Differences

This shows you the differences between two versions of the page.

Link to this comparison view

Both sides previous revision Previous revision
monad_._haskell [2014/09/13 18:11]
nikolaj
monad_._haskell [2015/01/30 12:51] (current)
nikolaj
Line 92: Line 92:
 Note that conversely, '<​*>'​ doesn'​t suffice to define a '>>​='​. Note that conversely, '<​*>'​ doesn'​t suffice to define a '>>​='​.
  
-I'm going to explain what this (<*>) does in mathematical terminology:​ The Yoneda lemma implies that there is an isomorphism from $FA$ to $\mathrm{nat}(\mathrm{Hom}(A,​B),​FB)$. Consider a the category with exponential objects and arrows corresponding to components of the natural transformations (such as is the case for Hask, where the above map is $at\mapsto g\mapsto T(g)(ta):​FA\to B^A\to FB$), then we can apply the functors arrow map to the latter and obtain a map $FA\to F\,B^A\to FFB)$ to. One could say that this is provides a way to "​apply"​ $tg:F\,B^A$ to $ta:FA$. Note, however, that $T\,B^A$ isn't necessarily a honest function space anymore, so that's abuse of terminology. (For example, '​Noting :: Maybe (Int->​Int)'​ isn't a function. Things still work out, however, because if you pass '​Noting',​ the outer arrow mapping in the described function takes care of it. In that case the '​tg'​ value doesn'​t even matter so this is hardly an "​evaluation"​ of '​Nothing'​.). Lastly, if the functor comes with a monadic '​join',​ then we can get rid of the second $F$ in $FFB$ in the type of the function, and then it's called '<​*>'​.+I'm going to explain what this (<*>) does in mathematical terminology:​ The Yoneda lemma implies that there is an isomorphism from $FA$ to $\mathrm{nat}(\mathrm{Hom}(A,​B),​FB)$. Consider a category with exponential objects and arrows corresponding to components of the natural transformations (such as is the case for Hask, where the above map is $at\mapsto g\mapsto T(g)(ta):​FA\to B^A\to FB$), then we can apply the functors arrow map to the latter and obtain a map $FA\to F\,B^A\to FFB)$ to. One could say that this is provides a way to "​apply"​ $tg:F\,B^A$ to $ta:FA$. Note, however, that $T\,B^A$ isn't necessarily a honest function space anymore, so that's abuse of terminology. (For example, '​Noting :: Maybe (Int->​Int)'​ isn't a function. Things still work out, however, because if you pass '​Noting',​ the outer arrow mapping in the described function takes care of it. In that case the '​tg'​ value doesn'​t even matter so this is hardly an "​evaluation"​ of '​Nothing'​.). Lastly, if the functor comes with a monadic '​join',​ then we can get rid of the second $F$ in $FFB$ in the type of the function, and then it's called '<​*>'​.
  
 == Alternative definitions == == Alternative definitions ==
Link to graph
Log In
Improvements of the human condition