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=41b17c5205de8518033ac047d3f2e452a8cc3777;hb=HEAD;hpb=31786296de9b1c4456c8b85c36c0897eaf02d1c2 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn deleted file mode 100644 index 41b17c52..00000000 --- a/advanced_topics/monads_in_category_theory.mdwn +++ /dev/null @@ -1,360 +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 these do 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 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:c1->c2: - id[c2] o f = f = f o id[c1] - -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 is the identity function over 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. - - - -3. 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: - -
    -
  1. associate with every element c1 of C an element F(c1) of D -
  2. associate with every morphism f:c1->c2 of C a morphism F(f):F(c1)->F(c2) of D -
  3. "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)]. -
  4. "distribute over composition", that is for any morphisms f and g in C: - F(g o f) = F(g) o 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. - - - -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. **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 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] - -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). - - -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: ------> | | ------+ +--------+ +------------+ +------- - -Then (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. - -And (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, where f:c1->c2: - 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] - -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: - - (phi -h- eta)[c1] = L(eta[c1]) o psi[G(c1)] - = psi[H(c1)] o K(eta[c1]) - -Horizontal composition is also associative, and has the same identity as vertical composition. - - - -5. 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 on 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. In other words: - - 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. - - - -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 <=<. - -(* - 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)]. -*) - -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]. - - -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] o MM(f) = M(f) o 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)] o MM(q*). - -Next, consider the composite transformation ((M q) -v- (join Q)). - (3) This assigns to c1 the morphism M(q*) o join[Q(c1)]. - -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] - -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] o f = M(f) o unit[a] - -Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to c1 the morphism M(q*) o unit[Q(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: - ((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] - -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] o f = M(f) o 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. - -