Mercurial > hg > Members > kono > Proof > category
view src/Definitions.agda @ 1110:45de2b31bf02
add original library and fix for safe mode
| author | Shinji KONO <kono@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Sat, 07 Oct 2023 19:43:31 +0900 |
| parents | src/cat-utility.agda@4a6d3d27a9fb |
| children | 0e750446e463 |
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{-# OPTIONS --cubical-compatible --safe #-} module Definitions where -- Shinji KONO <kono@ie.u-ryukyu.ac.jp> open import Level open import Category open import HomReasoning open Functor id1 : ∀{c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (a : Obj A ) → Hom A a a id1 A a = (Id {_} {_} {_} {A} a) -- We cannot make A implicit -- -- one to one (without naturality) -- record Iso {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (x y : Obj C ) : Set ( suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁)) where field ≅→ : Hom C x y ≅← : Hom C y x iso→ : C [ C [ ≅← o ≅→ ] ≈ id1 C x ] iso← : C [ C [ ≅→ o ≅← ] ≈ id1 C y ] ≡-iso : {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) (x : Obj C ) → Iso C x x ≡-iso C x = record { ≅→ = id1 C _ ; ≅← = id1 C _ ; iso→ = ≈-Reasoning.idL C ; iso← = ≈-Reasoning.idL C } sym-iso : {c₁ c₂ ℓ : Level} (C : Category c₁ c₂ ℓ) {x y : Obj C } → Iso C x y → Iso C y x sym-iso C i = record { ≅→ = Iso.≅← i ; ≅← = Iso.≅→ i ; iso→ = Iso.iso← i ; iso← = Iso.iso→ i } -- Iso with cong -- record IsoS {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') (a b : Obj A) ( a' b' : Obj B ) : Set ( c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' ) where field ≅→ : Hom A a b → Hom B a' b' ≅← : Hom B a' b' → Hom A a b iso→ : {f : Hom B a' b' } → B [ ≅→ ( ≅← f) ≈ f ] iso← : {f : Hom A a b } → A [ ≅← ( ≅→ f) ≈ f ] cong→ : {f g : Hom A a b } → A [ f ≈ g ] → B [ ≅→ f ≈ ≅→ g ] cong← : {f g : Hom B a' b'} → B [ f ≈ g ] → A [ ≅← f ≈ ≅← g ] -- F(f) -- F(a) ---→ F(b) -- | | -- |t(a) |t(b) G(f)t(a) = t(b)F(f) -- | | -- v v -- G(a) ---→ G(b) -- G(f) record IsNTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (D : Category c₁ c₂ ℓ) (C : Category c₁′ c₂′ ℓ′) ( F G : Functor D C ) (TMap : (A : Obj D) → Hom C (FObj F A) (FObj G A)) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where field commute : {a b : Obj D} {f : Hom D a b} → C [ C [ ( FMap G f ) o ( TMap a ) ] ≈ C [ (TMap b ) o (FMap F f) ] ] record NTrans {c₁ c₂ ℓ c₁′ c₂′ ℓ′ : Level} (domain : Category c₁ c₂ ℓ) (codomain : Category c₁′ c₂′ ℓ′) (F G : Functor domain codomain ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁′ ⊔ c₂′ ⊔ ℓ′)) where field TMap : (A : Obj domain) → Hom codomain (FObj F A) (FObj G A) isNTrans : IsNTrans domain codomain F G TMap ---- -- C is locally small i.e. Hom C i j is a set c₁ -- open import Relation.Binary.PropositionalEquality using (_≡_) record Small { c₁ c₂ ℓ : Level} ( C : Category c₁ c₂ ℓ ) ( I : Set c₁ ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field hom→ : {i j : Obj C } → Hom C i j → I hom← : {i j : Obj C } → ( f : I ) → Hom C i j hom-iso : {i j : Obj C } → { f : Hom C i j } → C [ hom← ( hom→ f ) ≈ f ] hom-rev : {i j : Obj C } → { f : I } → hom→ ( hom← {i} {j} f ) ≡ f ≡←≈ : {i j : Obj C } → { f g : Hom C i j } → C [ f ≈ g ] → f ≡ g record IsUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') ( U : Functor B A ) ( F : Obj A → Obj B ) ( η : (a : Obj A) → Hom A a ( FObj U (F a) ) ) ( _* : { a : Obj A}{ b : Obj B} → ( Hom A a (FObj U b) ) → Hom B (F a ) b ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where field universalMapping : {a : Obj A} { b : Obj B } → { f : Hom A a (FObj U b) } → A [ A [ FMap U ( f * ) o η a ] ≈ f ] uniquness : {a : Obj A} { b : Obj B } → { f : Hom A a (FObj U b) } → { g : Hom B (F a) b } → A [ A [ FMap U g o η a ] ≈ f ] → B [ f * ≈ g ] record UniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') ( U : Functor B A ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where infixr 11 _* field F : Obj A → Obj B η : (a : Obj A) → Hom A a ( FObj U (F a) ) _* : { a : Obj A}{ b : Obj B} → ( Hom A a (FObj U b) ) → Hom B (F a ) b isUniversalMapping : IsUniversalMapping A B U F η _* record coIsUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') ( F : Functor A B ) ( U : Obj B → Obj A ) ( ε : (b : Obj B) → Hom B ( FObj F (U b) ) b ) ( _* : { b : Obj B}{ a : Obj A} → ( Hom B (FObj F a) b ) → Hom A a (U b ) ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where field couniversalMapping : {b : Obj B} { a : Obj A } → { f : Hom B (FObj F a) b } → B [ B [ ε b o FMap F ( f * ) ] ≈ f ] uniquness : {b : Obj B} { a : Obj A } → { f : Hom B (FObj F a) b } → { g : Hom A a (U b) } → B [ B [ ε b o FMap F g ] ≈ f ] → A [ f * ≈ g ] record coUniversalMapping {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') ( F : Functor A B ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where infixr 11 _* field U : Obj B → Obj A ε : (b : Obj B) → Hom B ( FObj F (U b) ) b _* : { b : Obj B}{ a : Obj A} → ( Hom B (FObj F a) b ) → Hom A a (U b ) iscoUniversalMapping : coIsUniversalMapping A B F U ε _* open NTrans open import Category.Cat record IsAdjunction {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') ( U : Functor B A ) ( F : Functor A B ) ( η : NTrans A A identityFunctor ( U ○ F ) ) ( ε : NTrans B B ( F ○ U ) identityFunctor ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where field adjoint1 : { b : Obj B } → A [ A [ ( FMap U ( TMap ε b )) o ( TMap η ( FObj U b )) ] ≈ id1 A (FObj U b) ] adjoint2 : {a : Obj A} → B [ B [ ( TMap ε ( FObj F a )) o ( FMap F ( TMap η a )) ] ≈ id1 B (FObj F a) ] record Adjunction {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) (B : Category c₁' c₂' ℓ') : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where field U : Functor B A F : Functor A B η : NTrans A A identityFunctor ( U ○ F ) ε : NTrans B B ( F ○ U ) identityFunctor isAdjunction : IsAdjunction A B U F η ε U-functor = U F-functor = F Eta = η Epsiron = ε record IsMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( T : Functor