PS

Writer monad

圏論 Monad Haskell

Monoid lifting

Monoidal functors send monoids to monoids:

Monoidal functor:

$(F, \phi, u) : (\mathcal{C}, \otimes, I) \to (\mathcal{C'}, \otimes', I')$

について、functor:

$F _ \ast : \mathcal{Mon} _ \mathcal{C} \to \mathcal{Mon} _ \mathcal{C'}$ *1
$F _ \ast( m, \eta, \mu ) := (Fm, I' \overset{u}{\to} FI \overset{F \eta}{\to} Fm, Fm \otimes' Fm \overset{\phi _ {m, m}}{\to} F(m \otimes m) \overset{F \mu}{\to} Fm)$
$F _ \ast(f) = F(f)$

を定義できる。

Monads from monoids

特に、 $F$ のcodomainがmonoidal category of endofunctors:

$(\mathcal{C'}, \otimes', I') := (\mathcal{C} ^ \mathcal{C}, \circ, 1 _ \mathcal{C})$

のとき

$F _ \ast : \mathcal{Mon} _ \mathcal{C} \to \mathcal{Mon} _ { \mathcal{C} ^ \mathcal{C} } = \mathcal{Mnd}(\mathcal{C})$ *2

Writer monad

$F(m) := (\unicode{x2013}) \times m$
$(\phi _ {a, b}) _ c : ( c \times b) \times a \cong c \times (a \times b)$
$u _ a : a \cong a \times \lbrace \ast \rbrace$

とすると、monoidal functor:

$(F, \phi, u) : ( \mathcal{Hask}, \times, \lbrace \ast \rbrace) \to ( \mathcal{Hask} ^ \mathcal{Hask}, \circ, 1 _ \mathcal{Hask} )$

になるので、あるmonoid:

$(m, \mathtt{mempty}, \mathtt{mappend})$

が存在すれば

$F _ \ast (m, \mathtt{mempty}, \mathtt{mappend}) \in \mathcal{Mnd}(\mathcal{Hask})$

具体的には

$(\unicode{x2013}) \times m$
$a \mapsto (a, \mathtt{mempty})$
$( (a, w'), w) \mapsto (a, \mathtt{mappend}(w, w') )$

はmonadを成す。

参考文献

*1:category of monoid objects

*2:category of monads