-------
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:
+
+<pre>
+ 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
+</pre>
Some examples of monoids are:
----------
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.
+<pre>
+ (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]
+ (iii) every element E of the category has to have an identity morphism 1<sub>E</sub>, which is such that for every morphism f:C1->C2: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>
+</pre>
-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.
+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.)
--------
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(id[c1]) = id[F(c1)].
+<pre>
+ (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(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
(iv) "distribute over composition", that is for any morphisms f and g in **C**: F(g o f) = F(g) o F(f)
+</pre>
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.
----------------------
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)`.
+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:
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`:
+`(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]
+ φ[C2] o H(f) o η[C1] = φ[C2] o H(f) o η[C1]
by naturalness of φ, is:
- φ[c2] o H(f) o η[c1] = J(f) o φ[c1] o η[c1]
+ φ[C2] o H(f) o η[C1] = J(f) o φ[C1] o η[C1]
by naturalness of η, is:
- φ[c2] o η[c2] o G(f) = J(f) o φ[c1] o η[c1]
+ φ[C2] o η[C2] o G(f) = J(f) o φ[C1] o η[C1]
Hence, we can define `(φ -v- η)[x]` as: φ[x] o η[x]` and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
- (φ -v- η)[c2] o G(f) = J(f) o (φ -v- η)[c1]
+ (φ -v- η)[C2] o G(f) = J(f) o (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:
`(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]) o ψ[G(c1)]
- = ψ[H(c1)] o K(η[c1])
+ (φ -h- η)[C1] = L(η[C1]) o ψ[G(C1)]
+ = ψ[H(C1)] o K(η[C1])
Horizontal composition is also associative, and has the same identity as vertical composition.
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`.
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)].
+ P2. every element C1 of a category **C** has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1->C2 in **C**: 1<sub>C2</sub> o f = f = f o 1<sub>C1</sub>.
+ P3. functors "preserve identity", that is for every element C1 in F's source category: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
*)
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*).
- (join MQ') is a transformation from MMMQ' to MMQ' that assigns c1 the morphism join[MQ'(c1)].
+ 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*).
+ (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)].
+ (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 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**:
+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 ((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*.
+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 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'.
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:
- t[c2] o f = M(f) o t[c1]
+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).
+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:
+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.
+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.