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 4941e80..f05747f 100644 (file)
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -33,8 +33,8 @@ Some examples of monoids are:
 
 *      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`
 
 *      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.
+*      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
 ----------
 
 Categories
 ----------


@@ -58,7 +58,7 @@ To have a category, the elements and morphisms have to satisfy some constraints:
 
 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 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. Morphisms correspond to directed paths of length &ge; 0 in the graph.

 
 
 Some examples of categories are:
 
 
 Some examples of categories are:


@@ -67,7 +67,7 @@ Some examples of categories are:
 
 *      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`.
 
 
 *      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 <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:
+*      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 `s1`,`s2` of `S` such that neither <code>s1&le;s2</code> nor <code>s2&le;s1</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)


@@ -136,7 +136,7 @@ Then <code>(&eta; F)</code> is a natural transformation from the (composite) fun
 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>.
 
 
 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>.
 
 

-<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>:
+<code>(&phi; -v- &eta;)</code> is a natural transformation from `G` to `J`; this is known as a "vertical composition". For any morphism <code>f:C1&rarr;C2</code> in <b>C</b>:

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


@@ -186,22 +186,20 @@ In earlier days, these were also called "triples."
 
 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.
 
 
 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 <code>&phi;</code>, each being between some (variable) functor `F` and another functor which is the composite `MF'` of `M` and a (variable) functor `F'`. That is, for each element `C1` in <b>C</b>, <code>&phi;</code> assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is, <code>&phi;</code> is a transformation from functor `F` to `MF'`, <code>&gamma;</code> is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` need be the same.
+Let `T` be a set of natural transformations <code>&phi;</code>, each being between some arbitrary endofunctor `F` on <b>C</b> and another functor which is the composite `MF'` of `M` and another arbitrary endofunctor `F'` on <b>C</b>. That is, for each element `C1` in <b>C</b>, <code>&phi;</code> assigns `C1` a morphism from element `F(C1)` to element `MF'(C1)`, satisfying the constraints detailed in the previous section. For different members of `T`, the relevant functors may differ; that is, <code>&phi;</code> is a transformation from functor `F` to `MF'`, <code>&gamma;</code> is a transformation from functor `G` to `MG'`, and none of `F`, `F'`, `G`, `G'` 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 <b>C</b> 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`.
+We also need to designate for `M` a `join` transformation, which is a natural transformation from the (composite) functor `MM` to `M`.

 
 These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
 
 
 These two natural transformations have to satisfy some constraints ("the monad laws") which are most easily stated if we can introduce a defined notion.
 

-Let <code>&phi;</code> and <code>&gamma;</code> be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now <code>(M &gamma;)</code> will also be a natural transformation, formed by composing the functor `M` with the natural transformation <code>&gamma;</code>. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, <code>(M &gamma;)</code>, and <code>&phi;</code>, and abbreviate it as follows:
+Let <code>&phi;</code> and <code>&gamma;</code> be members of `T`, that is they are natural transformations from `F` to `MF'` and from `G` to `MG'`, respectively. Let them be such that `F' = G`. Now <code>(M &gamma;)</code> will also be a natural transformation, formed by composing the functor `M` with the natural transformation <code>&gamma;</code>. Similarly, `(join G')` will be a natural transformation, formed by composing the natural transformation `join` with the functor `G'`; it will transform the functor `MMG'` to the functor `MG'`. Now take the vertical composition of the three natural transformations `(join G')`, <code>(M &gamma;)</code>, and <code>&phi;</code>, and abbreviate it as follows. Since composition is associative I don't specify the order of composition on the rhs.

 
 <pre>
        &gamma; <=< &phi;  =def.  ((join G') -v- (M &gamma;) -v- &phi;)
 </pre>
 
 
 <pre>
        &gamma; <=< &phi;  =def.  ((join G') -v- (M &gamma;) -v- &phi;)
 </pre>
 

-Since composition is associative I don't specify the order of composition on the rhs.
-

 In other words, `<=<` is a binary operator that takes us from two members <code>&phi;</code> and <code>&gamma;</code> of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written <code>&phi; >=> &gamma;</code> where that's the same as <code>&gamma; &lt;=&lt; &phi;</code>.)
 
 <code>&phi;</code> is a transformation from `F` to `MF'`, where the latter = `MG`; <code>(M &gamma;)</code> is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite <code>&gamma; &lt;=&lt; &phi;</code> will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.
 In other words, `<=<` is a binary operator that takes us from two members <code>&phi;</code> and <code>&gamma;</code> of `T` to a composite natural transformation. (In functional programming, at least, this is called the "Kleisli composition operator". Sometimes it's written <code>&phi; >=> &gamma;</code> where that's the same as <code>&gamma; &lt;=&lt; &phi;</code>.)
 
 <code>&phi;</code> is a transformation from `F` to `MF'`, where the latter = `MG`; <code>(M &gamma;)</code> is a transformation from `MG` to `MMG'`; and `(join G')` is a transformation from `MMG'` to `MG'`. So the composite <code>&gamma; &lt;=&lt; &phi;</code> will be a transformation from `F` to `MG'`, and so also eligible to be a member of `T`.


@@ -212,48 +210,66 @@ Now we can specify the "monad laws" governing a monad as follows:
        (T, <=<, unit) constitute a monoid
 </pre>
 
        (T, <=<, unit) constitute a monoid
 </pre>
 

-That's it. Well, there may be a wrinkle here.
-
-I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, <code>&gamma; <=< &phi;</code> isn't fully defined on `T`, but only when <code>&phi;</code> is a transformation to some `MF'` and <code>&gamma;</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws are satisfied:
+That's it. Well, there may be a wrinkle here. I don't know whether the definition of a monoid requires the operation to be defined for every pair in its set. In the present case, <code>&gamma; &lt;=&lt; &phi;</code> isn't fully defined on `T`, but only when <code>&phi;</code> is a transformation to some `MF'` and <code>&gamma;</code> is a transformation from `F'`. But wherever `<=<` is defined, the monoid laws must hold:

 
 <pre>
            (i) &gamma; <=< &phi; is also in T
 
           (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
 
 
 <pre>
            (i) &gamma; <=< &phi; is also in T
 
           (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
 

-       (iii.1) unit <=< &phi;  =  &phi;                 (here &phi; has to be a natural transformation to M(1C))
+       (iii.1) unit <=< &phi;  =  &phi;
+               (here &phi; has to be a natural transformation to M(1C))

 
 

-       (iii.2)                &phi;  =  &phi; <=< unit  (here &phi; has to be a natural transformation from 1C)
+       (iii.2)                &rho;  =  &rho; <=< unit
+               (here &rho; has to be a natural transformation from 1C)

 </pre>
 
 </pre>
 

-If <code>&phi;</code> is a natural transformation from `F` to `M(1C)` and <code>&gamma;</code> is <code>(&phi; G')</code>, that is, a natural transformation from `FG` to `MG`, then we can extend (iii.1) as follows:
+If <code>&phi;</code> is a natural transformation from `F` to `M(1C)` and <code>&gamma;</code> is <code>(&phi; G')</code>, that is, a natural transformation from `FG'` to `MG'`, then we can extend (iii.1) as follows:

 
 <pre>
        &gamma; = (&phi; G')
          = ((unit <=< &phi;) G')
 
 <pre>
        &gamma; = (&phi; G')
          = ((unit <=< &phi;) G')

-         = ((join -v- (M unit) -v- &phi;) G')
-         = (join G') -v- ((M unit) G') -v- (&phi; G')
-         = (join G') -v- (M (unit G')) -v- &gamma;
-         ??
+         = (((join 1C) -v- (M unit) -v- &phi;) G')
+         = (((join 1C) G') -v- ((M unit) G') -v- (&phi; G'))
+         = ((join (1C G')) -v- (M (unit G')) -v- &gamma;)
+         = ((join G') -v- (M (unit G')) -v- &gamma;)
+         since (unit G') is a natural transformation to MG',
+         this satisfies the definition for &lt;=&lt;:

          = (unit G') <=< &gamma;
 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from some `FG'` to `MG'`.
 