A A ) ( η : NTrans A A identityFunctor T ) ( μ : NTrans A A (T ○ T) T) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field assoc : {a : Obj A} → A [ A [ TMap μ a o TMap μ ( FObj T a ) ] ≈ A [ TMap μ a o FMap T (TMap μ a) ] ] unity1 : {a : Obj A} → A [ A [ TMap μ a o TMap η ( FObj T a ) ] ≈ Id {_} {_} {_} {A} (FObj T a) ] unity2 : {a : Obj A} → A [ A [ TMap μ a o (FMap T (TMap η a ))] ≈ Id {_} {_} {_} {A} (FObj T a) ] record Monad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field T : Functor A A η : NTrans A A identityFunctor T μ : NTrans A A (T ○ T) T isMonad : IsMonad A T η μ -- g ○ f = μ(c) T(g) f join : { a b : Obj A } → { c : Obj A } → ( Hom A b ( FObj T c )) → ( Hom A a ( FObj T b)) → Hom A a ( FObj T c ) join {_} {_} {c} g f = A [ TMap μ c o A [ FMap T g o f ] ] record IsCoMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) ( S : Functor A A ) ( ε : NTrans A A S identityFunctor ) ( δ : NTrans A A S (S ○ S) ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field assoc : {a : Obj A} → A [ A [ TMap δ (FObj S a) o TMap δ a ] ≈ A [ FMap S (TMap δ a) o TMap δ a ] ] unity1 : {a : Obj A} → A [ A [ TMap ε ( FObj S a ) o TMap δ a ] ≈ Id {_} {_} {_} {A} (FObj S a) ] unity2 : {a : Obj A} → A [ A [ (FMap S (TMap ε a )) o TMap δ a ] ≈ Id {_} {_} {_} {A} (FObj S a) ] record coMonad {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (S : Functor A A) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ )) where field ε : NTrans A A S identityFunctor δ : NTrans A A S (S ○ S) isCoMonad : IsCoMonad A S ε δ -- g ○ f = g S(f) δ(a) coJoin : { a b : Obj A } → { c : Obj A } → ( Hom A (FObj S b ) c ) → ( Hom A ( FObj S a) b ) → Hom A ( FObj S a ) c coJoin {a} {_} {_} g f = A [ A [ g o FMap S f ] o TMap δ a ] open NTrans nat : { c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} {a b : Obj D} {f : Hom D a b} {F G : Functor D A } → (η : NTrans D A F G ) → A [ A [ FMap G f o TMap η a ] ≈ A [ TMap η b o FMap F f ] ] nat η = IsNTrans.commute ( isNTrans η ) nat1 : { c₁ c₂ ℓ : Level} {A : Category c₁ c₂ ℓ} { c₁′ c₂′ ℓ′ : Level} {D : Category c₁′ c₂′ ℓ′} {a b : Obj D} {F G : Functor D A } → (η : NTrans D A F G ) → (f : Hom D a b) → A [ A [ FMap G f o TMap η a ] ≈ A [ TMap η b o FMap F f ] ] nat1 η f = IsNTrans.commute ( isNTrans η ) Functor*Nat : {c₁ c₂ ℓ c₁' c₂' ℓ' c₁'' c₂'' ℓ'' : Level} (A : Category c₁ c₂ ℓ) {B : Category c₁' c₂' ℓ'} (C : Category c₁'' c₂'' ℓ'') (F : Functor B C) → { G H : Functor A B } → ( n : NTrans A B G H ) → NTrans A C (F ○ G) (F ○ H) Functor*Nat A {B} C F {G} {H} n = record { TMap = λ a → FMap F (TMap n a) ; isNTrans = record { commute = commute } } where commute : {a b : Obj A} {f : Hom A a b} → C [ C [ (FMap F ( FMap H f )) o ( FMap F (TMap n a)) ] ≈ C [ (FMap F (TMap n b )) o (FMap F (FMap G f)) ] ] commute {a} {b} {f} = let open ≈-Reasoning (C) in begin (FMap F ( FMap H f )) o ( FMap F (TMap n a)) ≈⟨ sym (distr F) ⟩ FMap F ( B [ (FMap H f) o TMap n a ]) ≈⟨ IsFunctor.