X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=d5c0941da9a74e539a9e4944ca686ed0473ae91e;hp=24670796b21e1fc3cd230352e0049772bf17a2b3;hb=115b2bc2456adb01295e074ccc825942e45c46c0;hpb=ac47fea0dc57a0f496d14dba80383206397a35f9;ds=sidebyside
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 24670796..d5c0941d 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -476,34 +476,126 @@ Collecting the results, our monad laws turn out in this format to be:
Getting to the functional programming presentation of the monad laws
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-In functional programming, unit is usually called "return" and the monad laws are usually stated in terms of return and an operation called "bind" which is interdefinable with <=< or with join.
-
-Additionally, whereas in category-theory one works "monomorphically", in functional programming one usually works with "polymorphic" functions.
+In functional programming, `unit` is sometimes called `return` and the monad laws are usually stated in terms of `unit`/`return` and an operation called `bind` which is interdefinable with `<=<` or with `join`.
The base category C will have types as elements, and monadic functions as its morphisms. The source and target of a morphism will be the types of its argument and its result. (As always, there can be multiple distinct morphisms from the same source to the same target.)
-A monad M will consist of a mapping from types C1 to types M(C1), and a mapping from functions f:C1→C2 to functions M(f):M(C1)→M(C2). This is also known as "fmap f" or "liftM f" for M, and is called "function f lifted into the monad M." For example, where M is the list monad, M maps every type X into the type "list of Xs", and maps every function f:x→y into the function that maps [x1,x2...] to [y1,y2,...].
+A monad `M` will consist of a mapping from types `'t` to types `M('t)`, and a mapping from functions f:C1→C2
to functions M(f):M(C1)→M(C2)
. This is also known as liftM f
for `M`, and is pronounced "function f lifted into the monad M." For example, where `M` is the list monad, `M` maps every type `'t` into the type `'t list`, and maps every function f:x→y
into the function that maps `[x1,x2...]` to `[y1,y2,...]`.
+
+In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.
+A "monadic value" is any member of a type M('t), for any type 't. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function (φ : F('t) → M(F'('t)))
to an argument `a` of type `F('t)`.
-A natural transformation t assigns to each type C1 in C a morphism t[C1]: C1→M(C1) such that, for every f:C1→C2:
- t[C2] ∘ f = M(f) ∘ t[C1]
+Let `'t` be a type variable, and `F` and `F'` be functors, and let `phi` be a polymorphic function that takes arguments of type `F('t)` and yields results of type `MF'('t)` in the monad `M`. An example with `M` being the list monad:
-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
+
+ let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)] ++Here phi is defined when `'t = 't1*'t2`, `F('t1*'t2) = char * 't1 * 't2`, and `F'('t1 * 't2) = int * 't1 * 't2`. -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. +Now where `gamma` is another function into monad `M` of type
F'('t) → MG'('t)
, we define:
+
++ gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a) + + = ((join G') -v- (M gamma) -v- phi) a + = (gamma <=< phi) a ++ +Hence: + +
+ gamma <=< phi = fun a -> (gamma =<< phi a) ++ +`gamma =<< phi a` is called the operation of "binding" the function gamma to the monadic value `phi a`, and is usually written as `phi a >>= gamma`. + +With these definitions, our monadic laws become: + + +
+ Where phi is a polymorphic function from type F('t) -> M F'('t) + and gamma is a polymorphic function from type G('t) -> M G' ('t) + and rho is a polymorphic function from type R('t) -> M R' ('t) + and F' = G and G' = R, + and a ranges over values of type F('t) for some type 't, + and b ranges over values of type G('t): + + (i) γ <=< φ is defined, + and is a natural transformation from F to MG' + ==> + (i'') fun a -> gamma =<< phi a is defined, + and is a function from type F('t) -> M G' ('t) + + + + (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ) + ==> + (fun a -> (rho <=< gamma) =<< phi a) = (fun a -> rho =<< (gamma <=< phi) a) + (fun a -> (fun b -> rho =<< gamma b) =<< phi a) = (fun a -> rho =<< (gamma =<< phi a)) + + (ii'') (fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a) + + + + (iii.1) (unit G') <=< γ = γ + when γ is a natural transformation from some FG' to MG' + + (unit G') <=< gamma = gamma + when gamma is a function of type FQ'('t) -> M G'('t) + + fun b -> (unit G') =<< gamma b = gamma + + (unit G') =<< gamma b = gamma b + + As below, return will map arguments c of type G'('t) + to the monadic value (unit G') b, of type M G'('t). + + (iii.1'') return =<< gamma b = gamma b + + + + (iii.2) γ = γ <=< (unit G) + when γ is a natural transformation from G to some MR'G + ==> + gamma = gamma <=< (unit G) + when gamma is a function of type G('t) -> M R' G('t) + + gamma = fun b -> gamma =<< ((unit G) b) + + Let return be a polymorphic function mapping arguments + of any type 't to M('t). In particular, it maps arguments + b of type G('t) to the monadic value (unit G) b, of + type M G('t). + + gamma = fun b -> gamma =<< return b + + (iii.2'') gamma b = gamma =<< return b ++ +Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature: + +* `fun b -> rho =<< gamma b) =<< phi a = rho =<< (gamma =<< phi a)` + + Usually written reversed, and with a monadic variable `u` standing in + for `phi a`: + + `u >>= (fun b -> gamma b >>= rho) = (u >>= gamma) >>= rho` -For an example of the latter, let φ be a function that takes arguments of some (schematic, polymorphic) type C1 and yields results of some (schematic, polymorphic) type M(C2). An example with M being the list monad, and C2 being the tuple type schema int * C1: +* `return =<< gamma b = gamma b` - let φ = fun c → [(1,c), (2,c)] + Usually written reversed, and with `u` standing in for `phi a`: -φ is polymorphic: when you apply it to the int 0 you get a result of type "list of int * int": [(1,0), (2,0)]. When you apply it to the char 'e' you get a result of type "list of int * char": [(1,'e'), (2,'e')]. + `u >>= return = u` -However, to keep things simple, we'll work instead with functions whose type is settled. So instead of the polymorphic φ, we'll work with (φ : C1 → M(int * C1)). This only accepts arguments of type C1. For generality, I'll talk of functions with the type (φ : C1 → M(C1')), where we assume that C1' is a function of C1. +* `gamma b = gamma =<< return b` -A "monadic value" is any member of a type M(C1), for any type C1. For example, a list is a monadic value for the list monad. We can think of these monadic values as the result of applying some function (φ : C1 → M(C1')) to an argument of type C1. + Usually written reversed: + `return b >>= gamma = gamma b` +