[英]Is list a monad and comonad?
The list monad is given here . 单子列表在这里给出 。 Also see Spivak's paper here .
另请参见Spivak的论文 。 So list is a monad.
所以清单是单子。 Is it a comonad?
是一个笨蛋吗? How would you prove that?
您将如何证明?
The list type constructor a ↦ μ L. 1 + a * L
doesn't admit a comonad structure. 列表类型构造函数a↦μL。1
a ↦ μ L. 1 + a * L
不允许comonad结构。 Recall that if it were a comonad, we'd have (using the names from Haskell's Functor
and Comonad
typeclasses) 回想一下,如果它是一个普通符号,我们将拥有(使用Haskell的
Functor
和Comonad
类型类的名称)
fmap :: ∀ a b. (a → b) → [a] → [b]
extract :: ∀ a. [a] → a
duplicate :: ∀ a. [a] → [[a]]
However, even without going into any required laws, extract
cannot be implemented, since its input could be the empty list, giving no way to come up with an a
. 但是,即使不遵循任何必需的法律,
extract
也无法实现,因为extract
的输入可能是空列表,无法给出a
。
The nonempty list type constructor a ↦ μ NE. a + a * NE
非空列表类型构造函数
a ↦ μ NE. a + a * NE
a ↦ μ NE. a + a * NE
does admit a comonad structure, with extract
returning the first element, and duplicate
mapping [x, y, ..., z]
to [[x], [x, y], ..., [x, y, ..., z]]
(note that each of them are non-empty by construction). a ↦ μ NE. a + a * NE
确实接受了共纳结构,其中extract
返回第一个元素,并将[x, y, ..., z]
duplicate
到[[x], [x, y], ..., [x, y, ..., z]]
(请注意,它们在构造上都是非空的)。
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