          = (unit G') <=< &gamma;
 </pre>
 
 where as we said <code>&gamma;</code> is a natural transformation from some `FG'` to `MG'`.
 

-Similarly, if <code>&phi;</code> is a natural transformation from `1C` to `MF'`, and <code>&gamma;</code> is <code>(&phi; G)</code>, that is, a natural transformation from `G` to `MF'G`, then we can extend (iii.2) as follows:
+Similarly, if <code>&rho;</code> is a natural transformation from `1C` to `MR'`, and <code>&gamma;</code> is <code>(&rho; G)</code>, that is, a natural transformation from `G` to `MR'G`, then we can extend (iii.2) as follows:

 
 <pre>
 
 <pre>

-       &gamma; = (&phi; G)
-         = ((&phi; <=< unit) G)
-         = (((join F') -v- (M &phi;) -v- unit) G)
-         = ((join F'G) -v- ((M &phi;) G) -v- (unit G))
-         = ((join F'G) -v- (M (&phi; G)) -v- (unit G))
-         ??
+       &gamma; = (&rho; G)
+         = ((&rho; <=< unit) G)
+         = (((join R') -v- (M &rho;) -v- unit) G)
+         = (((join R') G) -v- ((M &rho;) G) -v- (unit G))
+         = ((join (R'G)) -v- (M (&rho; G)) -v- (unit G))
+         since &gamma; = (&rho; G) is a natural transformation to MR'G,
+         this satisfies the definition &lt;=&lt;:

          = &gamma; <=< (unit G)
 </pre>
 
          = &gamma; <=< (unit G)
 </pre>
 

-where as we said <code>&gamma;</code> is a natural transformation from `G` to some `MF'G`.
+where as we said <code>&gamma;</code> is a natural transformation from `G` to some `MR'G`.

 
 

+Summarizing then, the monad laws can be expressed as:
+
+<pre>
+       For all &rho;, &gamma;, &phi; in T for which &rho; <=< &gamma; and &gamma; <=< &phi; are defined:
+
+           (i) &gamma; <=< &phi; etc are also in T
+
+          (ii) (&rho; <=< &gamma;) <=< &phi;  =  &rho; <=< (&gamma; <=< &phi;)
+
+       (iii.1) (unit G') <=< &gamma;  =  &gamma;
+               when &gamma; is a natural transformation from some FG' to MG'
+
+       (iii.2)                     &gamma;  =  &gamma; <=< (unit G)
+               when &gamma; is a natural transformation from G to some MR'G
+</pre>

 
 
 
 
 
 


@@ -272,40 +288,47 @@ Let's remind ourselves of some principles:
 
 *      functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: <code>F(g &#8728; f) = F(g) &#8728; F(f)</code>
 
 
 *      functors "distribute over composition", that is for any morphisms `f` and `g` in `F`'s source category: <code>F(g &#8728; f) = F(g) &#8728; F(f)</code>
 

-*      if <code>&eta;</code> is a natural transformation from `F` to `G`, then for every <code>f:C1&rarr;C2</code> in `F` and `G`'s source category <b>C</b>: <code>&eta;[C2] &#8728; F(f) = G(f) &#8728; &eta;[C1]</code>.
+*      if <code>&eta;</code> is a natural transformation from `G` to `H`, then for every <code>f:C1&rarr;C2</code> in `G` and `H`'s source category <b>C</b>: <code>&eta;[C2] &#8728; G(f) = H(f) &#8728; &eta;[C1]</code>.
+
+*      <code>(&eta; F)[E] = &eta;[F(E)]</code> 
+
+*      <code>(K &eta;)[E} = K(&eta;[E])</code>
+
+*      <code>((&phi; -v- &eta;) F) = ((&phi; F) -v- (&eta; F))</code>

 
 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.
 