≈-cong (isFunctor F) ( nat n ) ⟩ FMap F ( B [ (TMap n b ) o FMap G f ] ) ≈⟨ distr F ⟩ (FMap F (TMap n b )) o (FMap F (FMap G f)) ∎ Nat*Functor : {c₁ c₂ ℓ c₁' c₂' ℓ' c₁'' c₂'' ℓ'' : Level} (A : Category c₁ c₂ ℓ) {B : Category c₁' c₂' ℓ'} (C : Category c₁'' c₂'' ℓ'') { G H : Functor B C } → ( n : NTrans B C G H ) → (F : Functor A B) → NTrans A C (G ○ F) (H ○ F) Nat*Functor A {B} C {G} {H} n F = record { TMap = λ a → TMap n (FObj F a) ; isNTrans = record { commute = commute } } where commute : {a b : Obj A} {f : Hom A a b} → C [ C [ ( FMap H (FMap F f )) o ( TMap n (FObj F a)) ] ≈ C [ (TMap n (FObj F b )) o (FMap G (FMap F f)) ] ] commute {a} {b} {f} = IsNTrans.commute ( isNTrans n) -- T ≃ (U_R ○ F_R) -- μ = U_R ε F_R -- nat-ε -- nat-η -- same as η but has different types record MResolution {c₁ c₂ ℓ c₁' c₂' ℓ' : Level} (A : Category c₁ c₂ ℓ) ( B : Category c₁' c₂' ℓ' ) ( M : Monad A ) : Set (suc (c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁' ⊔ c₂' ⊔ ℓ' )) where field UR : Functor B A FR : Functor A B ηR : NTrans A A identityFunctor ( UR ○ FR ) εR : NTrans B B ( FR ○ UR ) identityFunctor μR : NTrans A A ( (UR ○ FR) ○ ( UR ○ FR )) ( UR ○ FR ) Adj : IsAdjunction A B UR FR ηR εR T=UF : Monad.T M ≃ (UR ○ FR) μ=UεF : {x : Obj A } → A [ TMap μR x ≈ FMap UR ( TMap εR ( FObj FR x ) ) ] -- ηR=η : {x : Obj A } → A [ TMap ηR x ≈ TMap η x ] -- We need T → UR FR conversion -- μR=μ : {x : Obj A } → A [ TMap μR x ≈ TMap μ x ] -- -- e f e -- c -------→ a ---------→ b -------→ c -- ↑ . ---------→ . | -- | . g . | -- |k . . | k -- | . h h . ↓ -- d d record IsEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A c a) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field fe=ge : A [ A [ f o e ] ≈ A [ g o e ] ] k : {d : Obj A} (h : Hom A d a) → A [ A [ f o h ] ≈ A [ g o h ] ] → Hom A d c ek=h : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → A [ A [ e o k {d} h eq ] ≈ h ] uniqueness : {d : Obj A} → ∀ {h : Hom A d a} → {eq : A [ A [ f o h ] ≈ A [ g o h ] ] } → {k' : Hom A d c } → A [ A [ e o k' ] ≈ h ] → A [ k {d} h eq ≈ k' ] equalizer1 : Hom A c a equalizer1 = e record Equalizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field equalizer-c : Obj A equalizer : Hom A equalizer-c a isEqualizer : IsEqualizer A equalizer f g record IsCoEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {c a b : Obj A} (e : Hom A b c) (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field ef=eg : A [ A [ e o f ] ≈ A [ e o g ] ] k : {d : Obj A} (h : Hom A b d) → A [ A [ h o f ] ≈ A [ h o g ] ] → Hom A c d ke=h : {d : Obj A} → ∀ {h : Hom A b d } → {eq : A [ A [ h o f ] ≈ A [ h o g ] ] } → A [ A [ k {d} h eq o e ] ≈ h ] uniqueness : {d : Obj A} → ∀ {h : Hom A b d } → {eq : A [ A [ h o f ] ≈ A [ h o g ] ] } → {k' : Hom A c d} → A [ A [ k' o e ] ≈ h ] → A [ k {d} h eq ≈ k' ] equalizer1 : Hom A b c equalizer1 = e record CoEqualizer { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b : Obj A} (f g : Hom A a b) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field coEqualizer-c : Obj A coEqualizer : Hom A coEqualizer-c a isCoEqualizer : IsEqualizer A coEqualizer f g -- -- Product -- -- c -- f | g -- |f×g -- v -- a <-------- ab ---------→ b -- π1 π2 record IsProduct { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a b ab : Obj A) ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field _×_ : {c : Obj A} ( f : Hom A c a ) → ( g : Hom A c b ) → Hom A c ab π1fxg=f : {c : Obj A} { f : Hom A c a } → { g : Hom A c b } → A [ A [ π1 o ( f × g ) ] ≈ f ] π2fxg=g : {c : Obj A} { f : Hom A c a } → { g : Hom A c b } → A [ A [ π2 o ( f × g ) ] ≈ g ] uniqueness : {c : Obj A} { h : Hom A c ab } → A [ ( A [ π1 o h ] ) × ( A [ π2 o h ] ) ≈ h ] ×-cong : {c : Obj A} { f f' : Hom A c a } → { g g' : Hom A c b } → A [ f ≈ f' ] → A [ g ≈ g' ] → A [ f × g ≈ f' × g' ] record Product { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( a b : Obj A ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field product : Obj A π1 : Hom A product a π2 : Hom A product b isProduct : IsProduct A a b product π1 π2 record IsCoProduct { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a b ab : Obj A) ( κ1 : Hom A a ab ) ( κ2 : Hom A b ab ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field _+_ : {c : Obj A} ( f : Hom A a c ) → ( g : Hom A b c ) → Hom A ab c κ1f+g=f : {c : Obj A} { f : Hom A a c } → { g : Hom A b c } → A [ A [ ( f + g ) o κ1 ] ≈ f ] κ2f+g=g : {c : Obj A} { f : Hom A a c } → { g : Hom A b c } → A [ A [ ( f + g ) o κ2 ] ≈ g ] uniqueness : {c : Obj A} { h : Hom A ab c } → A [ ( A [ h o κ1 ] ) + ( A [ h o κ2 ] ) ≈ h ] +-cong : {c : Obj A} { f f' : Hom A a c } → { g g' : Hom A b c } → A [ f ≈ f' ] → A [ g ≈ g' ] → A [ f + g ≈ f' + g' ] record coProduct { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( a b : Obj A ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field coproduct : Obj A κ1 : Hom A a coproduct κ2 : Hom A b coproduct isProduct : IsCoProduct A a b coproduct κ1 κ2 ----- -- -- product on arbitrary index -- record IsIProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ ) ( p : Obj A ) -- product ( ai : I → Obj A ) -- families ( pi : (i : I ) → Hom A p ( ai i ) ) -- projections : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where field iproduct : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → Hom A q p pif=q : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → ∀ { i : I } → A [ A [ ( pi i ) o ( iproduct qi ) ] ≈ (qi i) ] ip-uniqueness : {q : Obj A} { h : Hom A q p } → A [ iproduct ( λ (i : I) → A [ (pi i) o h ] ) ≈ h ] ip-cong : {q : Obj A} → { qi : (i : I) → Hom A q (ai i) } → { qi' : (i : I) → Hom A q (ai i) } → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ iproduct qi ≈ iproduct qi' ] -- another form of uniquness ip-uniqueness1 : {q : Obj A} → ( qi : (i : I) → Hom A q (ai i) ) → ( product' : Hom A q p ) → ( ∀ { i : I } → A [ A [ ( pi i ) o product' ] ≈ (qi i) ] ) → A [ product' ≈ iproduct qi ] ip-uniqueness1 {a} qi product' eq = let open ≈-Reasoning (A) in begin product' ≈↑⟨ ip-uniqueness ⟩ iproduct (λ i₁ → A [ pi i₁ o product' ]) ≈⟨ ip-cong ( λ i → begin pi i o product' ≈⟨ eq {i} ⟩ qi i ∎ ) ⟩ iproduct qi ∎ record IProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ ) (ai : I → Obj A) : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where field iprod : Obj A pi : (i : I ) → Hom A iprod ( ai i ) isIProduct : IsIProduct I A iprod ai pi record IsICoProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ ) ( p : Obj A ) -- coproduct ( ci : I → Obj A ) -- cases ( ki : (i : I ) → Hom A ( ci i ) p ) -- coprojections : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where field icoproduct : {q : Obj A} → ( qi : (i : I) → Hom A (ci i) q ) → Hom A p q kif=q : {q : Obj A} → { qi : (i : I) → Hom A (ci i) q } → ∀ { i : I } → A [ A [ ( icoproduct qi ) o ( ki i ) ] ≈ (qi i) ] icp-uniqueness : {q : Obj A} { h : Hom A p q } → A [ icoproduct ( λ (i : I) → A [ h o (ki i) ] ) ≈ h ] icp-cong : {q : Obj A} → { qi : (i : I) → Hom A (ci i) q} → { qi' : (i : I) → Hom A (ci i) q } → ( ∀ (i : I ) → A [ qi i ≈ qi' i ] ) → A [ icoproduct qi ≈ icoproduct qi' ] -- another form of uniquness icp-uniqueness1 : {q : Obj A} → ( qi : (i : I) → Hom A (ci i) q ) → ( product' : Hom A p q ) → ( ∀ { i : I } → A [ A [ product' o ( ki i )] ≈ (qi i) ] ) → A [ product' ≈ icoproduct qi ] icp-uniqueness1 {a} qi product' eq = let open ≈-Reasoning (A) in begin product' ≈↑⟨ icp-uniqueness ⟩ icoproduct (λ i₁ → A [ product' o ki i₁ ]) ≈⟨ icp-cong ( λ i → begin product' o ki i ≈⟨ eq {i} ⟩ qi i ∎ ) ⟩ icoproduct qi ∎ record ICoProduct { c c₁ c₂ ℓ : Level} ( I : Set c) ( A : Category c₁ c₂ ℓ ) (ci : I → Obj A) : Set (c ⊔ ℓ ⊔ (c₁ ⊔ c₂)) where field icoprod : Obj A ki : (i : I ) → Hom A ( ci i ) icoprod isICoProduct : IsICoProduct I A icoprod ci ki -- Pullback -- f -- a ------→ c -- ^ ^ -- π1 | |g -- | | -- ab ------→ b -- ^ π2 -- | -- d -- record IsPullback { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c ab : Obj A} ( f : Hom A a c ) ( g : Hom A b c ) ( π1 : Hom A ab a ) ( π2 : Hom A ab b ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field commute : A [ A [ f o π1 ] ≈ A [ g o π2 ] ] pullback : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → A [ A [ f o π1' ] ≈ A [ g o π2' ] ] → Hom A d ab π1p=π1 : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → { eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] } → A [ A [ π1 o pullback eq ] ≈ π1' ] π2p=π2 : { d : Obj A } → { π1' : Hom A d a } { π2' : Hom A d b } → { eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] } → A [ A [ π2 o pullback eq ] ≈ π2' ] uniqueness : { d : Obj A } → ( p' : Hom A d ab ) → ( π1' : Hom A d a ) ( π2' : Hom A d b ) → ( eq : A [ A [ f o π1' ] ≈ A [ g o π2' ] ] ) → ( π1p=π1' : A [ A [ π1 o p' ] ≈ π1' ] ) → ( π2p=π2' : A [ A [ π2 o p' ] ≈ π2' ] ) → A [ pullback eq ≈ p' ] record Pullback { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) {a b c : Obj A} ( f : Hom A a c ) ( g : Hom A b c ) : Set (ℓ ⊔ (c₁ ⊔ c₂)) where field ab : Obj A π1 : Hom A ab