In the list monad, your map $TT \rightarrow T$ takes a list of lists and concatenates them to form a list. There is another way to take a list of lists and create a list, which is to shuffle randomly the two lists together. You can also imagine taking a list of lists and mapping that to a set of every shuffle. I do not think you can create a Monad with this map because there is a number of ways to shuffle two lists. I guess I want to know if there is a shuffle monad, or how would you model this with functors and natural transformations? Interestingly, people doing functional programming have done something like this
$\begingroup$
$\endgroup$
1
-
1$\begingroup$ There exists an operad closely related to the shuffle product, usually called the Zinbiel operad. $\endgroup$– F. C.Commented Jul 29, 2016 at 18:12
Add a comment
|
2 Answers
$\begingroup$
$\endgroup$
The haskell monad you mention models the process of drawing from a sequence at random. It is a state monad--with underlying map of objects $X \mapsto (S\times X)^S$--where the state includes the remaining list of elements and a 'source of randomness'. Every object $S$ of a Cartesian closed category induces a state monad on that category.
answered Jul 30, 2016 at 7:16
$\begingroup$
$\endgroup$