X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?a=blobdiff_plain;ds=sidebyside;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=07977f29b4328b4f46a2d3e5efda8f227bba1d11;hb=9e80b25daf1427ebcde2715e35de10dc1a2dfa78;hp=666257667dd6849677dcc428b8b30b6f1f3cffa8;hpb=5b391a18cbbaa7234a3f84e47bb8cc8ac0babc01;p=lambda.git
diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn
index 66625766..07977f29 100644
--- a/advanced_topics/monads_in_category_theory.mdwn
+++ b/advanced_topics/monads_in_category_theory.mdwn
@@ -24,15 +24,15 @@ A **monoid** is a structure (S,⋆,z)
consisting of an associat
for all s1, s2, s3 in S: - (i) s1⋆s2 etc are also in S - (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) + (i) s1⋆s2 etc are also in S + (ii) (s1⋆s2)⋆s3 = s1⋆(s2⋆s3) (iii) z⋆s1 = s1 = s1⋆zSome 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`
+* 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.
@@ -40,14 +40,20 @@ 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 ∘ 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 1E, which is such that for every morphism f:C1→C2: 1C2 ∘ f = f = f ∘ 1C1 + (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 1E, which is such that for every morphism f:C1→C2: + 1C2 ∘ f = f = f ∘ 1C1These 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: -* 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`.
-* 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 (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:
* 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 C to category D must:
- (i) associate with every element C1 of C an element F(C1) of D - (ii) associate with every morphism f:C1→C2 of C a morphism F(f):F(C1)→F(C2) of D - (iii) "preserve identity", that is, for every element C1 of C: F of C1's identity morphism in C must be the identity morphism of F(C1) in D: F(1C1) = 1F(C1). - (iv) "distribute over composition", that is for any morphisms f and g in C: F(g ∘ f) = F(g) ∘ 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(1C1) = 1F(C1). + + (iv) "distribute over composition", that is for any morphisms f and g in C: + F(g ∘ f) = F(g) ∘ 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`. @@ -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. -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 C to category D, a natural transformation η between `G` and `H` is a family of morphisms
η[C1]:G(C1)→H(C1)
in D for each element `C1` of C. That is, η[C1]
has as source `C1`'s image under `G` in D, and as target `C1`'s image under `H` in D. The morphisms in this family must also satisfy the constraint:
- for every morphism f:C1→C2 in C: η[C2] ∘ G(f) = H(f) ∘ η[C1]
++ for every morphism f:C1→C2 in C: + 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)`. +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:
+- B -+ +--- C --+ +---- D -----+ +-- E -- | | | | | | - 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, (η F)[B1] = η[F(B1)]
---that is, the morphism in D that η
assigns to the element `F(B1)` of C.
-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 C, (K η)[C1] = K(η[C1])
---that is, the morphism in E that `K` assigns to the morphism η[C1]
of D.
-`(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] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] +-by naturalness of φ, is: +by naturalness of
φ
, is:
+φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1] +-by naturalness of η, is: +by naturalness of
η
, is:
+φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1] +-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`: +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] ∘ 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: +
((φ -v- η) F) = ((φ F) -v- (η F)) +I'll assert without proving that vertical composition is associative and has an identity, which we'll call "the identity transformation." -`(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: +
(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]) ∘ ψ[G(C1)] = ψ[H(C1)] ∘ K(η[C1]) +Horizontal composition is also associative, and has the same identity as vertical composition.