CategoryTheory/agda-kleisli
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity

added comments

Browse source
This commit is contained in:
Leon Vatthauer 2023-07-15 12:29:26 +02:00
parent f2cccdbdf0
commit c521bf635a
Signed by: leonv
SSH key fingerprint: SHA256:G4+ddwoZmhLPRB1agvXzZMXIzkVJ36dUYZXf5NxT+u8
1 changed files with 12 additions and 7 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

19
ExtensionSystem.agda
View file

@ -32,8 +32,11 @@ ExtensionSystem→Monad {𝒞 = 𝒞} 𝐾 = record
{ F = T { F = T
; η = η' ; η = η'
; μ = μ' ; μ = μ'
-- M3
; identityˡ = Identityˡ ; identityˡ = Identityˡ
-- M2
; identityʳ = K2 ; identityʳ = K2
-- M1
; assoc = assoc' ; assoc = assoc'
; sym-assoc = sym assoc' ; sym-assoc = sym assoc'
} }
@ -45,20 +48,22 @@ ExtensionSystem→Monad {𝒞 = 𝒞} 𝐾 = record
T = RMonad⇒Functor 𝐾 T = RMonad⇒Functor 𝐾
open Functor T renaming (F₁ to T₁) open Functor T renaming (F₁ to T₁)
open Equiv open Equiv
-- constructing the natural transformation η from the given family of morphisms 'unit'
η' = ntHelper {F = Id} {G = T} record η' = ntHelper {F = Id} {G = T} record
{ η = λ X → unit { η = λ X → unit
; commute = λ {X} {Y} f → sym K2 ; commute = λ {X} {Y} f → sym K2
} }
-- constructing the natural transformation μ
μ' = ntHelper {F = T ∘F T} {G = T} record μ' = ntHelper {F = T ∘F T} {G = T} record
{ η = λ X → (idC {A = T₀ X})ᵀ { η = λ X → (idC {A = T₀ X})ᵀ
; commute = λ {X} {Y} f → begin ; commute = λ {X} {Y} f → begin
((idC ᵀ) ∘ (unit ∘ (unit ∘ f)ᵀ)ᵀ) ≈⟨ (sym $ K3) ⟩ ((idC ᵀ) ∘ (unit ∘ (unit ∘ f)ᵀ)ᵀ) ≈⟨ (sym $ K3) ⟩
(((idC ᵀ) ∘ unit ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ sym-assoc ⟩ (((idC ᵀ) ∘ unit ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ sym-assoc ⟩
((((idC ᵀ) ∘ unit) ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ (∘-resp-≈ˡ K2) ⟩ ((((idC ᵀ) ∘ unit) ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ (∘-resp-≈ˡ K2) ⟩
((idC ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ identityˡ ⟩ ((idC ∘ ((unit ∘ f) ᵀ)) ᵀ) ≈⟨ ᵀ-≈ identityˡ ⟩
(((unit ∘ f) ᵀ) ᵀ) ≈⟨ ᵀ-≈ (sym identityʳ) ⟩ (((unit ∘ f) ᵀ) ᵀ) ≈⟨ ᵀ-≈ (sym identityʳ) ⟩
((((unit ∘ f) ᵀ) ∘ idC) ᵀ) ≈⟨ K3 ⟩ ((((unit ∘ f) ᵀ) ∘ idC) ᵀ) ≈⟨ K3 ⟩
(unit ∘ f)ᵀ ∘ (idC ᵀ) ∎ (unit ∘ f)ᵀ ∘ (idC ᵀ) ∎
} }
open NaturalTransformation η' using () renaming (η to η) open NaturalTransformation η' using () renaming (η to η)
open NaturalTransformation μ' using () renaming (η to μ) open NaturalTransformation μ' using () renaming (η to μ)