven't yet been able to understand them well.

Solution

  • Your problem is that λ s a → (Store s a) (or, eta-contracted, Store) is not a comonad; its type (or, for your Haskell intuition, we could say its kind) is not right.

    However, for any choice of s, Store s is! So let's write that:

    StoreComonad : ∀ {s : Set} → RawComonad (Store s)

    The rest of the definition of StoreComonad will follow easily.

    As an aside, you can make the definition of StoreComonad nicer by using pattern-matching lambdas instead of using explicit projections (and please do read that link because it seems you have mixed up normal lambdas with pattern-matching ones); e.g:

      extract   = λ { (f , a) → f a }

    and so on.