(iii.2') (join (unit M)) = the identity transformation
</pre>
+In category-theory presentations, you may see `unit` referred to as <code>η</code>, and `join` referred to as <code>μ</code>. Also, instead of the monad `(M, unit, join)`, you may sometimes see discussion of the "Kleisli triple" `(M, unit, =<<)`. Alternatively, `=<<` may be called <code>⋆</code>. These are interdefinable (see below).
Getting to the functional programming presentation of the monad laws
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,...]`.
+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."
-A "monadic value" is any member of a type `M('t)`, for any type `'t`. For example, any `int list` is a monadic value for the list monad. We can think of these monadic values as the result of applying some function `phi`, whose type is `F('t)->M(F'('t))`. `'t` here is any collection of free type variables, and `F('t)` and `F'('t)` are types parameterized on `'t`. An example, with `M` being the list monad, `'t` being `('t1,'t2)`, `F('t1,'t2)` being `char * 't1 * 't2`, and `F'('t1,'t2)` being `int * 't1 * 't2`:
+A "monadic value" is any member of a type `M('t)`, for any type `'t`. For example, any `int list` is a monadic value for the List monad. We can think of these monadic values as the result of applying some function `phi`, whose type is `F('t) -> M(F'('t))`. `'t` here is any collection of free type variables, and `F('t)` and `F'('t)` are types parameterized on `'t`. An example, with `M` being the List monad, `'t` being `('t1,'t2)`, `F('t1,'t2)` being `char * 't1 * 't2`, and `F'('t1,'t2)` being `int * 't1 * 't2`:
<pre>
- let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
+ let phi = fun ((_:char), x, y) -> [(1,x,y),(2,x,y)]
</pre>
+[-- I intentionally chose this polymorphic function because simpler ways of mapping the polymorphic monad operations from functional programming onto the category theory notions can't accommodate it. We have all the F, MF' (unit G') and so on in order to be able to be handle even phis like this. --]
-Now where `gamma` is another function of type <code>F'('t) → M(G'('t))</code>, we define:
+Now where `gamma` is another function of type <code>F'('t) -> M(G'('t))</code>, we define:
<pre>
gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
Hence:
<pre>
- gamma <=< phi = fun a -> (gamma =<< phi a)
+ 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`.
rho is a polymorphic function of type R('t) -> M(R'('t))
and F' = G and G' = R,
and a ranges over values of type F('t),
- b ranges over values of type G('t),
+ and b ranges over values of type G('t),
and c ranges over values of type G'('t):
(i) γ <=< φ is defined,
(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))
+ (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)
</pre>
<pre>
(iii.1) (unit G') <=< γ = γ
- when γ is a natural transformation from some FG' to MG'
+ whenever γ is a natural transformation from some FG' to MG'
==>
(unit G') <=< gamma = gamma
- when gamma is a function of type F(G'('t)) -> M(G'('t))
+ whenever gamma is a function of type F(G'('t)) -> M(G'('t))
- fun b -> (unit G') =<< gamma b = gamma
+ (fun b -> (unit G') =<< gamma b) = gamma
(unit G') =<< gamma b = gamma b
<pre>
(iii.2) γ = γ <=< (unit G)
- when γ is a natural transformation from G to some MR'G
+ whenever 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)))
+ whenever gamma is a function of type G('t) -> M(R'(G('t)))
- gamma = fun b -> gamma =<< (unit G) b
+ gamma = (fun b -> gamma =<< (unit G) b)
As above, return will map arguments b of type G('t) to the
monadic value (unit G) b, of type M(G('t)).
- gamma = fun b -> gamma =<< return b
+ 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)`
+* `(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`: