Читач, Письменницькі монади

Нехай $C$ - CCC . Нехай $(\times)$ буде біфунктор продукт на $C$ . Оскільки Кіт є CCC, ми можемо каррі $(\times)$ :

$curry (\times) : C \rightarrow(C \Rightarrow C)$

$curry (\times) A = \lambda B. A \times B$

Функціональна категорія $C \Rightarrow C$ має звичну моноїдну структуру. Моноїд в $C \Rightarrow C$ є монадой в $C$ . Розглядаються кінцеві продукти , як моноідальние структури на $C$ .

$curry (\times) 1 \cong id$

$\forall A\ B. curry (\times) (A\times B) \cong (curry (\times) A) \circ (curry (\times) B)$

Тому $(curry (\times))$ зберігає моноїдну структуру, тому він транспортує моноїд до монади, а комоноїд - до комонади. А саме, він транспортує довільну моноїду $w$ до $(Writer\ w)$ монади (подивіться на визначення - $w$ повинен бути моноїдом). Аналогічно він транспортує діагональний комоноїд до комонади Кореадера.

Тепер, для конкретності, я розгортаю конструкцію Writer.

Почніть. Насправді , вони просто мають чіткі імена в Haskell. Ми маємо Haskell моноїд ⟨ ш , м а р р е н д , м е т р т у ⟩ :$Writer = Coreader = curry(\times)$ $\langle w, mappend, mempty\rangle$

$mappend : w\times w \to w$

$mempty : 1 \to w$

Writer is a functor, so it must map also morphisms, such as $mappend$ and $mempty$. I write this as below, though it is invalid in Haskell:

$Writer\ mappend : Writer(w\times w) \to Writer\ w$

$Writer\ mappend$ is a natural transformation, a morphism in $C\Rightarrow C$. By properties of $curry(\times)$ it is a function, which takes $a \in Ob(C)$ and gives a morphism in $C$:

$Writer\ mappend\ a = mappend\times(id(a)) : Writer (w\times w) a \to Writer\ w\ a$

Неформально, сум компоненти типу ш і насоси недоторканими. Це саме визначення Writer в Haskell. Одним з перешкод є те , що для монади ⟨ W г я т е р ш , ц , п ⟩ Нам потрібно$Writer\ mappend\ a$$w$$a$$\langle Writer\ w,\mu,\eta\rangle$

$\mu : Writer\ w \circ Writer\ w \to Writer\ w$

i.e. incompatibility of types. But these functors are isomorphic: $Writer (w\times w) = \lambda a. (w\times w)\times a$ by the usual associator for finite products which is a natural isomorphism $\cong \lambda a. w\times(w\times a) = Writer\ w \circ Writer\ w$. Then we define $\mu$ via $Writer\ mappend$. I omit a construction of $\eta$ via $mempty$.

Writer, being a functor, preserves commutative diagrams, i.e. preserves monoid equalities, so we have for granted proved equalities for $\langle Writer\ w,\mu,\eta\rangle$ = a monoid in $(C\Rightarrow C)$ = a monad in $C$. End.

What about Reader and Cowriter? Reader is adjoint to Coreader, as explained in the definition of Coreader, see link above. Similarly, Cowriter is adjoint to Writer. I did not find a definition of Cowriter, so I made it up by analogy shown in the table:

{- base, Hackage.category-extras -}
import Control.Comonad
import Data.Monoid
data Cowriter w a = Cowriter (w -> a)
instance Functor (Cowriter w) where
    fmap f (Cowriter g) = Cowriter (f . g)
instance Monoid w => Copointed (Cowriter w) where
    extract (Cowriter g) = g mempty
instance Monoid w => Comonad (Cowriter w) where
    duplicate (Cowriter g) = Cowriter
        (\w' -> Cowriter (\w -> g (w `mappend` w')))

Below are the simplified definitions of those (co)monads. fr_ob F denotes a mapping of a functor F on objects, fr_mor F denotes a mapping of a functor F on morphisms. There is a monoid object $\langle w,\hat{+},\hat{0}\rangle$ in $C$.

