cat theory tweaks

[lambda.git] / advanced_topics / monads_in_category_theory.mdwn

diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index f26e2cd..29b6feb 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -19,35 +19,41 @@ corrections.
 
 Monoids
 -------
 
 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,&#8902;,z)</code> consisting of an associative binary operation <code>&#8902;</code> over some set `S`, which is closed under <code>&#8902;</code>, and which contains an identity element `z` for <code>&#8902;</code>. That is:

 
 
 <pre>
        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
+         (i) s1&#8902;s2 etc are also in S
+        (ii) (s1&#8902;s2)&#8902;s3 = s1&#8902;(s2&#8902;s3)
+       (iii) z&#8902;s1 = s1 = s1&#8902;z

 </pre>
 
 Some examples of monoids are:
 
 </pre>
 
 Some examples of monoids are:
 

-*      finite strings of an alphabet `A`, with `*` being concatenation and `z` being the empty string
-*      all functions `X&rarr;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>&#8902;</code> being concatenation and `z` being the empty string
+*      all functions <code>X&rarr;X</code> over a set `X`, with <code>&#8902;</code> being composition and `z` being the identity function over `X`
+*      the natural numbers with <code>&#8902;</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>&#8902;</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."
 
 
 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 <b>C</b> has source `C1` and target `C2`, we'll write `f:C1&rarr;C2`.
+When a morphism `f` in category <b>C</b> has source `C1` and target `C2`, we'll write <code>f:C1&rarr;C2</code>.

 
 To have a category, the elements and morphisms have to satisfy some constraints:
 
 <pre>
 
 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&rarr;C2 and g:C2&rarr;C3, g &ordm; f is also a morphism of the category, which maps C1&rarr;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&rarr;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&rarr;C2 and g:C2&rarr;C3, g &#8728; f is also a
+             morphism of the category, which maps C1&rarr;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&rarr;C2:
+             1<sub>C2</sub> &#8728; f = f = f &#8728; 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.
 </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.


@@ -57,11 +63,11 @@ A good intuitive picture of a category is as a generalized directed graph, where
 
 Some examples of categories are:
 
 
 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,&#8902;,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&#8902;s2</code>. The identity morphism for the (single) category element `x` is the monoid's identity `z`.

 
 

-*      a **preorder** is a structure `(S, &le;)` 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&le;y` nor `y&le;x`). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that `s1&le;s2` and `s2&le;s1` but `s1` and `s2` are not identical). Some examples:
+*      a **preorder** is a structure <code>(S, &le;)</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&le;y</code> nor <code>y&le;x</code>). It need not be anti-symmetric (that is, there may be members `s1`,`s2` of `S` such that <code>s1&le;s2</code> and <code>s2&le;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)
 
        *       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)


@@ -74,10 +80,16 @@ Functors
 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>
 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>
-       (ii) associate with every morphism f:C1&rarr;C2 of <b>C</b> a morphism F(f):F(C1)&rarr;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 o f) = F(g) o F(f)
+         (i) associate with every element C1 of <b>C</b> an element F(C1) of <b>D</b>
+
+        (ii) associate with every morphism f:C1&rarr;C2 of <b>C</b> a morphism F(f):F(C1)&rarr;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 &#8728; f) = F(g) &#8728; 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 <b>C</b> to itself is denoted `1C`.
 </pre>
 
 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`.


@@ -92,60 +104,77 @@ 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.
 
 ----------------------
 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 <b>C</b> to category <b>D</b>, a natural transformation &eta; between `G` and `H` is a family of morphisms &eta;[C1]:G(C1)&rarr;H(C1)` in <b>D</b> for each element `C1` of <b>C</b>. That is, &eta;[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:
+Where `G` and `H` are functors from category <b>C</b> to category <b>D</b>, a natural transformation &eta; between `G` and `H` is a family of morphisms <code>&eta;[C1]:G(C1)&rarr;H(C1)</code> in <b>D</b> for each element `C1` of <b>C</b>. That is, <code>&eta;[C1]</code> 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&rarr;C2 in <b>C</b>: &eta;[C2] o G(f) = H(f) o &eta;[C1]
+<pre>
+       for every morphism f:C1&rarr;C2 in <b>C</b>:
+       &eta;[C2] &#8728; G(f) = H(f) &#8728; &eta;[C1]
+</pre>

 
 

-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 <code>&eta;[C2]</code> to `H(C2)`, is identical to the morphism from `G(C1)` via <code>&eta;[C1]</code> to `H(C1)`, and then via `H(f)` from `H(C1)` to `H(C2)`.

