Initial commit
This commit is contained in:
commit
61f84701f6
|
|
@ -0,0 +1,60 @@
|
|||
|
||||
# Created by https://www.toptal.com/developers/gitignore/api/emacs,agda
|
||||
# Edit at https://www.toptal.com/developers/gitignore?templates=emacs,agda
|
||||
|
||||
### Agda ###
|
||||
*.agdai
|
||||
MAlonzo/**
|
||||
|
||||
### Emacs ###
|
||||
# -*- mode: gitignore; -*-
|
||||
*~
|
||||
\#*\#
|
||||
/.emacs.desktop
|
||||
/.emacs.desktop.lock
|
||||
*.elc
|
||||
auto-save-list
|
||||
tramp
|
||||
.\#*
|
||||
|
||||
# Org-mode
|
||||
.org-id-locations
|
||||
*_archive
|
||||
|
||||
# flymake-mode
|
||||
*_flymake.*
|
||||
|
||||
# eshell files
|
||||
/eshell/history
|
||||
/eshell/lastdir
|
||||
|
||||
# elpa packages
|
||||
/elpa/
|
||||
|
||||
# reftex files
|
||||
*.rel
|
||||
|
||||
# AUCTeX auto folder
|
||||
/auto/
|
||||
|
||||
# cask packages
|
||||
.cask/
|
||||
dist/
|
||||
|
||||
# Flycheck
|
||||
flycheck_*.el
|
||||
|
||||
# server auth directory
|
||||
/server/
|
||||
|
||||
# projectiles files
|
||||
.projectile
|
||||
|
||||
# directory configuration
|
||||
.dir-locals.el
|
||||
|
||||
# network security
|
||||
/network-security.data
|
||||
|
||||
|
||||
# End of https://www.toptal.com/developers/gitignore/api/emacs,agda
|
||||
|
|
@ -0,0 +1,121 @@
|
|||
{-# OPTIONS --type-in-type #-}
|
||||
|
||||
module Playground.Category where
|
||||
|
||||
open import Agda.Builtin.Equality public
|
||||
|
||||
_∘_ : {A B C : Set} → (B → C) → (A → B) → (A → C)
|
||||
_∘_ f g x = f (g x)
|
||||
|
||||
postulate
|
||||
funext : (A : Set)(B : A → Set)(f g : (a : A) → B a) → ((a : A) → f a ≡ g a) → f ≡ g
|
||||
|
||||
record Category (Obj : Set) (Hom : Obj → Obj → Set) : Set where
|
||||
field
|
||||
id : (A : Obj) → Hom A A
|
||||
comp : {A B C : Obj} → Hom B C → Hom A B → Hom A C
|
||||
left-id : {A B : Obj}(f : Hom A B) → comp (id B) f ≡ f
|
||||
right-id : {A B : Obj}(f : Hom A B) → comp f (id A) ≡ f
|
||||
comp-assoc : {A B C D : Obj}(f : Hom C D)(g : Hom B C)(h : Hom A B)
|
||||
→ comp (comp f g) h ≡ comp f (comp g h)
|
||||
|
||||
open Category
|
||||
|
||||
ArrowCategory : Category Set (\x y → x → y)
|
||||
id ArrowCategory A = λ x → x
|
||||
comp ArrowCategory {A} {B} {C} f g = f ∘ g
|
||||
left-id ArrowCategory {A} {B} f =
|
||||
funext A (\a → B) (comp ArrowCategory (id ArrowCategory B) f) f (\a → refl)
|
||||
right-id ArrowCategory {A} {B} f =
|
||||
funext A (\a → B) (comp ArrowCategory f (id ArrowCategory A)) f (\a → refl)
|
||||
comp-assoc ArrowCategory {A} {B} {C} {D} f g h =
|
||||
funext A (\a → D) (comp ArrowCategory (comp ArrowCategory f g) h) (comp ArrowCategory f (comp ArrowCategory g h)) (\a → refl)
|
||||
|
||||
data Void : Set where
|
||||
|
||||
record Unit : Set where
|
||||
constructor top
|
||||
|
||||
Unit-singleton : (x : Unit) → x ≡ top