• Writer
  • $fr\_ob (Writer\ w) a = a\times w$
  • $fr\_mor (Writer\ w) f = \lambda \langle a_0,w_2\rangle. \langle a_0,f\ w_2\rangle$
  • $\eta a = \lambda a_0. \langle a_0, \hat{0} \rangle$
  • $\mu a = \lambda \langle \langle a_0,w_1\rangle,w_0\rangle. \langle a_0, w_0\hat{+}w_1\rangle$
• Reader
  • $fr\_ob (Reader\ r) a = r\to a$
  • $fr\_mor (Reader\ r) f = \lambda g\ r_0. f (g\ r_0)$
  • $\eta a = \lambda a_0\ r_0. a_0$
  • $\mu a = \lambda f\ r_0. f\ r_0\ r_0$
• Coreader
  • $fr\_ob (Coreader\ r) a = r\times a$
  • $fr\_mor (Coreader\ r) f = \lambda \langle r_0,a_0\rangle. \langle f\ r_0,a_0\rangle$
  • $\eta a = \lambda \langle r_0,a_0\rangle. a_0$
  • $\mu a = \lambda \langle r_0,a_0\rangle. \langle r_0,\langle r_0,a_0\rangle\rangle$
• Cowriter
  • $fr\_ob (Cowriter\ w) a = w\to a$
  • $fr\_mor (Cowriter\ w) f = \lambda g\ r_0. f (g\ r_0)$
  • $\eta a = \lambda f. f\ \hat{0}$
  • $\mu a = \lambda f\ w_1 w_0. f (w_0\hat{+}w_1)$

The question is that the adjunction in $C$ relates functors, not monads. I do not see how the adjunction implies "Coreader is a comonad" $\to$ "Reader is a monad" and "Writer is a monad" $\to$ "Cowriter is a comonad".

Remark. I am struggling to provide more context. It requires some work. Especially, if you require categorical purity and those (co)monads were introduced for programmers. Keep nagging! ;)

Yes, if a monad $M : C \to C$ has a right adjoint $N$, then $N$ automatically inherits a comonad structure.

The general category-theoretic setting to understand this is as follows. Let $C$ and $D$ be two categories. Write $\mathrm{Fun}(C, D)$ for the categeory of functors from $C$ to $D$; Its objects are functors and its morphisms natural transformations. Write $\mathrm{Fun}^L(C, D)$ for the full subcategory of $\mathrm{Fun}(C, D)$ on the functors which have right adjoints (in other words, we consider functors $C \to D$ with right adjoints and arbitrary natural transformations between them). Write $F^R : D \to C$ for the right adjoint of a functor $F : C \to D$. Then $-^R : \mathrm{Fun}^L(C, D) \to \mathrm{Fun}(D, C)$ is a contravariant functor: if $\alpha : F \to G$ is a natural transformation then there is an induced natural transformation $\alpha^R : G^R \to F^R$.

If $C = D$, then $\mathrm{Fun}(C, C)$ has a monoidal structure given by composition and so does $\mathrm{Fun}^L(C, D)$, because the composition of left adjoints is a left adjoint. Specifically, $(FG)^R = G^R F^R$, so $-^R$ is an antimonoidal contravariant functor. If you apply $-^R$ to the structural natural transformations which equip a functor $M$ with the structure of a monad, what you get out is a comonad.

By the way, this:

Let $(\times)$ be a product bifunctor in $C$. As $C$ is CCC, we can curry $(\times)$

is slightly incorrect. For one, the usual terminology would be (if I'm not mistaken) that $\times$ is a bifunctor over or on $C$. "In" typically means constructions using the arrows and objects of a category, whereas functors "on" categories refer to constructions relating multiple categories. And the product bifunctor isn't a construction within a Cartesian category.

And this relates to the larger inaccuracy: the ability to curry the product bifunctor has nothing to do with $C$ being Cartesian closed. Rather, it is possible because $Cat$, the category of categories (insert caveats) is Cartesian closed. So the currying in question is given by:

where $C \times D$ is a product of categories, and $E^D$ is the category of functors $F : D \to E$. This works regardless of whether $C$, $D$ and $E$ are Cartesian closed, though. When we let $C = D = E$, we get:

But this is merely a special case of:

Consider the adjunction $\langle F,G,\epsilon,\eta \rangle$. For every such adjunction we have a monad $\langle GF, \eta, G\epsilon F\rangle$ and also a comonad $\langle FG, \epsilon, F\eta G\rangle$. Notably, $F$ and $G$ need not be endofunctors, and in general they aren't (e.g., the list monad is an adjuction between the free and forgetful functors between $\mathbf{Set}$ and $\mathbf{Mon}$).

So, what you want to do is take Reader (or Writer) and decompose it into the adjoint functors which give rise to the monad and the corresponding comonad. Which is to say that the connection between Reader and Coreader (or Writer and Cowriter) isn't the one you're looking for.

And it's probably better to think of currying as $-^\ast : \text{hom}({-}\times A, {=}) \cong \text{hom}({-}, {=}^A)$, i.e. $\forall X,Y.~ \lbrace f : X\times A \to Y \rbrace \cong \lbrace f^\ast : X \to Y^A \rbrace$. Or if it helps, $-^\ast : \text{hom}({-}\times A, {=}\times 1) \cong \text{hom}({-}^1, {=}^A)$