= (((join 1C) G') -v- ((M unit) G') -v- (φ G'))
= ((join (1C G')) -v- (M (unit G')) -v- γ)
= ((join G') -v- (M (unit G')) -v- γ)
- since (unit G') is a natural transformation to MG', this satisfies the definition for <=<:
+ since (unit G') is a natural transformation to MG',
+ this satisfies the definition for <=<:
= (unit G') <=< γ
</pre>
= (((join R') -v- (M ρ) -v- unit) G)
= (((join R') G) -v- ((M ρ) G) -v- (unit G))
= ((join (R'G)) -v- (M (ρ G)) -v- (unit G))
- since γ = (ρ G) is a natural transformation to MR'G, this satisfies the definition for <=<:
+ since γ = (ρ G) is a natural transformation to MR'G,
+ this satisfies the definition <=<:
= γ <=< (unit G)
</pre>
where as we said <code>γ</code> is a natural transformation from `G` to some `MR'G`.
+Summarizing then, the monad laws can be expressed as:
+
+<pre>
+ For all ρ, γ, φ in T for which ρ <=< γ and γ <=< φ are defined:
+
+ (i) γ <=< φ etc are also in T
+
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
+
+ (iii.1) (unit G') <=< γ = γ
+ when γ is a natural transformation from some FG' to MG'
+
+ (iii.2) γ = γ <=< (unit G)
+ when γ is a natural transformation from G to some MR'G
+</pre>
-The standard category-theory presentation of the monad laws
------------------------------------------------------------
+Getting to the standard category-theory presentation of the monad laws
+----------------------------------------------------------------------
In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`.
<!--
* functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: <code>F(g ∘ f) = F(g) ∘ F(f)</code>
-* if <code>η</code> is a natural transformation from `F` to `G`, then for every <code>f:C1→C2</code> in `F` and `G`'s source category <b>C</b>: <code>η[C2] ∘ F(f) = G(f) ∘ η[C1]</code>.
+* if <code>η</code> is a natural transformation from `G` to `H`, then for every <code>f:C1→C2</code> in `G` and `H`'s source category <b>C</b>: <code>η[C2] ∘ G(f) = H(f) ∘ η[C1]</code>.
+
+* <code>(η F)[E] = η[F(E)]</code>
+
+* <code>(K η)[E} = K(η[E])</code>
+
+* <code>((φ -v- η) F) = ((φ F) -v- (η F))</code>
Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
-Recall that join is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in <b>C</b>, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
+Recall that `join` is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in <b>C</b>, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
<pre>
(1) join[C2] ∘ MM(f) = M(f) ∘ join[C1]
</pre>
-Next, consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
+Next, let <code>γ</code> be a transformation from `G` to `MG'`, and
+ consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
-* <code>γ</code> is a transformation from `G` to `MG'`, and assigns elements `C1` in <b>C</b> a morphism <code>γ\*: G(C1) → MG'(C1)</code>. <code>(MM γ)</code> is a transformation that instead assigns `C1` the morphism <code>MM(γ\*)</code>.
+* <code>γ</code> assigns elements `C1` in <b>C</b> a morphism <code>γ\*:G(C1) → MG'(C1)</code>. <code>(MM γ)</code> is a transformation that instead assigns `C1` the morphism <code>MM(γ\*)</code>.
-* `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
+* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
Composing them:
(2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
</pre>
-Next, consider the composite transformation <code>((M γ) -v- (join G))</code>.
+Next, consider the composite transformation <code>((M γ) -v- (join G))</code>:
<pre>
- (3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
+ (3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
</pre>
So for every element `C1` of <b>C</b>:
<pre>
((join MG') -v- (MM γ))[C1], by (2) is:
- join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*: G(C1)→MG'(C1) is:
+ join[MG'(C1)] ∘ MM(γ*), which by (1), with f=γ*:G(C1)→MG'(C1) is:
M(γ*) ∘ join[G(C1)], which by 3 is:
((M γ) -v- (join G))[C1]
</pre>
So our **(lemma 1)** is:
<pre>
- ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'.
+ ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)),
+ where as we said γ is a natural transformation from G to MG'.
</pre>
-Next recall that unit is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:a→b</code> in <b>C</b>:
+Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in <b>C</b>, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism <code>f:C1→C2</code> in <b>C</b>:
<pre>
- (4) unit[b] ∘ f = M(f) ∘ unit[a]
+ (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
</pre>
-Next consider the composite transformation <code>((M γ) -v- (unit G))</code>:
+Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
<pre>
- (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+ (5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
</pre>
-Next consider the composite transformation <code>((unit MG') -v- γ)</code>.
+Next, consider the composite transformation <code>((unit MG') -v- γ)</code>:
<pre>
- (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+ (6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
</pre>
So for every element C1 of <b>C</b>:
<pre>
((M γ) -v- (unit G))[C1], by (5) =
- M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*: G(C1)→MG'(C1) is:
+ M(γ*) ∘ unit[G(C1)], which by (4), with f=γ*:G(C1)→MG'(C1) is:
unit[MG'(C1)] ∘ γ*, which by (6) =
((unit MG') -v- γ)[C1]
</pre>
So our **(lemma 2)** is:
<pre>
- (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'.
+ (((M γ) -v- (unit G)) = ((unit MG') -v- γ)),
+ where as we said γ is a natural transformation from G to MG'.
</pre>
Finally, we substitute <code>((join G') -v- (M γ) -v- φ)</code> for <code>γ <=< φ</code> in the monad laws. For simplicity, I'll omit the "-v-".
<pre>
- for all φ,γ,ρ in T, where φ is a transformation from F to MF', γ is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R:
+ For all ρ, γ, φ in T,
+ where φ is a transformation from F to MF',
+ γ is a transformation from G to MG',
+ ρ is a transformation from R to MR',
+ and F'=G and G'=R:
- (i) γ <=< φ etc are also in T
+ (i) γ <=< φ etc are also in T
==>
- (i') ((join G') (M γ) φ) etc are also in T
+ (i') ((join G') (M γ) φ) etc are also in T
+
- (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
==>
- (ρ <=< γ) is a transformation from G to MR', so:
- (ρ <=< γ) <=< φ becomes: (join R') (M (ρ <=< γ)) φ
- which is: (join R') (M ((join R') (M ρ) γ)) φ
- substituting in (ii), and helping ourselves to associativity on the rhs, we get:
+ (ρ <=< γ) is a transformation from G to MR', so
+ (ρ <=< γ) <=< φ becomes: ((join R') (M (ρ <=< γ)) φ)
+ which is: ((join R') (M ((join R') (M ρ) γ)) φ)
- ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
- ---------------------
- which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
- ------------------------
- ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
- ---------------
- which by lemma 1, with ρ a transformation from G' to MR', yields:
- -----------------
- ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
+ similarly, ρ <=< (γ <=< φ) is:
+ ((join R') (M ρ) ((join G') (M γ) φ))
- which will be true for all ρ,γ,φ just in case:
+ substituting these into (ii), and helping ourselves to associativity on the rhs, we get:
+ ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
+
+ which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields:
+ ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (M ρ) (join G') (M γ) φ)
+
+ which by lemma 1, with ρ a transformation from G' to MR', yields:
+ ((join R') (M join R') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
- ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+ which will be true for all ρ,γ,φ only when:
+ ((join R') (M join R')) = ((join R') (join MR')), for any R'.
