-**Don't try to read this yet!!! Many substantial edits are still in process.
-Will be ready soon.**
-
Caveats
-------
I really don't know much category theory. Just enough to put this
authoritative source (that's accessible to a category theory beginner like
myself) that discusses the correspondence between the category-theoretic and
functional programming uses of these notions in enough detail to be sure that
-none of the pieces here is misguided. In particular, it wasn't completely
-obvious how to map the polymorphism on the programming theory side into the
-category theory. And I'm bothered by the fact that our `<=<` operation is only
-partly defined on our domain of natural transformations. But this does seem to
-me to be the reasonable way to put the pieces together. We very much welcome
+none of the pieces here is mistaken.
+In particular, it wasn't completely obvious how to map the polymorphism on the
+programming theory side into the category theory. The way I accomplished this
+may be more complex than it needs to be.
+Also I'm bothered by the fact that our `<=<` operation is only partly defined
+on our domain of natural transformations.
+There are three additional points below that I wonder whether may be too
+cavalier.
+But all considered, this does seem to
+me to be a reasonable way to put the pieces together. We very much welcome
feedback from anyone who understands these issues better, and will make
corrections.
(ii) composition of morphisms has to be associative
- (iii) every element E of the category has to have an identity
- morphism 1<sub>E</sub>, which is such that for every morphism f:C1→C2:
+ (iii) every element X of the category has to have an identity
+ morphism 1<sub>X</sub>, which is such that for every morphism f:C1→C2:
1<sub>C2</sub> ∘ f = f = f ∘ 1<sub>C1</sub>
</pre>
-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.
+These parallel the constraints for monoids. Note that there can be multiple distinct morphisms between an element `X` 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. Morphisms correspond to directed paths of length ≥ 0 in the graph.
* Categories whose elements are sets and whose morphisms are functions between those sets. Here the source and target of a function are its domain and range, so distinct functions sharing a domain and range (e.g., `sin` and `cos`) are distinct morphisms between the same source and target elements. The identity morphism for any element/set is just the identity function for that set.
-* 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`.
+* any monoid <code>(S,⋆,z)</code> generates a category with a single element `Q`; this `Q` 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 `Q` 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 `Q` 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 `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:
+* 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)
- Any pre-order <code>(S,≤)</code> generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff <code>s1≤s2</code>.
+ Any pre-order <code>(S,≤)</code> generates a category whose elements are the members of `S` and which has only a single morphism between any two elements `s1` and `s2`, iff <code>s1 ≤ s2</code>.
Functors
<pre>
γ = (φ G')
= ((unit <=< φ) G')
+ since unit is a natural transformation to M(1C), this is:
= (((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 <=<:
+ since (unit G') is a natural transformation to MG', this is:
= (unit G') <=< γ
</pre>
<pre>
γ = (ρ G)
= ((ρ <=< unit) G)
+ = since ρ is a natural transformation to MR', this is:
= (((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 <=<:
+ since γ = (ρ G) is a natural transformation to MR'G, this is:
= γ <=< (unit G)
</pre>
(ii) (ρ <=< γ) <=< φ = ρ <=< (γ <=< φ)
(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'
(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
</pre>
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:
+Let's remind ourselves of principles stated above:
* composition of morphisms, functors, and natural compositions is associative
* 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>(η F)[X] = η[F(X)]</code>
-* <code>(K η)[E} = K(η[E])</code>
+* <code>(K η)[X] = K(η[X])</code>
* <code>((φ -v- η) F) = ((φ F) -v- (η F))</code>
(i) γ <=< φ etc are also in T
==>
(i') ((join G') (M γ) φ) etc are also in T
+</pre>
-
-
+<pre>
(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)
+ similarly, ρ <=< (γ <=< φ) is:
+ ((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)
+ 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)
-
- which will be true for all ρ,γ,φ just in case:
-
- ((join R') (M join R')) = ((join R') (join MR')), for any R'.
+ 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)
- which will in turn be true just in case:
-
- (ii') (join (M join)) = (join (join M))
+ [-- Are the next two steps too cavalier? --]
+ 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 when:
+ (ii') (join (M join)) = (join (join M))
+</pre>
+<pre>
(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:
- (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
-
- which will be true for all φ just in case:
+ (unit G') is a transformation from G' to MG', so:
+ (unit G') <=< γ becomes: ((join G') (M (unit G')) γ)
+ which is: ((join G') ((M unit) G') γ)
- ((join F') (M unit F')) = the identity transformation, for any F'
+ substituting in (iii.1), we get:
+ ((join G') ((M unit) G') γ) = γ
- which will in turn be true just in case:
-
- (iii.1') (join (M unit) = the identity transformation
+ which is:
+ (((join (M unit)) G') γ) = γ
+ [-- Are the next two steps too cavalier? --]
+ which will be true for all γ just in case:
+ for any G', ((join (M unit)) G') = the identity transformation
+ which will in turn be true just in case:
+ (iii.1') (join (M unit)) = the identity transformation
+</pre>
- (iii.2) γ = γ <=< (unit G)
+<pre>
+ (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:
- 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)
+ γ <=< (unit G) becomes: ((join R'G) (M γ) (unit G))
+
+ 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 φ just in case:
+ which is:
+ γ = (((join (unit M)) R'G) γ)
- ((join F') (unit MF')) = the identity transformation, for any F'
+ [-- Are the next two steps too cavalier? --]
- which will in turn be true just in case:
+ which will be true for all γ just in case:
+ for any R'G, ((join (unit M)) R'G) = the identity transformation
+ 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:
+<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
+ (i') ((join G') (M γ) φ) etc also in T
- (ii') (join (M join)) = (join (join M))
+ (ii') (join (M join)) = (join (join M))
(iii.1') (join (M unit)) = the identity transformation
- (iii.2')(join (unit M)) = the identity transformation
+ (iii.2') (join (unit M)) = the identity transformation
</pre>
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 <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,...].
+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 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]
+<pre>
+ let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
+</pre>
-The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
+Now where `gamma` is another function of type <code>F'('t) → M(G'('t))</code>, we define:
-In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question.
+<pre>
+ gamma =<< phi a =def. ((join G') -v- (M gamma)) (phi a)
+ = ((join G') -v- (M gamma) -v- phi) a
+ = (gamma <=< phi) a
+</pre>
-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:
+Hence:
- let φ = fun c → [(1,c), (2,c)]
+<pre>
+ gamma <=< phi = fun a -> (gamma =<< phi a)
+</pre>
-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')].
+`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`.
-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.
+With these definitions, our monadic laws become:
-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.
+<pre>
+ Where phi is a polymorphic function of type F('t) -> M(F'('t))
+ gamma is a polymorphic function of type G('t) -> M(G'('t))
+ 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 c 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))
+</pre>
+
+<pre>
+ (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)
+</pre>
+
+<pre>
+ (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 F(G'('t)) -> M(G'('t))
+
+ fun b -> (unit G') =<< gamma b = gamma
+
+ (unit G') =<< gamma b = gamma b
+
+ Let return be a polymorphic function mapping arguments of any
+ type 't to M('t). In particular, it maps arguments c of type
+ G'('t) to the monadic value (unit G') c, of type M(G'('t)).
+
+ (iii.1'') return =<< gamma b = gamma b
+</pre>
+
+<pre>
+ (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
+
+ 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
+
+ (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`
+
+* `return =<< gamma b = gamma b`
+
+ Usually written reversed, and with `u` standing in for `gamma b`:
+
+ `u >>= return = u`
+
+* `gamma b = gamma =<< return b`
+
+ Usually written reversed:
+
+ `return b >>= gamma = gamma b`
+