 
 
 How natural transformations compose:
 
 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 &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:
 
 
 
 How natural transformations compose:
 
 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 &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:
 

+<pre>

        - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
                 | |        | |            | |
        - <b>B</b> -+ +--- <b>C</b> --+ +---- <b>D</b> -----+ +-- <b>E</b> --
                 | |        | |            | |

-        F: -----&rarr; G: -----&rarr;     K: -----&rarr;
-                | |        | |  | &eta;     | |  | &psi;
+        F: ------> G: ------>     K: ------>
+                | |        | |  | &eta;       | |  | &psi;

                 | |        | |  v         | |  v
                 | |        | |  v         | |  v

-                | |    H: -----&rarr;     L: -----&rarr;
-                | |        | |  | &phi;     | |
+                | |    H: ------>     L: ------>
+                | |        | |  | &phi;       | |

                 | |        | |  v         | |
                 | |        | |  v         | |

-                | |    J: -----&rarr;         | |
+                | |    J: ------>         | |

        -----+ +--------+ +------------+ +-------
        -----+ +--------+ +------------+ +-------

+</pre>

 
 

-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>B</b>, `(&eta; F)[b1] = &eta;[F(b1)]`---that is, the morphism in <b>D</b> that &eta; assigns to the element `F(b1)` of <b>C</b>.
+Then <code>(&eta; F)</code> 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>, <code>(&eta; F)[B1] = &eta;[F(B1)]</code>---that is, the morphism in <b>D</b> that <code>&eta;</code> assigns to the element `F(B1)` of <b>C</b>.

 
 

-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 <b>C</b>, `(K &eta;)[C1] = K(&eta;[C1])`---that is, the morphism in <b>E</b> that `K` assigns to the morphism &eta;[C1]` of <b>D</b>.
+And <code>(K &eta;)</code> 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>, <code>(K &eta;)[C1] = K(&eta;[C1])</code>---that is, the morphism in <b>E</b> that `K` assigns to the morphism <code>&eta;[C1]</code> of <b>D</b>.

 
 
 
 

-`(&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&rarr;C2`:
+<code>(&phi; -v- &eta;)</code> is a natural transformation from `G` to `J`; this is known as a "vertical composition". We will rely later on this, where <code>f:C1&rarr;C2</code>:

 
 

-       &phi;[C2] o H(f) o &eta;[C1] = &phi;[C2] o H(f) o &eta;[C1]
+<pre>
+       &phi;[C2] &#8728; H(f) &#8728; &eta;[C1] = &phi;[C2] &#8728; H(f) &#8728; &eta;[C1]
+</pre>

 
 

-by naturalness of &phi;, is:
+by naturalness of <code>&phi;</code>, is:

 
 

-       &phi;[C2] o H(f) o &eta;[C1] = J(f) o &phi;[C1] o &eta;[C1]
+<pre>
+       &phi;[C2] &#8728; H(f) &#8728; &eta;[C1] = J(f) &#8728; &phi;[C1] &#8728; &eta;[C1]
+</pre>

 
 

-by naturalness of &eta;, is:
+by naturalness of <code>&eta;</code>, is:

 
 

-       &phi;[C2] o &eta;[C2] o G(f) = J(f) o &phi;[C1] o &eta;[C1]
+<pre>
+       &phi;[C2] &#8728; &eta;[C2] &#8728; G(f) = J(f) &#8728; &phi;[C1] &#8728; &eta;[C1]
+</pre>

 
 

-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 <code>(&phi; -v- &eta;)[\_]</code> as: <code>&phi;[\_] &#8728; &eta;[\_]</code> 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]
+<pre>
+       (&phi; -v- &eta;)[C2] &#8728; G(f) = J(f) &#8728; (&phi; -v- &eta;)[C1]
+</pre>

 
 An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that:
 
 
 An observation we'll rely on later: given the definitions of vertical composition and of how natural transformations compose with functors, it follows that:
 

+<pre>

        ((&phi; -v- &eta;) F) = ((&phi; F) -v- (&eta; F))
        ((&phi; -v- &eta;) F) = ((&phi; F) -v- (&eta; F))

+</pre>

 
 I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation."
 