- which will in turn be true just in case:
+ which will in turn be true when:
+ (ii') (join (M join)) = (join (join M))
- (ii') (join (M join)) = (join (join M))
- (iii.1) (unit F') <=< φ = φ
+ (iii.1) (unit G') <=< γ = γ
+ when γ is a natural transformation from some FG' to MG'
==>
- (unit F') is a transformation from F' to MF', so:
- (unit F') <=< φ becomes: (join F') (M unit F') φ
- which is: (join F') (M unit F') φ
- substituting in (iii.1), we get:
- ((join F') (M unit F') φ) = φ
+ (unit G') is a transformation from G' to MG', so:
+ (unit G') <=< γ becomes: ((join G') (M unit G') γ)
- which will be true for all φ just in case:
+ substituting in (iii.1), we get:
+ ((join G') (M unit G') γ) = γ
- ((join F') (M unit F')) = the identity transformation, for any F'
-
- which will in turn be true just in case:
+ which will be true for all γ just in case:
+ ((join G') (M unit G')) = the identity transformation, for any G'
+ which will in turn be true just in case:
(iii.1') (join (M unit) = the identity transformation
- (iii.2) φ = φ <=< (unit F)
+
+
+ (iii.2) γ = γ <=< (unit G)
+ when γ is a natural transformation from G to some MR'G
==>
- φ is a transformation from F to MF', so:
- unit <=< φ becomes: (join F') (M φ) unit
- substituting in (iii.2), we get:
- φ = ((join F') (M φ) (unit F))
- --------------
- which by lemma (2), yields:
- ------------
- φ = ((join F') ((unit MF') φ)
+ unit <=< γ becomes: ((join R'G) (M γ) unit)
+
+ substituting in (iii.2), we get:
+ γ = ((join R'G) (M γ) (unit G))
+
+ which by lemma 2, yields:
+ γ = ((join R'G) ((unit MR'G) γ)
+
+ which will be true for all γ just in case:
+ ((join R'G) (unit MR'G)) = the identity transformation, for any R'G
+
+ which will in turn be true just in case:
+ (iii.2') (join (unit M)) = the identity transformation
+</pre>
+
+
+Collecting the results, our monad laws turn out in this format to be:
- which will be true for all φ just in case:
+<pre>
+ For all ρ, γ, φ in T,
+ where φ is a transformation from F to MF',
+ γ is a transformation from G to MG',
+ ρ is a transformation from R to MR',
+ and F'=G and G'=R:
+
+ (i') ((join G') (M γ) φ) etc also in T
- ((join F') (unit MF')) = the identity transformation, for any F'
+ (ii') (join (M join)) = (join (join M))
- which will in turn be true just in case:
+ (iii.1') (join (M unit)) = the identity transformation
(iii.2') (join (unit M)) = the identity transformation
</pre>
-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 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 <b>C</b> 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 `'t` to types `M('t)`, and a mapping from functions <code>f:C1→C2</code> to functions <code>M(f):M(C1)→M(C2)</code>. This is also known as <code>lift<sub>M</sub> f</code> 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 <code>f:x→y</code> 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 <code>(φ : F('t) → M(F'('t)))</code> to an argument `a` of type `F('t)`.
+
+
+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:
+
+<pre>
+ let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
</pre>
- when φ a transformation from F to MF', γ a transformation from F' to MG', ρ a transformation from G' to MR' all in T:
- (i') ((join G') (M γ) φ) etc also in T
+Here phi is defined when `'t = 't1*'t2`, `F('t1*'t2) = char * 't1 * 't2`, and `F'('t1 * 't2) = int * 't1 * 't2`.
- (ii') (join (M join)) = (join (join M))
- (iii.1') (join (M unit)) = the identity transformation
+Now where `gamma` is another function into monad `M` of type <code>F'('t) → MG'('t)</code>, we define:
+
+<pre>
+ gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
+
+ = ((join G') -v- (M gamma) -v- phi) a
+ = (gamma <=< phi) a
+</pre>
+
+Hence:
- (iii.2')(join (unit M)) = the identity transformation
+<pre>
+ gamma <=< phi = fun a -> (gamma =<< phi a)
</pre>
+`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:
-7. 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.
+<pre>
+ 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)
-The base category <b>C</b> 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,...].
+ (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)
-A natural transformation t assigns to each type C1 in <b>C</b> a morphism t[C1]: C1→M(C1) such that, for every f:C1→C2:
- t[C2] ∘ f = M(f) ∘ t[C1]
-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
+ (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
-In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.
+ (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
+</pre>
+
+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`
+