X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=173160e51c310018c87b43984820f3ffcffd176e;hp=56c897a6b11dbad51d410319831975ecba0e421a;hb=446376aee15c6fbc9d71383d1c7654fd02ebb8f6;hpb=ac1f590813b0da4eb32727b544c5be957cb79fe9 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 56c897a6..173160e5 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -1,343 +1,610 @@ +Caveats +------- +I really don't know much category theory. Just enough to put this +together. Also, this really is "put together." I haven't yet found an +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 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. + + +Monoids +------- +A **monoid** is a structure (S,⋆,z) consisting of an associative binary operation ⋆ over some set `S`, which is closed under ⋆, and which contains an identity element `z` for ⋆. That is: + + +
+	for all s1, s2, s3 in S:
+	  (i) s1⋆s2 etc are also in S
+	 (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3)
+	(iii) z⋆s1 = s1 = s1⋆z
+
-**Don't try to read this yet!!! Many substantial edits are still in process. Will be ready soon.** +Some examples of monoids are: + +* finite strings of an alphabet `A`, with ⋆ being concatenation and `z` being the empty string +* all functions X→X over a set `X`, with ⋆ being composition and `z` being the identity function over `X` +* the natural numbers with ⋆ 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 ⋆ be multiplication and `z` be 1, we get different monoids over the same sets as in the previous item. -1. Monoids +Categories ---------- -A is a structure consisting of an associative binary operation * over some set S, which is closed under *, and which contains an identity element z for *. That is: - for all s1,s2,s3 in S: - (i) s1*s2 etc are also in S - (ii) (s1*s2)*s3 = s1*(s2*s3) - (iii) z*s1 = s1 = s1*z +A **category** is a generalization of a monoid. A category consists of a class of **elements**, and a class of **morphisms** between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a single morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension." -Some examples of monoids are: +When a morphism `f` in category C has source `C1` and target `C2`, we'll write f:C1→C2. - (a) finite strings of an alphabet A, with * being concatenation and z being the empty string +To have a category, the elements and morphisms have to satisfy some constraints: - (b) all functions X->X over a set X, with * being composition and z being the identity function over X +
+	  (i) the class of morphisms has to be closed under composition:
+	      where f:C1→C2 and g:C2→C3, g ∘ f is also a
+	      morphism of the category, which maps C1→C3.
 
-	(c) the natural numbers with * being plus and z being 0 (in particular, this is a ). 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 .)
+	 (ii) composition of morphisms has to be associative
 
-	(d) the natural numbers with * being multiplication and z being 1 constitute a different monoid over the same set as in (c).
+	(iii) every element X of the category has to have an identity
+	      morphism 1X, which is such that for every morphism f:C1→C2:
+	      1C2 ∘ f = f = f ∘ 1C1
+
+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. -2. Categories -------------- -A is a generalization of a monoid. A category consists of a class of elements, and a class of between those elements. Morphisms are sometimes also called maps or arrows. They are something like functions (and as we'll see below, given a set of functions they'll determine a category). However, a given morphism only maps between a single source element and a single target element. Also, there can be multiple distinct morphisms between the same source and target, so the identity of a morphism goes beyond its "extension." -When a morphism f in category C has source c1 and target c2, we'll write f:c1->c2. +Some examples of categories are: -To have a category, the elements and morphisms have to satisfy some constraints: - (i) the class of morphisms has to be closed under composition: where f:c1->c2 and g:c2->c3, g o f is also a morphism of the category, which maps c1->c3. - (ii) composition of morphisms has to be associative - (iii) every element e of the category has to have an identity morphism id[e], which is such that for every morphism f:a->b: - id[b] o f = f = f o id[a] +* 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. -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. +* any monoid (S,⋆,z) 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 s3=s1⋆s2. The identity morphism for the (single) category element `Q` is the monoid's identity `z`. +* a **preorder** is a structure (S, ≤) 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 s1 ≤ s2 nor s2 ≤ s1). