From 95e37df6ddcd64a91b2e9e49531933bad2084c7a Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 13:00:11 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 118 ++++++++++++++++++++++--- 1 file changed, 105 insertions(+), 13 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 24670796..51832424 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 -------------------------------------------------------------------- -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 + -- 2.11.0