* finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
* all functions <code>X→X</code> over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
-* the natural numbers with <code>⋆</code> being plus and `z` being `0` (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.)
-* if we let <code>⋆</code> be multiplication and `z` be `1`, we get different monoids over the same sets as in the previous item.
+* the natural numbers with <code>⋆</code> being plus and `z` being 0 (in particular, this is a **commutative monoid**). If we use the integers, or the naturals mod n, instead of the naturals, then every element will have an inverse and so we have not merely a monoid but a **group**.
+* if we let <code>⋆</code> be multiplication and `z` be 1, we get different monoids over the same sets as in the previous item.
Categories
----------
These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `E` and itself; they need not all be identity morphisms. Indeed from (iii) it follows that each element can have only a single identity morphism.
-A good intuitive picture of a category is as a generalized directed graph, where the category elements are the graph's nodes, and there can be multiple directed edges between a given pair of nodes, and nodes can also have multiple directed edges to themselves. (Every node must have at least one such, which is that node's identity morphism.)
+A good intuitive picture of a category is as a generalized directed graph, where the category elements are the graph's nodes, and there can be multiple directed edges between a given pair of nodes, and nodes can also have multiple directed edges to themselves. Morphisms correspond to directed paths of length ≥ 0 in the graph.
Some examples of categories are:
* any monoid <code>(S,⋆,z)</code> generates a category with a single element `x`; this `x` need not have any relation to `S`. The members of `S` play the role of *morphisms* of this category, rather than its elements. All of these morphisms are understood to map `x` to itself. The result of composing the morphism consisting of `s1` with the morphism `s2` is the morphism `s3`, where <code>s3=s1⋆s2</code>. The identity morphism for the (single) category element `x` is the monoid's identity `z`.
-* a **preorder** is a structure <code>(S, ≤)</code> consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `x`,`y` of `S` such that neither <code>x≤y</code> nor <code>y≤x</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1≤s2</code> and <code>s2≤s1</code> but `s1` and `s2` are not identical). Some examples:
+* a **preorder** is a structure <code>(S, ≤)</code> consisting of a reflexive, transitive, binary relation on a set `S`. It need not be connected (that is, there may be members `s1`,`s2` of `S` such that neither <code>s1≤s2</code> nor <code>s2≤s1</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1≤s2</code> and <code>s2≤s1</code> but `s1` and `s2` are not identical). Some examples:
* sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a pre-order without anti-symmetry)
* sets ordered by size (this illustrates it too)
And <code>(K η)</code> is a natural transformation from the (composite) functor `KG` to the (composite) functor `KH`, such that where `C1` is an element of category <b>C</b>, <code>(K η)[C1] = K(η[C1])</code>---that is, the morphism in <b>E</b> that `K` assigns to the morphism <code>η[C1]</code> of <b>D</b>.
-<code>(φ -v- η)</code> is a natural transformation from `G` to `J`; this is known as a "vertical composition". We will rely later on this, where <code>f:C1→C2</code>:
+<code>(φ -v- η)</code> is a natural transformation from `G` to `J`; this is known as a "vertical composition". For any morphism <code>f:C1→C2</code> in <b>C</b>:
<pre>
φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1]
A **monad** is a structure consisting of an (endo)functor `M` from some category <b>C</b> to itself, along with some natural transformations, which we'll specify in a moment.
-Let `T` be a set of natural transformations <code>φ</code>, each being between some (variable) functor `F` and another functor which is the composite `MF'` of `M` and a (variable) functor `F'`. That is, for each element `C1` in <b>C</b>, <code>φ</code> assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is, <code>φ</code> is a transformation from functor `F` to `MF'`, <code>γ</code> is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` need be the same.
+Let `T` be a set of natural transformations <code>φ</code>, each being between some arbitrary endofunctor `F` on <b>C</b> and another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` on <b>C</b>. That is, for each element `C1` in <b>C</b>, <code>φ</code> assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is, <code>φ</code> is a transformation from functor `F` to `MF'`, <code>γ</code> is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` need be the same.
-One of the members of `T` will be designated the "unit" transformation for `M`, and it will be a transformation from the identity functor `1C` for <b>C</b> to `M(1C)`. So it will assign to `C1` a morphism from `C1` to `M(C1)`.