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that s1 ≤ s2 and s2 ≤ s1 but `s1` and `s2` are not identical). Some examples: -Some examples of categories are: + * 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 (S,≤) 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 s1 ≤ s2. - (a) any category 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 is the identity function over that set. - (b) any monoid (S,*,z) 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. The result of composing the morphism consisting of s1 with the morphism s2 is the morphism s3, where s3=s1+s2. The identity morphism on the (single) category element x is the monoid's identity z. +Functors +-------- +A **functor** is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor `F` from category C to category D must: - (c) a <= is a binary relation on a set S which is reflexive and transitive. It need not be connected (that is, there may be members x,y of S such that neither x<=y nor y<=x). It need not be anti-symmetric (that is, there may be members s1,s2 of S such that s1<=s2 and s2<=s1 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). - * sets ordered by size - Any pre-order (S,<=) 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 s1<=s2. +
+	  (i) associate with every element C1 of C an element F(C1) of D
 
+	 (ii) associate with every morphism f:C1→C2 of C a morphism F(f):F(C1)→F(C2) of D
 
+	(iii) "preserve identity", that is, for every element C1 of C:
+	      F of C1's identity morphism in C must be the identity morphism of F(C1) in D:
+	      F(1C1) = 1F(C1).
 
-3. Functors
------------
-A  is a "homomorphism", that is, a structure-preserving mapping, between categories. In particular, a functor F from category C to category D must:
-	(i) associate with every element c1 of C an element F(c1) of D
-	(ii) associate with every morphism f:c1->c2 of C a morphism F(f):F(c1)->F(c2) of D
-	(iii) "preserve identity", that is, for every element c1 of C: F of c1's identity morphism in C must be the identity morphism of F(c1) in D:
-		F(id[c1]) = id[F(c1)]. 
-	(iv) "distribute over composition", that is for any morphisms f and g in C:
-		F(g o f) = F(g) o F(f)
+	 (iv) "distribute over composition", that is for any morphisms f and g in C:
+	      F(g ∘ f) = F(g) ∘ F(f)
+
-A functor that maps a category to itself is called an . The (endo)functor that maps every element and morphism of C to itself is denoted 1C. +A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of C to itself is denoted `1C`. -How functors compose: -If F is a functor from category C to category D, and H is a functor from category D to category E, then HF is a functor which maps every element c1 of C to element H(F(c1)) of E, and maps every morphism f of C to morphism H(F(f)) of E. +How functors compose: If `G` is a functor from category C to category D, and `K` is a functor from category D to category E, then `KG` is a functor which maps every element `C1` of C to element `K(G(C1))` of E, and maps every morphism `f` of C to morphism `K(G(f))` of E. I'll assert without proving that functor composition is associative. -4. Natural Transformation -------------------------- -So categories include elements and morphisms. Functors consist of mappings from the elements and morphisms of one category to those of another (or the same) category. are a third level of mappings, from one functor to another. +Natural Transformation +---------------------- +So categories include elements and morphisms. Functors consist of mappings from the elements and morphisms of one category to those of another (or the same) category. **Natural transformations** are a third level of mappings, from one functor to another. + +Where `G` and `H` are functors from category C to category D, a natural transformation η between `G` and `H` is a family of morphisms η[C1]:G(C1)→H(C1) in D for each element `C1` of C. That is, η[C1] has as source `C1`'s image under `G` in D, and as target `C1`'s image under `H` in D. The morphisms in this family must also satisfy the constraint: -Where G and H are functors from category C to category D, a natural transformation eta between G and H is a family of morphisms eta[c1]:G(c1)->H(c1) in D for each element c1 of C. That is, eta[c1] has as source c1's image under G in D, and as target c1's image under H in D. The morphisms in this family must also satisfy the constraint: - for every morphism f:c1->c2 in C: - eta[c2] o G(f) = H(f) o eta[c1] +
