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.gitignore
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.gitignore
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# Created by https://www.toptal.com/developers/gitignore/api/emacs,agda
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# Edit at https://www.toptal.com/developers/gitignore?templates=emacs,agda
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### Agda ###
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*.agdai
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MAlonzo/**
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### Emacs ###
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# -*- mode: gitignore; -*-
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*~
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\#*\#
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/.emacs.desktop
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/.emacs.desktop.lock
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*.elc
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auto-save-list
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tramp
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.\#*
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# Org-mode
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.org-id-locations
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*_archive
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# flymake-mode
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*_flymake.*
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# eshell files
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/eshell/history
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/eshell/lastdir
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# elpa packages
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/elpa/
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# reftex files
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*.rel
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# AUCTeX auto folder
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/auto/
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# cask packages
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.cask/
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dist/
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# Flycheck
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flycheck_*.el
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# server auth directory
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/server/
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# projectiles files
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.projectile
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# directory configuration
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.dir-locals.el
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# network security
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/network-security.data
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# End of https://www.toptal.com/developers/gitignore/api/emacs,agda
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121
Playground/Category.agda
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Playground/Category.agda
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{-# OPTIONS --type-in-type #-}
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module Playground.Category where
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open import Agda.Builtin.Equality public
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_∘_ : {A B C : Set} → (B → C) → (A → B) → (A → C)
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_∘_ f g x = f (g x)
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postulate
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funext : (A : Set)(B : A → Set)(f g : (a : A) → B a) → ((a : A) → f a ≡ g a) → f ≡ g
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record Category (Obj : Set) (Hom : Obj → Obj → Set) : Set where
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field
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id : (A : Obj) → Hom A A
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comp : {A B C : Obj} → Hom B C → Hom A B → Hom A C
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left-id : {A B : Obj}(f : Hom A B) → comp (id B) f ≡ f
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right-id : {A B : Obj}(f : Hom A B) → comp f (id A) ≡ f
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comp-assoc : {A B C D : Obj}(f : Hom C D)(g : Hom B C)(h : Hom A B)
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→ comp (comp f g) h ≡ comp f (comp g h)
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open Category
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ArrowCategory : Category Set (\x y → x → y)
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id ArrowCategory A = λ x → x
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comp ArrowCategory {A} {B} {C} f g = f ∘ g
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left-id ArrowCategory {A} {B} f =
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funext A (\a → B) (comp ArrowCategory (id ArrowCategory B) f) f (\a → refl)
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right-id ArrowCategory {A} {B} f =
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funext A (\a → B) (comp ArrowCategory f (id ArrowCategory A)) f (\a → refl)
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comp-assoc ArrowCategory {A} {B} {C} {D} f g h =
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funext A (\a → D) (comp ArrowCategory (comp ArrowCategory f g) h) (comp ArrowCategory f (comp ArrowCategory g h)) (\a → refl)
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data Void : Set where
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record Unit : Set where
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constructor top
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Unit-singleton : (x : Unit) → x ≡ top
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Unit-singleton top = refl
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Unit-Terminal : {A : Set} → A → Unit
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Unit-Terminal {A} a = top
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Void-Initial : {A : Set} → Void → A
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Void-Initial {A} ()
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VoidHom : Void → Void → Set
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VoidHom ()
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VoidCat : Category Void VoidHom
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id VoidCat ()
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comp VoidCat {()}
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left-id VoidCat {()}
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right-id VoidCat {()}
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comp-assoc VoidCat {()}
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UnitHom : Unit → Unit → Set
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UnitHom top top = Unit
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UnitCat : Category Unit UnitHom
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id UnitCat A = top
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comp UnitCat top top = top
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left-id UnitCat top = refl
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right-id UnitCat top = refl
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comp-assoc UnitCat top top top = refl
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record Pair (A : Set) (B : Set) : Set where
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constructor _,_
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field
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fst : A
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snd : B
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data Either (A : Set) (B : Set) : Set where
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inl : A → Either A B
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inr : B → Either A B
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PairEqlProof : {A B : Set}(p : Pair A B)(q : Pair A B)(a : Pair.fst p ≡ Pair.fst q)(b : Pair.snd p ≡ Pair.snd q) → (p ≡ q)
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PairEqlProof {A} {B} (p , q) (.p , .q) refl refl = refl
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-- EitherInlInrProof : {A B : Set}(x : Either A B) → Either (a : A , x = Either.inl a) (b : B , x = Either.inr b)
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record IsTerminal {Obj : Set}{Hom : Obj -> Obj -> Set}(C : Category Obj Hom)(T : Obj) : Set where
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field
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terminate : (A : Obj) → Hom A T
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unique-terminal : {A : Obj}(a : Hom A T) → a ≡ (terminate A)
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open IsTerminal
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UnitTerminal : IsTerminal UnitCat top
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terminate UnitTerminal top = top
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unique-terminal UnitTerminal top = refl
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ArrowTerminal : IsTerminal ArrowCategory Unit
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terminate ArrowTerminal A x = top
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unique-terminal ArrowTerminal a = refl
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comm-eql : {A : Set}{x y : A}(a : x ≡ y) → (y ≡ x)
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comm-eql refl = refl
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OppositeCat : {Obj : Set}{Hom : Obj -> Obj -> Set}(C : Category Obj Hom) → (Category Obj (\x y → Hom y x))
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id (OppositeCat c) = id c
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comp (OppositeCat c) x y = comp c y x
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left-id (OppositeCat c) f = right-id c f
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right-id (OppositeCat c) f = left-id c f
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comp-assoc (OppositeCat c) f g h = comm-eql (comp-assoc c h g f)
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record IsInitial {Obj : Set}{Hom : Obj → Obj → Set}(C : Category Obj Hom)(I : Obj) : Set where
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field
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initiate : (A : Obj) → Hom I A
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unique-initial : {A : Obj}(a : Hom I A) → a ≡ (initiate A)
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open IsInitial
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UnitInitial : IsInitial UnitCat top
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initiate UnitInitial A = top
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unique-initial UnitInitial a = refl
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ArrowInitial : IsInitial ArrowCategory Void
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initiate ArrowInitial A = Void-Initial
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unique-initial ArrowInitial a = {!!}
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