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:
+A **monoid** is a structure <code>(S,⋆,z)</code> consisting of an associative binary operation <code>⋆</code> over some set `S`, which is closed under <code>⋆</code>, and which contains an identity element `z` for <code>⋆</code>. That is:
<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
+ (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:
-* 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.
+* finite strings of an alphabet `A`, with <code>⋆</code> being concatenation and `z` being the empty string
+* all functions <code>X→X</code> over a set `X`, with <code>⋆</code> being composition and `z` being the identity function over `X`
+* the natural numbers with <code>⋆</code> 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 <code>⋆</code> 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`.
+When a morphism `f` in category <b>C</b> has source `C1` and target `C2`, we'll write <code>f:C1→C2</code>.
To have a category, the elements and morphisms have to satisfy some constraints:
<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 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>
+ (i) the class of morphisms has to be closed under composition:
+ where f:C1→C2 and g:C2→C3, g ∘ 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 1<sub>E</sub>, which is such that for every morphism f:C1→C2:
+ 1<sub>C2</sub> ∘ f = f = f ∘ 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.
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/set is just the identity function for that set.
+* 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/set is just the identity function for 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`.
+* any monoid <code>(S,⋆,z)</code> 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 <code>s3=s1⋆s2</code>. 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:
+* a **preorder** is a structure <code>(S, ≤)</code> 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 <code>x≤y</code> nor <code>y≤x</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1≤s2</code> and <code>s2≤s1</code> 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`.
+ Any pre-order <code>(S,≤)</code> 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 <code>s1≤s2</code>.
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 <b>C</b> to category <b>D</b> must:
+
+<pre>
+ (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
- (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)
+ (ii) associate with every morphism f:C1→C2 of <b>C</b> a morphism
+ F(f):F(C1)→F(C2) of <b>D</b>
+
+ (iii) "preserve identity", that is, for every element C1 of <b>C</b>:
+ F of C1's identity morphism in <b>C</b> must be the identity morphism
+ of F(C1) in <b>D</b>: F(1<sub>C1</sub>) = 1<sub>F(C1)</sub>.
+
+ (iv) "distribute over composition", that is for any morphisms f and g in <b>C</b>:
+ F(g ∘ f) = F(g) ∘ 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`.
+A functor that maps a category to itself is called an **endofunctor**. The (endo)functor that maps every element and morphism of <b>C</b> 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 <b>C</b> to category <b>D</b>, and `K` is a functor from category <b>D</b> to category <b>E</b>, then `KG` is a functor which maps every element `C1` of <b>C</b> to element `K(G(C1))` of <b>E</b>, and maps every morphism `f` of <b>C</b> to morphism `K(G(f))` of <b>E</b>.
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 <b>C</b> to category <b>D</b>, a natural transformation η between `G` and `H` is a family of morphisms η[C1]:G(C1)→H(C1)` in <b>D</b> for each element `C1` of <b>C</b>. That is, η[C1]` has as source `C1`'s image under `G` in <b>D</b>, and as target `C1`'s image under `H` in <b>D</b>. 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 <b>C</b>: η[C2] ∘ G(f) = H(f) ∘ 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>B</b>, <b>C</b>, <b>D</b>, and <b>E</b>. Let `F` be a functor from <b>B</b> to <b>C</b>; `G`, `H`, and `J` be functors from <b>C</b> to <b>D</b>; and `K` and `L` be functors from <b>D</b> to <b>E</b>. 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>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
| | | | | |
- 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**.
+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>B</b>, `(η F)[b1] = η[F(b1)]`---that is, the morphism in <b>D</b> that η assigns to the element `F(b1)` of <b>C</b>.
-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 <b>C</b>, `(K η)[C1] = K(η[C1])`---that is, the morphism in <b>E</b> that `K` assigns to the morphism η[C1]` of <b>D</b>.
-`(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] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ 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] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ 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] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ 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`:
+Hence, we can define `(φ -v- η)[x]` as: φ[x] ∘ 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] ∘ G(f) = J(f) ∘ (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]) ∘ ψ[G(C1)]
+ = ψ[H(C1)] ∘ 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 <b>C</b> 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 <b>C</b>, `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 <b>C</b> 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 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>.
+ P2. every element C1 of a category <b>C</b> has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1→C2 in <b>C</b>: 1<sub>C2</sub> ∘ f = f = f ∘ 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].
+ * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g ∘ f) = F(g) ∘ 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 <b>C</b>: η[C2] ∘ F(f) = G(f) ∘ 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 <b>C</b>, join[C1] will be a morphism from MM(C1) to M(C1). And for any morphism f:a→b in <b>C</b>:
- (1) join[b] o MM(f) = M(f) o join[a]
+ (1) join[b] ∘ MM(f) = M(f) ∘ 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 <b>C</b> 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)] ∘ 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*) ∘ join[Q(C1)].
-So for every element C1 of **C**:
+So for every element C1 of <b>C</b>:
((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:
+ join[MQ'(C1)] ∘ MM(q*), which by (1), with f=q*: Q(C1)→MQ'(C1) is:
+ M(q*) ∘ 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 recall that unit is a natural transformation from 1C to M. So for elements C1 in <b>C</b>, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a→b in <b>C</b>:
+ (4) unit[b] ∘ f = M(f) ∘ 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*) ∘ 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)] ∘ q*.
-So for every element C1 of **C**:
+So for every element C1 of <b>C</b>:
((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) =
+ M(q*) ∘ unit[Q(C1)], which by (4), with f=q*: Q(C1)→MQ'(C1) is:
+ unit[MQ'(C1)] ∘ 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'.
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 <b>C</b> 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 <b>C</b> a morphism t[C1]: C1→M(C1) such that, for every f:C1→C2:
+ t[C2] ∘ f = M(f) ∘ t[C1]
The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).
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.