 

-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 <code>f:C1&rarr;C2</code> 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 <code>f:C1&rarr;C2</code> in <b>C</b>:

 
 <pre>
        (1) join[C2] &#8728; MM(f)  =  M(f) &#8728; join[C1]
 </pre>
 
 
 <pre>
        (1) join[C2] &#8728; MM(f)  =  M(f) &#8728; join[C1]
 </pre>
 

-Next, consider the composite transformation <code>((join MG') -v- (MM &gamma;))</code>.
+Next, let <code>&gamma;</code> be a transformation from `G` to `MG'`, and
+ consider the composite transformation <code>((join MG') -v- (MM &gamma;))</code>.

 
 

-*      <code>&gamma;</code> is a transformation from `G` to `MG'`, and assigns elements `C1` in <b>C</b> a morphism <code>&gamma;\*: G(C1) &rarr; MG'(C1)</code>. <code>(MM &gamma;)</code> is a transformation that instead assigns `C1` the morphism <code>MM(&gamma;\*)</code>.
+*      <code>&gamma;</code> assigns elements `C1` in <b>C</b> a morphism <code>&gamma;\*: G(C1) &rarr; MG'(C1)</code>. <code>(MM &gamma;)</code> is a transformation that instead assigns `C1` the morphism <code>MM(&gamma;\*)</code>.

 
 

-*      `(join MG')` is a transformation from `MMMG'` to `MMG'` that assigns `C1` the morphism `join[MG'(C1)]`.
+*      `(join MG')` is a transformation from `MM(MG')` to `M(MG')` that assigns `C1` the morphism `join[MG'(C1)]`.

 
 Composing them:
 
 <pre>
 
 Composing them:
 
 <pre>

-       (2) ((join MG') -v- (MM &gamma;)) assigns to `C1` the morphism join[MG'(C1)] &#8728; MM(&gamma;*).
+       (2) ((join MG') -v- (MM &gamma;)) assigns to C1 the morphism join[MG'(C1)] &#8728; MM(&gamma;*).

 </pre>
 
 </pre>
 

-Next, consider the composite transformation <code>((M &gamma;) -v- (join G))</code>.
+Next:

 
 <pre>
 
 <pre>

-       (3) This assigns to C1 the morphism M(&gamma;*) &#8728; join[G(C1)].
+       (3) Consider the composite transformation <code>((M &gamma;) -v- (join G))</code>. This assigns to C1 the morphism M(&gamma;*) &#8728; join[G(C1)].

 </pre>
 
 So for every element `C1` of <b>C</b>:
 
 <pre>
        ((join MG') -v- (MM &gamma;))[C1], by (2) is:
 </pre>
 
 So for every element `C1` of <b>C</b>:
 
 <pre>
        ((join MG') -v- (MM &gamma;))[C1], by (2) is:

-       join[MG'(C1)] &#8728; MM(&gamma;*), which by (1), with f=&gamma;*: G(C1)&rarr;MG'(C1) is:
+       join[MG'(C1)] &#8728; MM(&gamma;*), which by (1), with f=&gamma;*:G(C1)&rarr;MG'(C1) is:

        M(&gamma;*) &#8728; join[G(C1)], which by 3 is:
        ((M &gamma;) -v- (join G))[C1]
 </pre>
        M(&gamma;*) &#8728; join[G(C1)], which by 3 is:
        ((M &gamma;) -v- (join G))[C1]
 </pre>


@@ -313,33 +336,34 @@ So for every element `C1` of <b>C</b>:
 So our **(lemma 1)** is:
 
 <pre>
 So our **(lemma 1)** is:
 
 <pre>

-       ((join MG') -v- (MM &gamma;))  =  ((M &gamma;) -v- (join G)), where &gamma; is a transformation from G to MG'.
+       ((join MG') -v- (MM &gamma;))  =  ((M &gamma;) -v- (join G)),
+       where as we said &gamma; is a natural transformation from G to MG'.

