X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=blobdiff_plain;f=advanced_topics%2Fmonads_in_category_theory.mdwn;h=1590619a865d902c81862d5f6abdeb5c3d1a1940;hp=53a66112dbac9639b9f3b9bc78f364f2d330120c;hb=22d14fe68168c87b96699431b381ef6dcb816b4e;hpb=440829c5f3907c6274fd03218c50cacaf3a81f3e diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 53a66112..1590619a 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -19,35 +19,41 @@ corrections. 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 (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: 
 	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
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 ∘ 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 ∘ 1C1
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. @@ -59,7 +65,7 @@ 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. -* 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 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: