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=29b6feb3e250ad195f8884215570afa5fe639ecf;hp=8c5f4cd9b256f39c32dccf112098162a5f9fa78e;hb=c4eb20ae862369e97cadef43183d0663f3eddd11;hpb=3ca93ad8ccdf0572c3b803b86ea68dd2ad21a5f2 diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 8c5f4cd9..29b6feb3 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -82,12 +82,11 @@ A **functor** is a "homomorphism", that is, a structure-preserving mapping, betw 
 	  (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
+	 (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).
+	      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)
@@ -105,60 +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: -----→
-		 | |        | |  | η     | |  | ψ
+	 F: ------> G: ------>     K: ------>
+		 | |        | |  | η       | |  | ψ
 		 | |        | |  v         | |  v
-		 | |    H: -----→     L: -----→
-		 | |        | |  | φ     | |
+		 | |    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)[\_] as: φ[\_] ∘ η[\_] 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])
+				  =  ψ[H(C1)] ∘ K(η[C1])
+

 
 Horizontal composition is also associative, and has the same identity as vertical composition.