+	for every morphism f:C1→C2 in C:
+	η[C2] ∘ G(f) = H(f) ∘ η[C1]
+
-That is, the morphism via G(f) from G(c1) to G(c2), and then via eta[c2] to H(c2), is identical to the morphism from G(c1) via eta[c1] to H(c1), and then via H(f) from H(c1) to H(c2). +That is, the morphism via `G(f)` from `G(C1)` to `G(C2)`, and then via η[C2] to `H(C2)`, is identical to the morphism from `G(C1)` via η[C1] to `H(C1)`, and then via `H(f)` from `H(C1)` to `H(C2)`. How natural transformations compose: -Consider four categories B,C,D, and E. -Let F be a functor from B to C; G,H, and J be functors from C to D; and K and L be functors from D to E. Let eta be a natural transformation from G to H; phi be a natural transformation from H to J; and psi be a natural transformation from K to L. Pictorally: - -- B -+ +--- C --+ +---- D -----+ +-- E -- - | | | | | | - F: ------> G: ------> K: ------> - | | | | | eta | | | psi - | | | | v | | v - | | H: ------> L: ------> - | | | | | phi | | - | | | | v | | - | | J: ------> | | ------+ +--------+ +------------+ +------- - -(eta F) is a natural transformation from the (composite) functor GF to the composite functor HF, such that where b1 is an element of category B, (eta F)[b1] = eta[F(b1)]---that is, the morphism in D that eta assigns to the element F(b1) of C. - -(K eta) is a natural transformation from the (composite) functor KG to the (composite) functor KH, such that where c1 is an element of category C, (K eta)[c1] = K(eta[c1])---that is, the morphism in E that K assigns to the morphism eta[c1] of D. - - -(phi -v- eta) is a natural transformation from G to J; this is known as a "vertical composition". We will rely later on this: - phi[c2] o H(f) o eta[c1] = phi[c2] o H(f) o eta[c1] - ------------- - by naturalness of phi, is: - -------------- - phi[c2] o H(f) o eta[c1] = J(f) o phi[c1] o eta[c1] - -------------- - by naturalness of eta, is: - -------------- - phi[c2] o eta[c2] o G(f) = J(f) o phi[c1] o eta[c1] - ----------------- ----------------- -Hence, we can define (phi -v- eta)[c1] as: phi[c1] o eta[c1] and rely on it to satisfy the constraints for a natural transformation from G to J: - ----------------- ----------------- - (phi -v- eta)[c2] o G(f) = J(f) o (phi -v- eta)[c1] +Consider four categories B, C, D, and E. Let `F` be a functor from B to C; `G`, `H`, and `J` be functors from C to D; and `K` and `L` be functors from D to E. Let η be a natural transformation from `G` to `H`; φ be a natural transformation from `H` to `J`; and ψ be a natural transformation from `K` to `L`. Pictorally: + +
+	- B -+ +--- C --+ +---- D -----+ +-- E --
+		 | |        | |            | |
+	 F: ------> G: ------>     K: ------>
+		 | |        | |  | η       | |  | ψ
+		 | |        | |  v         | |  v
+		 | |    H: ------>     L: ------>
+		 | |        | |  | φ       | |
+		 | |        | |  v         | |
+		 | |    J: ------>         | |
+	-----+ +--------+ +------------+ +-------
+
+ +Then (η F) is a natural transformation from the (composite) functor `GF` to the composite functor `HF`, such that where `B1` is an element of category B, (η F)[B1] = η[F(B1)]---that is, the morphism in D that η assigns to the element `F(B1)` of C. + +And (K η) is a natural transformation from the (composite) functor `KG` to the (composite) functor `KH`, such that where `C1` is an element of category C, (K η)[C1] = K(η[C1])---that is, the morphism in E that `K` assigns to the morphism η[C1] of D. + + +(φ -v- η) is a natural transformation from `G` to `J`; this is known as a "vertical composition". For any morphism f:C1→C2 in C: + +
+	φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1]
+
+ +by naturalness of φ, is: + +
+	φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1]
+
+ +by naturalness of η, is: + +
+	φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1]
+