+One of the members of `T` will be designated the `unit` transformation for `M`, and it will be a transformation from the identity functor `1C` for <b>C</b> to `M(1C)`. So it will assign to `C1` a morphism from `C1` to `M(C1)`.
-We also need to designate for `M` a "join" transformation, which is a natural transformation from the (composite) functor `MM` to `M`.
+We also need to designate for `M` a `join` transformation, which is a natural transformation from the (composite) functor `MM` to `M`.
These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
-Let <code>φ</code> and <code>γ</code> be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now `(M γ)` will also be a natural transformation, formed by composing the functor `M` with the natural transformation <code>γ</code>. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, <code>(M γ)</code>, and <code>φ</code>, and abbreviate it as follows:
+Let <code>φ</code> and <code>γ</code> be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now <code>(M γ)</code> will also be a natural transformation, formed by composing the functor `M` with the natural transformation <code>γ</code>. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, <code>(M γ)</code>, and <code>φ</code>, and abbreviate it as follows. Since composition is associative I don't specify the order of composition on the rhs.
<pre>
γ <=< φ =def. ((join G') -v- (M γ) -v- φ)
</pre>
-Since composition is associative I don't specify the order of composition on the rhs.
+In other words, `<=<` is a binary operator that takes us from two members <code>φ</code> and <code>γ</code> of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written <code>φ >=> γ</code> where that's the same as <code>γ <=< φ</code>.)
-In other words, `<=<` is a binary operator that takes us from two members <code>φ</code> and <code>γ</code> of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written <code>φ >=> γ</code> where that's the same as <code>γ <=< φ</code>.)
-
-φ is a transformation from `F` to `MF'` which = `MG`; `(M γ)` is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite γ <=< φ will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
+<code>φ</code> is a transformation from `F` to `MF'`, where the latter = `MG`; <code>(M γ)</code> is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite <code>γ <=< φ</code> will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
Now we can specify the "monad laws" governing a monad as follows:
+<pre>
(T, <=<, unit) constitute a monoid
+</pre>
-That's it. (Well, perhaps we're cheating a bit, because γ <=< φ isn't fully defined on `T`, but only when `F` is a functor to `MF'` and `G` is a functor from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
+That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, <code>γ <=< φ</code> isn't fully defined on `T`, but only when <code>φ</code> is a transformation to some `MF'` and <code>γ</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws must hold:
- (i) γ <=< φ is also in T
- (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
- (iii.1) unit <=< φ = φ (here φ has to be a natural transformation to M(1C))
- (iii.2) φ = φ <=< unit (here φ has to be a natural transformation from 1C)
+<pre>
+ (i) γ <=< φ is also in T
-If φ is a natural transformation from `F` to `M(1C)` and γ is `(φ G')`, that is, a natural transformation from `PG` to `MG`, then we can extend (iii.1) as follows:
+ (ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
+ (iii.1) unit <=< φ = φ
+ (here φ has to be a natural transformation to M(1C))
+
+ (iii.2) ρ = ρ <=< unit
+ (here ρ has to be a natural transformation from 1C)
+</pre>
+
+If <code>φ</code> is a natural transformation from `F` to `M(1C)` and <code>γ</code> is <code>(φ G')</code>, that is, a natural transformation from `FG'` to `MG'`, then we can extend (iii.1) as follows:
+
+<pre>
γ = (φ G')
= ((unit <=< φ) G')
- = ((join -v- (M unit) -v- φ) G')
- = (join G') -v- ((M unit) G') -v- (φ G')
- = (join G') -v- (M (unit G')) -v- γ
- ??
+ = (((join 1C) -v- (M unit) -v- φ) G')
+ = (((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 <=<:
= (unit G') <=< γ
+</pre>
-where as we said γ is a natural transformation from some `PG'` to `MG'`.
+where as we said <code>γ</code> is a natural transformation from some `FG'` to `MG'`.
-Similarly, if φ is a natural transformation from `1C` to `MF'`, and γ is `(φ G)`, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:
+Similarly, if <code>ρ</code> is a natural transformation from `1C` to `MR'`, and <code>γ</code> is <code>(ρ G)</code>, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:
- γ = (φ G)
- = ((φ <=< unit) G)
- = (((join F') -v- (M φ) -v- unit) G)
- = ((join F'G) -v- ((M φ) G) -v- (unit G))
- = ((join F'G) -v- (M (φ G)) -v- (unit G))
- ??
