From: Jim Pryor Date: Tue, 2 Nov 2010 12:19:57 +0000 (-0400) Subject: cat theory tweaks X-Git-Url: http://lambda.jimpryor.net/git/gitweb.cgi?p=lambda.git;a=commitdiff_plain;h=440829c5f3907c6274fd03218c50cacaf3a81f3e cat theory tweaks Signed-off-by: Jim Pryor --- diff --git a/advanced_topics/monads_in_category_theory.mdwn b/advanced_topics/monads_in_category_theory.mdwn index f26e2cd6..53a66112 100644 --- a/advanced_topics/monads_in_category_theory.mdwn +++ b/advanced_topics/monads_in_category_theory.mdwn @@ -45,9 +45,9 @@ When a morphism `f` in category C has source `C1` and target `C2`, we'll 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 º f is also a morphism of the category, which maps C1→C3.
+	(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 o f = f = f o 1C1
+	(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. @@ -77,7 +77,7 @@ 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 (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 o f) = F(g) o F(f) + (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`. @@ -94,7 +94,7 @@ So categories include elements and morphisms. Functors consist of mappings from 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] o G(f) = H(f) o 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)`. @@ -121,19 +121,19 @@ And `(K η)` is a natural transformation from the (composite) functor `KG` to `(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] o H(f) o η[C1] = φ[C2] o H(f) o η[C1] + φ[C2] ∘ H(f) ∘ η[C1] = φ[C2] ∘ H(f) ∘ η[C1] by naturalness of φ, is: - φ[C2] o H(f) o η[C1] = J(f) o φ[C1] o η[C1] + φ[C2] ∘ H(f) ∘ η[C1] = J(f) ∘ φ[C1] ∘ η[C1] by naturalness of η, is: - φ[C2] o η[C2] o G(f) = J(f) o φ[C1] o η[C1] + φ[C2] ∘ η[C2] ∘ G(f) = J(f) ∘ φ[C1] ∘ η[C1] -Hence, we can define `(φ -v- η)[x]` as: φ[x] o 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] o G(f) = J(f) o (φ -v- η)[C1] + (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: @@ -144,8 +144,8 @@ I'll assert without proving that vertical composition is associative and has an `(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]) o ψ[G(C1)] - = ψ[H(C1)] o K(η[C1]) + (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. @@ -218,14 +218,14 @@ The standard category-theory presentation of the monad laws In category theory, the monad laws are usually stated in terms of `unit` and `join` instead of `unit` and `<=<`. (* - P2. every element C1 of a category C has an identity morphism 1C1 such that for every morphism f:C1→C2 in C: 1C2 o f = f = f o 1C1. + P2. every element C1 of a category C has an identity morphism 1C1 such that for every morphism f:C1→C2 in C: 1C2 ∘ f = f = f ∘ 1C1. P3. functors "preserve identity", that is for every element C1 in F's source category: F(1C1) = 1F(C1). *) Let's remind ourselves of some principles: * composition of morphisms, functors, and natural compositions is associative - * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g o f) = F(g) o F(f) - * if η is a natural transformation from F to G, then for every f:C1→C2 in F and G's source category C: η[C2] o F(f) = G(f) o η[C1]. + * functors "distribute over composition", that is for any morphisms f and g in F's source category: F(g ∘ f) = F(g) ∘ F(f) + * if η is a natural transformation from F to G, then for every f:C1→C2 in F and G's source category C: η[C2] ∘ F(f) = G(f) ∘ η[C1]. Let's use the definitions of naturalness, and of composition of natural transformations, to establish two lemmas. @@ -233,37 +233,37 @@ Let's use the definitions of naturalness, and of composition of natural transfor Recall that join is a natural transformation from the (composite) functor MM to M. So for elements C1 in C, join[C1] will be a morphism from MM(C1) to M(C1). And for any morphism f:a→b in C: - (1) join[b] o MM(f) = M(f) o join[a] + (1) join[b] ∘ MM(f) = M(f) ∘ join[a] Next, consider the composite transformation ((join MQ') -v- (MM q)). q is a transformation from Q to MQ', and assigns elements C1 in C a morphism q*: Q(C1) → MQ'(C1). (MM q) is a transformation that instead assigns C1 the morphism MM(q*). (join MQ') is a transformation from MMMQ' to MMQ' that assigns C1 the morphism join[MQ'(C1)]. Composing them: - (2) ((join MQ') -v- (MM q)) assigns to C1 the morphism join[MQ'(C1)] o MM(q*). + (2) ((join MQ') -v- (MM q)) assigns to C1 the morphism join[MQ'(C1)] ∘ MM(q*). Next, consider the composite transformation ((M q) -v- (join Q)). - (3) This assigns to C1 the morphism M(q*) o join[Q(C1)]. + (3) This assigns to C1 the morphism M(q*) ∘ join[Q(C1)]. So for every element C1 of C: ((join MQ') -v- (MM q))[C1], by (2) is: - join[MQ'(C1)] o MM(q*), which by (1), with f=q*: Q(C1)→MQ'(C1) is: - M(q*) o join[Q(C1)], which by 3 is: + join[MQ'(C1)] ∘ MM(q*), which by (1), with f=q*: Q(C1)→MQ'(C1) is: + M(q*) ∘ join[Q(C1)], which by 3 is: ((M q) -v- (join Q))[C1] So our (lemma 1) is: ((join MQ') -v- (MM q)) = ((M q) -v- (join Q)), where q is a transformation from Q to MQ'. Next recall that unit is a natural transformation from 1C to M. So for elements C1 in C, unit[C1] will be a morphism from C1 to M(C1). And for any morphism f:a→b in C: - (4) unit[b] o f = M(f) o unit[a] + (4) unit[b] ∘ f = M(f) ∘ unit[a] -Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to C1 the morphism M(q*) o unit[Q(C1)]. +Next consider the composite transformation ((M q) -v- (unit Q)). (5) This assigns to C1 the morphism M(q*) ∘ unit[Q(C1)]. -Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to C1 the morphism unit[MQ'(C1)] o q*. +Next consider the composite transformation ((unit MQ') -v- q). (6) This assigns to C1 the morphism unit[MQ'(C1)] ∘ q*. So for every element C1 of C: ((M q) -v- (unit Q))[C1], by (5) = - M(q*) o unit[Q(C1)], which by (4), with f=q*: Q(C1)→MQ'(C1) is: - unit[MQ'(C1)] o q*, which by (6) = + M(q*) ∘ unit[Q(C1)], which by (4), with f=q*: Q(C1)→MQ'(C1) is: + unit[MQ'(C1)] ∘ q*, which by (6) = ((unit MQ') -v- q)[C1] So our lemma (2) is: (((M q) -v- (unit Q)) = ((unit MQ') -v- q)), where q is a transformation from Q to MQ'. @@ -369,7 +369,7 @@ A monad M will consist of a mapping from types C1 to types M(C1), and a mapping A natural transformation t assigns to each type C1 in C a morphism t[C1]: C1→M(C1) such that, for every f:C1→C2: - t[C2] o f = M(f) o t[C1] + t[C2] ∘ f = M(f) ∘ t[C1] The composite morphisms said here to be identical are morphisms from the type C1 to the type M(C2).