+ +Hence, we can define (φ -v- η)[\_] as: φ[\_] ∘ η[\_] and rely on it to satisfy the constraints for a natural transformation from `G` to `J`: + +
+	(φ -v- η)[C2] ∘ G(f) = J(f) ∘ (φ -v- η)[C1]
+
+ +An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that: + +
+	((φ -v- η) F) = ((φ F) -v- (η F))
+
I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation." -(psi -h- eta) is natural transformation from the (composite) functor KG to the (composite) functor LH; this is known as a "horizontal composition." It's trickier to define, but we won't be using it here. For reference: +(ψ -h- η) is natural transformation from the (composite) functor `KG` to the (composite) functor `LH`; this is known as a "horizontal composition." It's trickier to define, but we won't be using it here. For reference: - (phi -h- eta)[c1] = L(eta[c1]) o psi[G(c1)] - = psi[H(c1)] o K(eta[c1]) +
+	(φ -h- η)[C1]  =  L(η[C1]) ∘ ψ[G(C1)]
+				   =  ψ[H(C1)] ∘ K(η[C1])
+
Horizontal composition is also associative, and has the same identity as vertical composition. -5. Monads ---------- +Monads +------ In earlier days, these were also called "triples." -A is a structure consisting of an (endo)functor M from some category C to itself, along with some natural transformations, which we'll specify in a moment. +A **monad** is a structure consisting of an (endo)functor `M` from some category C to itself, along with some natural transformations, which we'll specify in a moment. -Let T be a set of natural transformations p, each being between some (variable) functor P and another functor which is the composite MP' of M and a (variable) functor P'. That is, for each element c1 in C, p assigns c1 a morphism from element P(c1) to element MP'(c1), satisfying the constraints detailed in the previous section. For different members of T, the relevant functors may differ; that is, p is a transformation from functor P to MP', q is a transformation from functor Q to MQ', and none of P,P',Q,Q' need be the same. +Let `T` be a set of natural transformations φ, each being between some arbitrary endofunctor `F` on C and another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` on C. That is, for each element `C1` in C, φ 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, φ is a transformation from functor `F` to `MF'`, γ 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 on C 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 C 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 p and q be members of T, that is they are natural transformations from P to MP' and from Q to MQ', respectively. Let them be such that P' = Q. Now (M q) will also be a natural transformation, formed by composing the functor M with the natural transformation q. Similarly, (join Q') will be a natural transformation, formed by composing the natural transformation join with the functor Q'; it will transform the functor MMQ' to the functor MQ'. Now take the vertical composition of the three natural transformations (join Q'), (M q), and p, and abbreviate it as follows: +Let φ and γ 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 γ. 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')`, (M γ), and φ, and abbreviate it as follows. Since composition is associative I don't specify the order of composition on the rhs. - q <=< p =def. ((join Q') -v- (M q) -v- p) --- since composition is associative I don't specify the order of composition on the rhs +
+	γ <=< φ  =def.  ((join G') -v- (M γ) -v- φ)
+
-In other words, <=< is a binary operator that takes us from two members p and q of T to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes its written p >=> q where that's the same as q <=< p.) +In other words, `<=<` is a binary operator that takes us from two members φ and γ of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written φ >=> γ where that's the same as γ <=< φ.) -p is a transformation from P to MP' which = MQ; (M q) is a transformation from MQ to MMQ'; and (join Q') is a transformation from MMQ' to MQ'. So the composite q <=< p will be a transformation from P to MQ', and so also eligible to be a member of T. +φ is a transformation from `F` to `MF'`, where the latter = `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`. Now we can specify the "monad laws" governing a monad as follows: +
	
 	(T, <=<, unit) constitute a monoid
+