 
 
 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:
+<code>(&psi; -h- &eta;)</code> 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])
+<pre>
+       (&phi; -h- &eta;)[C1]  =  L(&eta;[C1]) &#8728; &psi;[G(C1)]
+                                 =  &psi;[H(C1)] &#8728; K(&eta;[C1])
+</pre>

 
 Horizontal composition is also associative, and has the same identity as vertical composition.
 
 
 Horizontal composition is also associative, and has the same identity as vertical composition.
 


@@ -218,14 +247,14 @@ 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 <b>C</b> has an identity morphism 1<sub>C1</sub> such that for every morphism f:C1&rarr;C2 in <b>C</b>: 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&rarr;C2 in <b>C</b>: 1<sub>C2</sub> &#8728; f = f = f &#8728; 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
        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 &eta; is a natural transformation from F to G, then for every f:C1&rarr;C2 in F and G's source category <b>C</b>: &eta;[C2] o F(f) = G(f) o &eta;[C1].
+       * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g &#8728; f) = F(g) &#8728; F(f)
+       * if &eta; is a natural transformation from F to G, then for every f:C1&rarr;C2 in F and G's source category <b>C</b>: &eta;[C2] &#8728; F(f) = G(f) &#8728; &eta;[C1].

 
 
 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.
 
 
 Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas.


@@ -233,37 +262,37 @@ Let's use the definitions of naturalness, and of composition of natural transfor
 
 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&rarr;b in <b>C</b>:
 
 
 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&rarr;b in <b>C</b>:
 

-       (1) join[b] o MM(f)  =  M(f) o join[a]
+       (1) join[b] &#8728; MM(f)  =  M(f) &#8728; 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 <b>C</b> a morphism q*: Q(C1) &rarr; 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:
 
 Next, consider the composite transformation ((join MQ') -v- (MM q)).
        q is a transformation from Q to MQ', and assigns elements C1 in <b>C</b> a morphism q*: Q(C1) &rarr; 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)] &#8728; MM(q*).

 
 Next, consider the composite transformation ((M q) -v- (join 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*) &#8728; join[Q(C1)].

 
 So for every element C1 of <b>C</b>:
        ((join MQ') -v- (MM q))[C1], by (2) is:
 
 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)&rarr;MQ'(C1) is:
-       M(q*) o join[Q(C1)], which by 3 is:
+       join[MQ'(C1)] &#8728; MM(q*), which by (1), with f=q*: Q(C1)&rarr;MQ'(C1) is:
+       M(q*) &#8728; 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 <b>C</b>, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a&rarr;b in <b>C</b>:
        ((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 <b>C</b>, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a&rarr;b in <b>C</b>:

-       (4) unit[b] o f = M(f) o unit[a]
+       (4) unit[b] &#8728; f = M(f) &#8728; 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*) &#8728; 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)] &#8728; q*.

 
 So for every element C1 of <b>C</b>:
        ((M q) -v- (unit Q))[C1], by (5) =
 
 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)&rarr;MQ'(C1) is:
-       unit[MQ'(C1)] o q*, which by (6) =
+       M(q*) &#8728; unit[Q(C1)], which by (4), with f=q*: Q(C1)&rarr;MQ'(C1) is:
+       unit[MQ'(C1)] &#8728; 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'.
        ((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'.


@@ -369,7 +398,7 @@ A monad M will consist of a mapping from types C1 to types M(C1), and a mapping
 
 
 A natural transformation t assigns to each type C1 in <b>C</b> a morphism t[C1]: C1&rarr;M(C1) such that, for every f:C1&rarr;C2:
 
 
 A natural transformation t assigns to each type C1 in <b>C</b> a morphism t[C1]: C1&rarr;M(C1) such that, for every f:C1&rarr;C2:

-       t[C2] o f = M(f) o t[C1]
+       t[C2] &#8728; f = M(f) &#8728; 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).