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annotate comparison-em.agda @ 122:f8fbd5ecec97

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no yellow on em-category
author Shinji KONO <kono@ie.u-ryukyu.ac.jp>
date Fri, 02 Aug 2013 08:21:32 +0900
parents 324511654f23
children 44c58c27d12d
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1 -- -- -- -- -- -- -- --
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2 -- Comparison Functor of Eilenberg-Moore Category
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3 -- defines U^K and F^K as a resolution of Monad
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4 -- checks Adjointness
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5 --
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6 -- Shinji KONO <kono@ie.u-ryukyu.ac.jp>
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7 -- -- -- -- -- -- -- --
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8
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9 open import Category -- https://github.com/konn/category-agda
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10 open import Level
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11 --open import Category.HomReasoning
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12 open import HomReasoning
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13 open import cat-utility
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14 open import Category.Cat
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15 open import Relation.Binary.Core
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16
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17 module comparison-em
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18 { c₁ c₂ ℓ : Level} { A : Category c₁ c₂ ℓ }
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19 { T : Functor A A }
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20 { η : NTrans A A identityFunctor T }
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21 { μ : NTrans A A (T ○ T) T }
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22 { M' : Monad A T η μ }
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23 {c₁' c₂' ℓ' : Level} ( B : Category c₁' c₂' ℓ' )
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24 { U^K : Functor B A } { F^K : Functor A B }
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25 { η^K : NTrans A A identityFunctor ( U^K ○ F^K ) }
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26 { ε^K : NTrans B B ( F^K ○ U^K ) identityFunctor }
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27 { μ^K : NTrans A A (( U^K ○ F^K ) ○ ( U^K ○ F^K )) ( U^K ○ F^K ) }
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28 ( Adj^K : Adjunction A B U^K F^K η^K ε^K )
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29 ( RK : MResolution A B T U^K F^K {η^K} {ε^K} {μ^K} Adj^K )
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30 where
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31
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32 open import adj-monad
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33
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34 T^K = U^K ○ F^K
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35
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36 M : Monad A (U^K ○ F^K ) η^K μ^K
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37 M = Adj2Monad A B {U^K} {F^K} {η^K} {ε^K} Adj^K
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38
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39 open import em-category {c₁} {c₂} {ℓ} {A} { U^K ○ F^K } { η^K } { μ^K } { M }
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40
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41 open Functor
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42 open NTrans
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43 open Adjunction
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44 open MResolution
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45 open Eilenberg-Moore-Hom
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46
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47 emkobj : Obj B -> EMObj
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48 emkobj b = record {
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49 a = FObj U^K b ; phi = A [ FMap U^K o TMap ε^K b ] ; isAlgebra = record { identity = identity1; eval = eval1 }
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50 } where
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51 identity1 : ?
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52 identity1 = ?
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53 eval1 : ?
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54 eval1 = ?
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55
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56 emkmap : {a b : Obj B} (f : Hom B a b) -> EMHom (emkobj a) (emkobj b)
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57 emkmap {a} {b} f = record { EMap = FMap U^K f ; homomorphism = homomorphism1
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58 } where
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59 homomorphism1 : ?
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60 homomorphism1 = ?
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61
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62 K^T : Functor B Eilenberg-MooreCategory
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63 K^T = record {
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64 FObj = emkobj
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65 ; FMap = emkmap
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66 ; isFunctor = record
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67 { ≈-cong = ≈-cong
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68 ; identity = identity
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69 ; distr = distr1
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70 }
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71 } where
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72 identity : {a : Obj A} → B [ emkmap (EM-id {a}) ≈ id1 B (FObj F^K a) ]
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73 identity {a} = let open ≈-Reasoning (B) in
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74 begin
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75 emkmap (EM-id {a})
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76 ≈⟨ ? ⟩
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77 id1 B (FObj F^K a)
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78 ∎
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79 ≈-cong : {a b : Obj A} -> {f g : EMHom a b} → A [ EMap f ≈ EMap g ] → B [ emkmap f ≈ emkmap g ]
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80 ≈-cong {a} {b} {f} {g} f≈g = let open ≈-Reasoning (B) in
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81 begin
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82 emkmap f
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83 ≈⟨ ? ⟩
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84 emkmap g
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85 ∎
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86 distr1 : {a b c : Obj A} {f : EMHom a b} {g : EMHom b c} → B [ emkmap (g ∙ f) ≈ (B [ emkmap g o emkmap f ] )]
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87 distr1 {a} {b} {c} {f} {g} = let open ≈-Reasoning (B) in
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88 begin
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89 emkmap (g ∙ f)
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90 ≈⟨ ? ⟩
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91 emkmap g o emkmap f
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92 ∎
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93