+ +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, γ <=< φ isn't fully defined on `T`, but only when φ is a transformation to some `MF'` and γ 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)                ρ  =  ρ <=< unit
+	        (here ρ has to be a natural transformation from 1C)
+
+ +If φ is a natural transformation from `F` to `M(1C)` and γ is (φ G'), that is, a natural transformation from `FG'` to `MG'`, then we can extend (iii.1) as follows: + +
+	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 is:
+	  = (unit G') <=< γ
+
+ +where as we said γ is a natural transformation from some `FG'` to `MG'`. + +Similarly, if ρ is a natural transformation from `1C` to `MR'`, and γ is (ρ G), that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows: + +
+	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 is:
+	  = γ <=< (unit G)
+
+ +where as we said γ is a natural transformation from `G` to some `MR'G`. + +Summarizing then, the monad laws can be expressed as: + +
+	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
+	        whenever γ is a natural transformation from some FG' to MG'
+
+	(iii.2)                     γ  =  γ <=< (unit G)
+	        whenever γ is a natural transformation from G to some MR'G
+
+ + -That's it. In other words: +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 `<=<`. - for all p,q,r in T: - (i) q <=< p etc are also in T - (ii) (r <=< q) <=< p = r <=< (q <=< p) - (iii.1) (unit P') <=< p = p - (iii.2) p = p <=< (unit P) + -A word about the P' and P in (iii.1) and (iii.2): since unit on its own is a transformation from 1C to M(1C), it doesn't have the appropriate "type" for unit <=< p or p <=< unit to be defined, for arbitrary p. However, if p is a transformation from P to MP', then (unit P') <=< p and p <=< (unit P) will both be defined. +Let's remind ourselves of principles stated above: +* 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) -6. 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 <=<. +* if η is a natural transformation from `G` to `H`, then for every f:C1→C2 in `G` and `H`'s source category C: η[C2] ∘ G(f) = H(f) ∘ η[C1]. -(* - P2. every element c1 of a category C has an identity morphism id[c1] such that for every morphism f:c1->c2 in C: id[c2] o f = f = f o id[c1]. - P3. functors "preserve identity", that is for every element c1 in F's source category: F(id[c1]) = id[F(c1)]. -*) +* (η F)[X] = η[F(X)] -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 o f) = F(g) o F(f) - * if eta is a natural transformation from F to G, then for every f:c1->c2 in F and G's source category C: eta[c2] o F(f) = G(f) o eta[c1]. +* (K η)[X] = K(η[X]) +* ((φ -v- η) F) = ((φ F) -v- (η F)) 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 C, join[c1] will be a morphism from MM(c1) to M(c1). And for any morphism f:a->b in C: +Recall that `join` is a natural transformation from the (composite) functor `MM` to `M`. So for elements `C1` in C, `join[C1]` will be a morphism from `MM(C1)` to `M(C1)`. And for any morphism f:C1→C2 in C: - (1) join[b] o MM(f) = M(f) o join[a] +
+	(1) join[C2] ∘ MM(f)  =  M(f) ∘ join[C1]
+
-Next, consider the composite transformation ((join MQ') -v- (MM q)). - q is a transformation from Q to MQ', and assigns elements c1 in C a morphism q*: Q(c1) -> MQ'(c1). (MM q) is a transformation that instead assigns c1 the morphism MM(q*). - (join MQ') is a transformation from MMMQ' to MMQ' that assigns c1 the morphism join[MQ'(c1)]. - Composing them: - (2) ((join MQ') -v- (MM q)) assigns to c1 the morphism join[MQ'(c1)] o MM(q*). +Next, let γ be a transformation from `G` to `MG'`, and + consider the composite transformation ((join MG') -v- (MM γ)). -Next, consider the composite transformation ((M q) -v- (join Q)). - (3) This assigns to c1 the morphism M(q*) o join[Q(c1)]. +* γ assigns elements `C1` in C a morphism γ\*:G(C1) → MG'(C1). (MM γ) is a transformation that instead assigns `C1` the morphism MM(γ\*). -So for every element c1 of C: - ((join MQ') -v- (MM q))[c1], by (2) is: - join[MQ'(c1)] o MM(q*), which by (1), with f=q*: Q(c1)->MQ'(c1) is: - M(q*) o join[Q(c1)], which by 3 is: - ((M q) -v- (join Q))[c1] +* `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`. -So our (lemma 1) is: ((join MQ') -v- (MM q)) = ((M q) -v- (join Q)), where q is a transformation from Q to MQ'. +Composing them: +
+	(2) ((join MG') -v- (MM γ)) assigns to C1 the morphism join[MG'(C1)] ∘ MM(γ*).