|
||||
Unit-singleton top = refl
|
||||
|
||||
Unit-Terminal : {A : Set} → A → Unit
|
||||
Unit-Terminal {A} a = top
|
||||
|
||||
Void-Initial : {A : Set} → Void → A
|
||||
Void-Initial {A} ()
|
||||
|
||||
VoidHom : Void → Void → Set
|
||||
VoidHom ()
|
||||
|
||||
VoidCat : Category Void VoidHom
|
||||
id VoidCat ()
|
||||
comp VoidCat {()}
|
||||
left-id VoidCat {()}
|
||||
right-id VoidCat {()}
|
||||
comp-assoc VoidCat {()}
|
||||
|
||||
UnitHom : Unit → Unit → Set
|
||||
UnitHom top top = Unit
|
||||
|
||||
UnitCat : Category Unit UnitHom
|
||||
id UnitCat A = top
|
||||
comp UnitCat top top = top
|
||||
left-id UnitCat top = refl
|
||||
right-id UnitCat top = refl
|
||||
comp-assoc UnitCat top top top = refl
|
||||
|
||||
record Pair (A : Set) (B : Set) : Set where
|
||||
constructor _,_
|
||||
field
|
||||
fst : A
|
||||
snd : B
|
||||
|
||||
data Either (A : Set) (B : Set) : Set where
|
||||
inl : A → Either A B
|
||||
inr : B → Either A B
|
||||
|
||||
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)
|
||||
PairEqlProof {A} {B} (p , q) (.p , .q) refl refl = refl
|
||||
|
||||
-- EitherInlInrProof : {A B : Set}(x : Either A B) → Either (a : A , x = Either.inl a) (b : B , x = Either.inr b)
|
||||
|
||||
record IsTerminal {Obj : Set}{Hom : Obj -> Obj -> Set}(C : Category Obj Hom)(T : Obj) : Set where
|
||||
field
|
||||
terminate : (A : Obj) → Hom A T
|
||||
unique-terminal : {A : Obj}(a : Hom A T) → a ≡ (terminate A)
|
||||
|
||||
open IsTerminal
|
||||
|
||||
UnitTerminal : IsTerminal UnitCat top
|
||||
terminate UnitTerminal top = top
|
||||
unique-terminal UnitTerminal top = refl
|
||||
|
||||
ArrowTerminal : IsTerminal ArrowCategory Unit
|
||||
terminate ArrowTerminal A x = top
|
||||
unique-terminal ArrowTerminal a = refl
|
||||
|
||||
comm-eql : {A : Set}{x y : A}(a : x ≡ y) → (y ≡ x)
|
||||
comm-eql refl = refl
|
||||
|
||||
OppositeCat : {Obj : Set}{Hom : Obj -> Obj -> Set}(C : Category Obj Hom) → (Category Obj (\x y → Hom y x))
|
||||
id (OppositeCat c) = id c
|
||||
comp (OppositeCat c) x y = comp c y x
|
||||
left-id (OppositeCat c) f = right-id c f
|
||||
right-id (OppositeCat c) f = left-id c f
|
||||
comp-assoc (OppositeCat c) f g h = comm-eql (comp-assoc c h g f)
|
||||
|
||||
record IsInitial {Obj : Set}{Hom : Obj → Obj → Set}(C : Category Obj Hom)(I : Obj) : Set where
|
||||
field
|
||||
initiate : (A : Obj) → Hom I A
|
||||
unique-initial : {A : Obj}(a : Hom I A) → a ≡ (initiate A)
|
||||
|
||||
open IsInitial
|
||||
|
||||
UnitInitial : IsInitial UnitCat top
|
||||
initiate UnitInitial A = top
|
||||
unique-initial UnitInitial a = refl
|
||||
|
||||
ArrowInitial : IsInitial ArrowCategory Void
|
||||
initiate ArrowInitial A = Void-Initial
|
||||
unique-initial ArrowInitial a = {!!}
|
||||
Loading…
Reference in New Issue