+<pre>
+ γ = (ρ G)
+ = ((ρ <=< unit) G)
+ = (((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 <=<:
= 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)
-where as we said γ is a natural transformation from `G` to some `MF'G`.
+ (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 `<=<`.
-(*
+<!--
P2. every element C1 of a category <b>C</b> has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1→C2 in <b>C</b>: 1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>.
P3. functors "preserve identity", that is for every element C1 in F's source category: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
-*)
+-->
Let's remind ourselves of some principles:
- * composition of morphisms, functors, and natural compositions is associative
- * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g ∘ f) = F(g) ∘ F(f)
- * if η is a natural transformation from F to G, then for every f:C1→C2 in F and G's source category <b>C</b>: η[C2] ∘ F(f) = G(f) ∘ η[C1].
+* composition of morphisms, functors, and natural compositions is associative
+
+* 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 `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 f:a→b 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>
- (1) join[b] ∘ MM(f) = M(f) ∘ join[a]
+Next, let <code>γ</code> be a transformation from `G` to `MG'`, and
+ consider the composite transformation <code>((join MG') -v- (MM γ))</code>.
-Next, consider the composite transformation ((join MG') -v- (MM γ)).
- γ is a transformation from G to MG', and assigns elements C1 in <b>C</b> a morphism γ*: G(C1) → MG'(C1). (MM γ) is a transformation that instead assigns C1 the morphism MM(γ*).
- (join MG') is a transformation from MMMG' to MMG' that assigns C1 the morphism join[MG'(C1)].
- Composing them:
+* <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 `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.
+
+Composing them:
+
+<pre>
(2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
+</pre>
-Next, consider the composite transformation ((M γ) -v- (join G)).
- (3) This assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
+Next, consider the composite transformation <code>((M γ) -v- (join G))</code>:
-So for every element C1 of <b>C</b>:
+<pre>
+ (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: ((join MG') -v- (MM γ)) = ((M γ) -v- (join G)), where γ is a transformation from G to MG'.
+So our **(lemma 1)** is:
+
+<pre>
+ ((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 f:a→b in <b>C</b>:
- (4) unit[b] ∘ f = M(f) ∘ unit[a]
+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>:
-Next consider the composite transformation ((M γ) -v- (unit G)). (5) This assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+<pre>
+ (4) unit[C2] ∘ f = M(f) ∘ unit[C1]
+</pre>
-Next consider the composite transformation ((unit MG') -v- γ). (6) This assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+Next, consider the composite transformation <code>((M γ) -v- (unit G))</code>:
+
+<pre>
+ (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>:
+
+<pre>
+ (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: (((M γ) -v- (unit G)) = ((unit MG') -v- γ)), where γ is a transformation from G to MG'.
+So our **(lemma 2)** is:
+
+<pre>
+ (((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 ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-".
+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-".
- 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:
+<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) 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)
+
+ similarly, ρ <=< (γ <=< φ) is:
+ ((join R') (M ρ) ((join G') (M γ) φ))
- ((join R') (M ((join R') (M ρ) γ)) φ) = ((join R') (M ρ) (join G') (M γ) φ)
- ---------------------
+ 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)
- ---------------
+ ((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') (MM ρ) (M γ) φ) = ((join R') (join MR') (MM ρ) (M γ) φ)
- which will be true for all ρ,γ,φ just in case:
+ which will be true for all ρ,γ,φ only when:
+ ((join R') (M join R')) = ((join R') (join MR')), for any R'.
- ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+ which will in turn be true when:
+ (ii') (join (M join)) = (join (join M))
- which will in turn be true just in case:
-
- (ii') (join (M join)) = (join (join M))
+ (iii.1) (unit G') <=< γ = γ
+ when γ is a natural transformation from some FG' to MG'
(iii.1) (unit F') <=< φ = φ
==>
(unit F') is a transformation from F' to MF', so:
(iii.1') (join (M unit) = the identity transformation
+
+
+ (iii.2) γ = γ <=< (unit G)
+ when γ is a natural transformation from G to some MR'G
(iii.2) φ = φ <=< (unit F)
==>
φ is a transformation from F to MF', so:
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:
+</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
(iii.1') (join (M unit)) = the identity transformation
(iii.2')(join (unit M)) = the identity transformation
+</pre>
-7. The functional programming presentation of the monad laws
-------------------------------------------------------------
+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.