+
-Next recall that unit is a natural transformation from 1C to M. So for elements c1 in C, unit[c1] will be a morphism from c1 to M(c1). And for any morphism f:a->b in C: - (4) unit[b] o f = M(f) o unit[a] +Next, consider the composite transformation ((M γ) -v- (join G)): -Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to c1 the morphism M(q*) o unit[Q(c1)]. +
+	(3) ((M γ) -v- (join G)) assigns to C1 the morphism M(γ*) ∘ join[G(C1)].
+
-Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to c1 the morphism unit[MQ'(c1)] o q*. +So for every element `C1` of C: -So for every element c1 of C: - ((M q) -v- (unit Q))[c1], by (5) = - M(q*) o unit[Q(c1)], which by (4), with f=q*: Q(c1)->MQ'(c1) is: - unit[MQ'(c1)] o q*, which by (6) = - ((unit MQ') -v- q)[c1] +
+	((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:
+	M(γ*) ∘ join[G(C1)], which by 3 is:
+	((M γ) -v- (join G))[C1]
+
-So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- q)), where q is a transformation from Q to MQ'. +So our **(lemma 1)** is: +
+	((join MG') -v- (MM γ))  =  ((M γ) -v- (join G)),
+	where as we said γ is a natural transformation from G to MG'.
+
-Finally, we substitute ((join Q') -v- (M q) -v- p) for q <=< p in the monad laws. For simplicity, I'll omit the "-v-". - for all p,q,r in T, where p is a transformation from P to MP', q is a transformation from Q to MQ', R is a transformation from R to MR', and P'=Q and Q'=R: +Next recall that `unit` is a natural transformation from `1C` to `M`. So for elements `C1` in C, `unit[C1]` will be a morphism from `C1` to `M(C1)`. And for any morphism f:C1→C2 in C: - (i) q <=< p etc are also in T - ==> - (i') ((join Q') (M q) p) etc are also in T +
+	(4) unit[C2] ∘ f = M(f) ∘ unit[C1]
+
+Next, consider the composite transformation ((M γ) -v- (unit G)): - (ii) (r <=< q) <=< p = r <=< (q <=< p) - ==> - (r <=< q) is a transformation from Q to MR', so: - (r <=< q) <=< p becomes: (join R') (M (r <=< q)) p - which is: (join R') (M ((join R') (M r) q)) p - substituting in (ii), and helping ourselves to associativity on the rhs, we get: +
+	(5) ((M γ) -v- (unit G)) assigns to C1 the morphism M(γ*) ∘ unit[G(C1)].
+
+ +Next, consider the composite transformation ((unit MG') -v- γ): - ((join R') (M ((join R') (M r) q)) p) = ((join R') (M r) (join Q') (M q) p) - --------------------- - which by the distributivity of functors over composition, and helping ourselves to associativity on the lhs, yields: - ------------------------ - ((join R') (M join R') (MM r) (M q) p) = ((join R') (M r) (join Q') (M q) p) - --------------- - which by lemma 1, with r a transformation from Q' to MR', yields: - ----------------- - ((join R') (M join R') (MM r) (M q) p) = ((join R') (join MR') (MM r) (M q) p) +
+	(6) ((unit MG') -v- γ) assigns to C1 the morphism unit[MG'(C1)] ∘ γ*.