 </pre>
 
 
 </pre>
 
 

-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 <code>f:a&rarr;b</code> in <b>C</b>:
+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 <code>f:C1&rarr;C2</code> in <b>C</b>:

 
 <pre>
 
 <pre>

-       (4) unit[b] &#8728; f = M(f) &#8728; unit[a]
+       (4) unit[C2] &#8728; f = M(f) &#8728; unit[C1]

 </pre>
 
 </pre>
 

-Next consider the composite transformation <code>((M &gamma;) -v- (unit G))</code>:
+Next:

 
 <pre>
 
 <pre>

-       (5) This assigns to C1 the morphism M(&gamma;*) &#8728; unit[G(C1)].
+       (5) Consider the composite transformation ((M &gamma;) -v- (unit G)). This assigns to C1 the morphism M(&gamma;*) &#8728; unit[G(C1)].

 </pre>
 
 </pre>
 

-Next consider the composite transformation <code>((unit MG') -v- &gamma;)</code>.
+Next:

 
 <pre>
 
 <pre>

-       (6) This assigns to C1 the morphism unit[MG'(C1)] &#8728; &gamma;*.
+       (6) Consider the composite transformation ((unit MG') -v- &gamma;). This assigns to C1 the morphism unit[MG'(C1)] &#8728; &gamma;*.

 </pre>
 
 So for every element C1 of <b>C</b>:
 
 <pre>
        ((M &gamma;) -v- (unit G))[C1], by (5) =
 </pre>
 
 So for every element C1 of <b>C</b>:
 
 <pre>
        ((M &gamma;) -v- (unit G))[C1], by (5) =

-       M(&gamma;*) &#8728; unit[G(C1)], which by (4), with f=&gamma;*: G(C1)&rarr;MG'(C1) is:
+       M(&gamma;*) &#8728; unit[G(C1)], which by (4), with f=&gamma;*:G(C1)&rarr;MG'(C1) is:

        unit[MG'(C1)] &#8728; &gamma;*, which by (6) =
        ((unit MG') -v- &gamma;)[C1]
 </pre>
        unit[MG'(C1)] &#8728; &gamma;*, which by (6) =
        ((unit MG') -v- &gamma;)[C1]
 </pre>


@@ -347,11 +371,12 @@ So for every element C1 of <b>C</b>:
 So our **(lemma 2)** is:
 
 <pre>
 So our **(lemma 2)** is:
 
 <pre>

-       (((M &gamma;) -v- (unit G))  =  ((unit MG') -v- &gamma;)), where &gamma; is a transformation from G to MG'.
+       (((M &gamma;) -v- (unit G))  =  ((unit MG') -v- &gamma;)),
+       where as we said &gamma; is a natural transformation from G to MG'.

 </pre>
 
 
 </pre>
 
 

-Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <code>&gamma; <=< &phi;</code> in the monad laws. For simplicity, I'll omit the "-v-".
+Finally, we substitute <code>((join G') -v- (M &gamma;) -v- &phi;)</code> for <code>&gamma; &lt;=&lt; &phi;</code> in the monad laws. For simplicity, I'll omit the "-v-".

 
 <pre>
        for all &phi;,&gamma;,&rho; in T, where &phi; is a transformation from F to MF', &gamma; is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R:
 
 <pre>
        for all &phi;,&gamma;,&rho; in T, where &phi; is a transformation from F to MF', &gamma; is a transformation from G to MG', R is a transformation from R to MR', and F'=G and G'=R: