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
+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.
----------
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:
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(id[c1]) = id[F(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(id[c1]) = id[F(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.
----------------------
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:
+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: eta[c2] o G(f) = H(f) o eta[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 `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)`.
+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 `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:
+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: ------>
- | | | | | eta | | | psi
+ | | | | | η | | | ψ
| | | | v | | v
| | H: ------> L: ------>
- | | | | | phi | |
+ | | | | | φ | |
| | | | 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`.
+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 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`.
+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**.
-`(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`:
+`(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`:
- phi[c2] o H(f) o eta[c1] = phi[c2] o H(f) o eta[c1]
+ φ[c2] o H(f) o η[c1] = φ[c2] o H(f) o η[c1]
-by naturalness of phi, is:
+by naturalness of φ, is:
- phi[c2] o H(f) o eta[c1] = J(f) o phi[c1] o eta[c1]
+ φ[c2] o H(f) o η[c1] = J(f) o φ[c1] o η[c1]
-by naturalness of eta, is:
+by naturalness of η, is:
- phi[c2] o eta[c2] o G(f) = J(f) o phi[c1] o eta[c1]
+ φ[c2] o η[c2] o G(f) = J(f) o φ[c1] o η[c1]
-Hence, we can define `(phi -v- eta)[x]` as: `phi[x] o eta[x]` and rely on it to satisfy the constraints for a natural transformation from `G` to `J`:
+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`:
- (phi -v- eta)[c2] o G(f) = J(f) o (phi -v- eta)[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:
- ((phi -v- eta) F) = ((phi F) -v- (eta F))
+ ((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."
-`(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:
+`(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:
- (phi -h- eta)[c1] = L(eta[c1]) o psi[G(c1)]
- = psi[H(c1)] o K(eta[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.
------
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`.
(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.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:
= ((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'`.
q = (p Q)
= ((p <=< unit) Q)
- = (((join P') (M p) unit) Q)
- = ((join P'Q) ((M p) Q) (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`.
+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 <=<.
+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].
+ 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].
+ * 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*).
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:
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) =
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
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).
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')].