+
- which will be true for all r,q,p just in case: +So for every element C1 of C: - ((join R') (M join R')) = ((join R') (join MR')), for any R'. +
+	((M γ) -v- (unit G))[C1], by (5) =
+	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]
+
- which will in turn be true just in case: +So our **(lemma 2)** is: - (ii') (join (M join)) = (join (join M)) +
+	(((M γ) -v- (unit G))  =  ((unit MG') -v- γ)),
+	where as we said γ is a natural transformation from G to MG'.
+
- (iii.1) (unit P') <=< p = p +Finally, we substitute ((join G') -v- (M γ) -v- φ) for γ <=< φ in the monad laws. For simplicity, I'll omit the "-v-". + +
+	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
 	==>
-			(unit P') is a transformation from P' to MP', so:
-				(unit P') <=< p becomes: (join P') (M unit P') p
-						   which is: (join P') (M unit P') p
-				substituting in (iii.1), we get:
-			((join P') (M unit P') p) = p
+	    (i') ((join G') (M γ) φ) etc are also in T
+
- which will be true for all p just in case: +
+	    (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)
+
+			 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 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)
+
+			 [-- 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))
+
+ +
+	 (iii.1) (unit G') <=< γ  =  γ
+	         when γ is a natural transformation from some FG' to MG'
+	==>
+			 (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)
+
+			 substituting in (iii.1), we get:
+			 ((join G') ((M unit) G') γ) = γ
 
-	         ((join P') (M unit P')) = the identity transformation, for any P'
+			 which is:
+			 (((join (M unit)) G') γ) = γ
 
-			which will in turn be true just in case:
+			 [-- Are the next two steps too cavalier? --]
 
-	(iii.1') (join (M unit) = the identity transformation
+			 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
+
- (iii.2) p = p <=< (unit P) +
+	 (iii.2) γ  =  γ <=< (unit G)
+	         when γ is a natural transformation from G to some MR'G
 	==>
-			p is a transformation from P to MP', so:
-				unit <=< p becomes: (join P') (M p) unit
-				substituting in (iii.2), we get:
-			p = ((join P') (M p) (unit P))
-						   --------------
-				which by lemma (2), yields:
-                            ------------
-			p = ((join P') ((unit MP') p)
+			 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 p just in case:
+			 which is:
+			 γ = (((join (unit M)) R'G) γ)
 
-	        ((join P') (unit MP')) = the identity transformation, for any P'
+			 [-- 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
+
Collecting the results, our monad laws turn out in this format to be: - - when p a transformation from P to MP', q a transformation from P' to MQ', r a transformation from Q' to MR' all in T: - (i') ((join Q') (M q) p) etc also in T +
+	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
+
+	   (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
+
+ + + +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 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 `'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." +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`: +
+	let phi = fun ((_:char, x y) -> [(1,x,y),(2,x,y)]
+
-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. -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.) +Now where `gamma` is another function of type F'('t) → M(G'('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 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))
+
+ +
+	     (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 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
+
+ +
+	  (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)))
 
-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,...].
+			  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
+
-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] o f = M(f) o t[c1] +Summarizing (ii''), (iii.1''), (iii.2''), these are the monadic laws as usually stated in the functional programming literature: -The composite morphisms said here to be identical are morphisms from the type c1 to the type M(c2). +* `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` -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. +* `return =<< gamma b = gamma b` -For an example of the latter, let p 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: - - let p = fun c -> [(1,c), (2,c)] + Usually written reversed, and with `u` standing in for `gamma b`: -p 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 p, we'll work with (p : c1 -> M(int * c1)). This only accepts arguments of type c1. For generality, I'll talk of functions with the type (p : 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 (p : c1 -> M(c1')) to an argument of type c1. + Usually written reversed: + `return b >>= gamma = gamma b` +