From 9583e70a7aa87c31b70285030ad3e988d1005ef8 Mon Sep 17 00:00:00 2001
From: Jim Pryor
Date: Mon, 1 Nov 2010 20:48:51 0400
Subject: [PATCH] tweak cat theory
Signedoffby: Jim Pryor

advanced_topics/monads_in_category_theory.mdwn  105 +++++++++++++
1 file changed, 53 insertions(+), 52 deletions()
diff git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 41b17c52..3c80bbbc 100644
 a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ 1,7 +1,9 @@
**Don't try to read this yet!!! Many substantial edits are still in process.
Will be ready soon.**
**Caveats**: I really don't know much category theory. Just enough to put this
+Caveats
+
+I really don't know much category theory. Just enough to put this
together. Also, this really is "put together." I haven't yet found an
authoritative source (that's accessible to a category theory beginner like
myself) that discusses the correspondence between the categorytheoretic and
@@ 26,24 +28,25 @@ A **monoid** is a structure `(S, *, z)` consisting of an associative binary oper
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 `*` 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.
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 `C` has source `c1` and target `c2`, we'll write `f:c1>c2`.
To have a category, the elements and morphisms have to satisfy some constraints:
+
(i) the class of morphisms has to be closed under composition: where f:c1>c2 and g:c2>c3, g o f is also a morphism of the category, which maps c1>c3.
(ii) composition of morphisms has to be associative
 (iii) every element e of the category has to have an identity morphism id[e], which is such that for every morphism f:c1>c2:
 id[c2] o f = f = f o id[c1]
+ (iii) every element e of the category has to have an identity morphism id[e], which is such that for every morphism f:c1>c2:
+ id[c2] o f = f = f o id[c1]
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.)
@@ 52,31 +55,28 @@ 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 is the identity function over 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 `(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`.
* 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 antisymmetric (that is, there may be members `s1`,`s2` of `S` such that `s1<=s2` and `s2<=s1` but `s1` and `s2` are not identical).
+* 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 antisymmetric (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:
* sentences ordered by logical implication ("p and p" implies and is implied by "p", but these sentences are not identical; so this illustrates a preorder without antisymmetry)
* sets ordered by size (this illustrates it too)
 Any preorder `(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 preorder `(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`.
3. Functors

A **functor** is a "homomorphism", that is, a structurepreserving mapping, between categories. In particular, a functor F from category C to category D must:
+A **functor** is a "homomorphism", that is, a structurepreserving mapping, between categories. In particular, a functor `F` from category `C` to category `D` must:

 associate with every element c1 of C an element F(c1) of D

 associate with every morphism f:c1>c2 of C a morphism F(f):F(c1)>F(c2) of D

 "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)].

 "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`.
@@ 90,53 +90,54 @@ 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 `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:
+
for every morphism f:c1>c2 in C:
eta[c2] o G(f) = H(f) o eta[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 `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)`.
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 `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:
+
+  B + + C + + D + + E 
+      
+ F: > G: > K: >
+      eta    psi
+     v   v
+   H: > L: >
+      phi  
+     v  
+   J: >  
+ + ++ ++ +
 B + + C + + D + + E 
      
 F: > G: > K: >
      eta    psi
     v   v
   H: > L: >
      phi  
     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 (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.
+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 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.
+`(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`:
(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:
phi[c2] o H(f) o eta[c1] = phi[c2] o H(f) o eta[c1]
 
 by naturalness of phi, is:
 
+
+by naturalness of phi, is:
+
phi[c2] o H(f) o eta[c1] = J(f) o phi[c1] o eta[c1]
 
 by naturalness of eta, is:
 
+
+by naturalness of eta, is:
+
phi[c2] o eta[c2] o G(f) = J(f) o phi[c1] o eta[c1]
  
Hence, we can define (phi v eta)[c1] as: phi[c1] o eta[c1] and rely on it to satisfy the constraints for a natural transformation from G to J:
  
+
+Hence, we can define `(phi v eta)[c1]` as: `phi[c1] o eta[c1]` 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]
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:
+`(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:
(phi h eta)[c1] = L(eta[c1]) o psi[G(c1)]
= psi[H(c1)] o K(eta[c1])
@@ 145,11 +146,11 @@ Horizontal composition is also associative, and has the same identity as vertica
5. Monads

+Monads
+
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.

2.11.0