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=8c5f4cd9b256f39c32dccf112098162a5f9fa78e;hp=666257667dd6849677dcc428b8b30b6f1f3cffa8;hb=3ca93ad8ccdf0572c3b803b86ea68dd2ad21a5f2;hpb=5b391a18cbbaa7234a3f84e47bb8cc8ac0babc01 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 66625766..8c5f4cd9 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⋆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` +* 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 ∘ 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. @@ -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 +80,17 @@ 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`.