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=0f84bb28e49392cdeb605e6a2e830a9ac3112ea4;hp=aef5538a7d799aa4f5aa7e1e0412fe0b3aa6b58c;hb=8076acb29d9f26cb505398e08d79bf5992584fc7;hpb=3900b155388b440877c55e853989c21e79a01248 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index aef5538a..0f84bb28 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -40,7 +40,7 @@ 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: @@ -71,18 +71,18 @@ Some examples of categories are: 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: +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 o f) = F(g) o F(f)
+	(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 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`. +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**. +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. @@ -92,18 +92,18 @@ 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)`. 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: +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** -- + - B -+ +--- C --+ +---- D -----+ +-- E -- | | | | | | F: ------> G: ------> K: ------> | | | | | η | | | ψ @@ -114,9 +114,9 @@ Consider four categories **B**, **C**, **D**, and **E**. Let `F` be a functor fr | | 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**. +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**. +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`: @@ -155,11 +155,11 @@ 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. +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 `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)`. +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`. @@ -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*). @@ -244,7 +244,7 @@ Next, consider the composite transformation ((join MQ') -v- (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**: +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: @@ -253,14 +253,14 @@ So for every element C1 of **C**: 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)]. 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) = @@ -361,14 +361,14 @@ In functional programming, unit is usually called "return" and the monad laws ar 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.) +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: +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).