From 7026bf51a9211af0586d30d49b61ed187f897a79 Mon Sep 17 00:00:00 2001 From: Jim Pryor Date: Tue, 2 Nov 2010 08:43:05 -0400 Subject: [PATCH] cat theory tweaks Signed-off-by: Jim Pryor --- advanced_topics/monads_in_category_theory.mdwn | 37 +++++++++++++++++++------- 1 file changed, 27 insertions(+), 10 deletions(-) diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index 078bec2f..3e34f85b 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -104,17 +104,21 @@ 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: -----→
@@ -125,39 +129,52 @@ Consider four categories B, C, D, and E. Let `F` be
 		 | |        | |  v         | |
 		 | |    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. -- 2.11.0