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=0cdb5ab10c16bc2f13351138d6e37e7496baba9b;hp=7b52c3a6fe699400241714549176eb154ada8439;hb=HEAD;hpb=50e06b4a50c0fcdc84f5cc94773316935871ceb1 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn deleted file mode 100644 index 7b52c3a6..00000000 --- a/advanced_topics/monads_in_category_theory.mdwn +++ /dev/null @@ -1,396 +0,0 @@ -**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 -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 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 -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
-
- -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. - -Categories ----------- -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." - -When a morphism `f` in category C has source `C1` and target `C2`, we'll write f:C1→C2. - -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 ∘ 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 1E, 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 `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.) - - -Some examples of categories are: - -* 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 (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. 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 s3=s1⋆s2. The identity morphism for the (single) category element `x` 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 `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; 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. - - -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: - -
-	(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).
-	(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 **endofunctor**. The (endo)functor that maps every element and morphism of C to itself is denoted `1C`. - -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. - - - -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: - - 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 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 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". We will rely later on this, where `f:C1→C2`: - - 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)[x]` as: φ[x] ∘ η[x]` 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." - - -`(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: - - (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. - - - -Monads ------- -In earlier days, these were also called "triples." - -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. - -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`. - -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: - - 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. - -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`.) - -`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`. - -Now we can specify the "monad laws" governing a monad as follows: - - (T, <=<, unit) constitute a monoid - -That's it. (Well, perhaps we're cheating a bit, because `q <=< p` isn't fully defined on `T`, but only when `P` is a functor to `MP'` and `Q` is a functor from `P'`. But wherever `<=<` is defined, the monoid laws are satisfied: - - (i) q <=< p is also in T - (ii) (r <=< q) <=< p = r <=< (q <=< p) - (iii.1) unit <=< p = p (here p has to be a natural transformation to M(1C)) - (iii.2) p = p <=< unit (here p has to be a natural transformation from 1C) - -If `p` is a natural transformation from `P` to `M(1C)` and `q` is `(p Q')`, that is, a natural transformation from `PQ` to `MQ`, then we can extend (iii.1) as follows: - - q = (p Q') - = ((unit <=< p) Q') - = ((join -v- (M unit) -v- p) Q') - = (join Q') -v- ((M unit) Q') -v- (p Q') - = (join Q') -v- (M (unit Q')) -v- q - ?? - = (unit Q') <=< q - -where as we said `q` is a natural transformation from some `PQ'` to `MQ'`. - -Similarly, if `p` is a natural transformation from `1C` to `MP'`, and `q` is `(p Q)`, that is, a natural transformation from `Q` to `MP'Q`, then we can extend (iii.2) as follows: - - q = (p Q) - = ((p <=< unit) Q) - = (((join P') -v- (M p) -v- unit) Q) - = ((join P'Q) -v- ((M p) Q) -v- (unit Q)) - = ((join P'Q) -v- (M (p Q)) -v- (unit Q)) - ?? - = q <=< (unit Q) - -where as we said `q` is a natural transformation from `Q` to some `MP'Q`. - - - - -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 C has an identity morphism 1C1 such that for every morphism f:C1→C2 in C: 1C2 ∘ f = f = f ∘ 1C1. - P3. functors "preserve identity", that is for every element C1 in F's source category: F(1C1) = 1F(C1). -*) - -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 C: η[C2] ∘ F(f) = G(f) ∘ η[C1]. - - -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: - - (1) join[b] ∘ MM(f) = M(f) ∘ join[a] - -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)] ∘ MM(q*). - -Next, consider the composite transformation ((M q) -v- (join Q)). - (3) This assigns to C1 the morphism M(q*) ∘ join[Q(C1)]. - -So for every element C1 of C: - ((join MQ') -v- (MM q))[C1], by (2) is: - join[MQ'(C1)] ∘ MM(q*), which by (1), with f=q*: Q(C1)→MQ'(C1) is: - M(q*) ∘ join[Q(C1)], which by 3 is: - ((M q) -v- (join Q))[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'. - - -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] ∘ f = M(f) ∘ unit[a] - -Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to C1 the morphism M(q*) ∘ unit[Q(C1)]. - -Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to C1 the morphism unit[MQ'(C1)] ∘ q*. - -So for every element C1 of C: - ((M q) -v- (unit Q))[C1], by (5) = - M(q*) ∘ unit[Q(C1)], which by (4), with f=q*: Q(C1)→MQ'(C1) is: - unit[MQ'(C1)] ∘ q*, which by (6) = - ((unit MQ') -v- q)[C1] - -So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- q)), where q is a transformation from Q to MQ'. - - -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: - - (i) q <=< p etc are also in T - ==> - (i') ((join Q') (M q) p) etc are also in T - - - (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: - - ((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) - - which will be true for all r,q,p just in case: - - ((join R') (M join R')) = ((join R') (join MR')), for any R'. - - which will in turn be true just in case: - - (ii') (join (M join)) = (join (join M)) - - - (iii.1) (unit P') <=< p = p - ==> - (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 - - which will be true for all p just in case: - - ((join P') (M unit P')) = the identity transformation, for any P' - - which will in turn be true just in case: - - (iii.1') (join (M unit) = the identity transformation - - - (iii.2) p = p <=< (unit P) - ==> - 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) - - which will be true for all p just in case: - - ((join P') (unit MP')) = the identity transformation, for any P' - - 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 - - (ii') (join (M join)) = (join (join M)) - - (iii.1') (join (M unit)) = the identity transformation - - (iii.2')(join (unit M)) = the identity transformation - - - -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.) - -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 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] ∘ f = M(f) ∘ t[C1] - -The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2). - - - -In functional programming, instead of working with natural transformations we work with "monadic values" and polymorphic functions "into the monad" in question. - -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)] - -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')]. - -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. - -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. - -