From: Jim Pryor Date: Tue, 2 Nov 2010 12:14:24 +0000 (-0400) Subject: cat: right arrow X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=81182da89b2e6a8896a477a8d966bf9bd42bc8ff;hp=8076acb29d9f26cb505398e08d79bf5992584fc7 cat: right arrow Signed-off-by: Jim Pryor --- diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 0f84bb28..a5e6f979 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -32,7 +32,7 @@ A **monoid** is a structure `(S, *, z)` consisting of an associative binary oper 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` +* 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. @@ -40,14 +40,14 @@ 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`. +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.
+	(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 1E, which is such that for every morphism f:C1->C2: 1C2 o f = f = f o 1C1
+	(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 o f = f = f o 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. @@ -75,7 +75,7 @@ A **functor** is a "homomorphism", that is, a structure-preserving mapping, betw 
 	(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
+	(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 o f) = F(g) o F(f)
@@ -92,9 +92,9 @@ 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 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] o G(f) = H(f) o η[C1] + for every morphism f:C1→C2 in C: η[C2] o G(f) = H(f) o 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)`. @@ -105,13 +105,13 @@ Consider four categories B, C, D, and E. Let `F` be - B -+ +--- C --+ +---- D -----+ +-- E -- | | | | | | - F: ------> G: ------> K: ------> + F: -----→ G: -----→ K: -----→ | | | | | η | | | ψ | | | | v | | v - | | H: ------> L: ------> + | | H: -----→ L: -----→ | | | | | φ | | | | | | v | | - | | J: ------> | | + | | 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. @@ -119,7 +119,7 @@ Then `(η F)` is a natural transformation from the (composite) functor `GF` t 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 -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] o H(f) o η[C1] = φ[C2] o H(f) o η[C1] @@ -218,25 +218,25 @@ 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 o f = f = f o 1C1. + P2. every element C1 of a category C has an identity morphism 1C1 such that for every morphism f:C1→C2 in C: 1C2 o f = f = f o 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 o f) = F(g) o 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] o F(f) = G(f) o η[C1]. + * 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] o F(f) = G(f) o 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: +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*). + 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*). @@ -246,14 +246,14 @@ Next, consider the composite transformation ((M q) -v- (join Q)). 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: + 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: +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)]. @@ -262,7 +262,7 @@ Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns 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: + 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] @@ -363,12 +363,12 @@ Additionally, whereas in category-theory one works "monomorphically", in functio 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 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: +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). @@ -379,12 +379,12 @@ In functional programming, instead of working with natural transformations we wo 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)] + 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. +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. +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.