a π2 : Hom A ab b isPullback : IsPullback A f g π1 π2 -- -- Limit -- ----- -- Constancy Functor K : { c₁' c₂' ℓ' : Level} (I : Category c₁' c₂' ℓ') { c₁'' c₂'' ℓ'' : Level} ( A : Category c₁'' c₂'' ℓ'' ) → ( a : Obj A ) → Functor I A K I A a = record { FObj = λ i → a ; FMap = λ f → id1 A a ; isFunctor = let open ≈-Reasoning (A) in record { ≈-cong = λ f=g → refl-hom ; identity = refl-hom ; distr = sym idL } } record IsLimit { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) (a0 : Obj A ) (t0 : NTrans I A ( K I A a0 ) Γ ) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where field limit : ( a : Obj A ) → ( t : NTrans I A ( K I A a ) Γ ) → Hom A a a0 t0f=t : { a : Obj A } → { t : NTrans I A ( K I A a ) Γ } → ∀ { i : Obj I } → A [ A [ TMap t0 i o limit a t ] ≈ TMap t i ] limit-uniqueness : { a : Obj A } → { t : NTrans I A ( K I A a ) Γ } → { f : Hom A a a0 } → ( ∀ { i : Obj I } → A [ A [ TMap t0 i o f ] ≈ TMap t i ] ) → A [ limit a t ≈ f ] record Limit { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' ) ( Γ : Functor I A ) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where field a0 : Obj A t0 : NTrans I A ( K I A a0 ) Γ isLimit : IsLimit I A Γ a0 t0 LimitNat : { c₁' c₂' ℓ' : Level} (I : Category c₁' c₂' ℓ') { c₁ c₂ ℓ : Level} ( B : Category c₁ c₂ ℓ ) { c₁'' c₂'' ℓ'' : Level} ( C : Category c₁'' c₂'' ℓ'' ) ( Γ : Functor I B ) ( lim : Obj B ) ( tb : NTrans I B ( K I B lim ) Γ ) → ( U : Functor B C) → NTrans I C ( K I C (FObj U lim) ) (U ○ Γ) LimitNat I B C Γ lim tb U = record { TMap = TMap (Functor*Nat I C U tb) ; isNTrans = record { commute = λ {a} {b} {f} → let open ≈-Reasoning (C) in begin FMap (U ○ Γ) f o TMap (Functor*Nat I C U tb) a ≈⟨ nat ( Functor*Nat I C U tb ) ⟩ TMap (Functor*Nat I C U tb) b o FMap (U ○ K I B lim) f ≈⟨ cdr (IsFunctor.identity (isFunctor U) ) ⟩ TMap (Functor*Nat I C U tb) b o FMap (K I C (FObj U lim)) f ∎ } } open Limit record LimitPreserve { c₁ c₂ ℓ : Level} { c₁' c₂' ℓ' : Level} ( I : Category c₁ c₂ ℓ ) ( A : Category c₁' c₂' ℓ' ) { c₁'' c₂'' ℓ'' : Level} ( C : Category c₁'' c₂'' ℓ'' ) (G : Functor A C) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ ⊔ c₁'' ⊔ c₂'' ⊔ ℓ'' )) where field preserve : ( Γ : Functor I A ) → ( limita : Limit I A Γ ) → IsLimit I C (G ○ Γ) (FObj G (a0 limita ) ) (LimitNat I A C Γ (a0 limita ) (t0 limita) G ) plimit : { Γ : Functor I A } → ( limita : Limit I A Γ ) → Limit I C (G ○ Γ ) plimit {Γ} limita = record { a0 = (FObj G (a0 limita )) ; t0 = LimitNat I A C Γ (a0 limita ) (t0 limita) G ; isLimit = preserve Γ limita } record Complete { c₁' c₂' ℓ' : Level} { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( I : Category c₁' c₂' ℓ' ) : Set (suc (c₁' ⊔ c₂' ⊔ ℓ' ⊔ c₁ ⊔ c₂ ⊔ ℓ )) where field climit : ( Γ : Functor I A ) → Limit I A Γ cproduct : ( I : Set c₁' ) (fi : I → Obj A ) → IProduct I A fi -- c₁ should be a free level cequalizer : {a b : Obj A} (f g : Hom A a b) → Equalizer A f g open Limit limit-c : ( Γ : Functor I A ) → Obj A limit-c Γ = a0 ( climit Γ) limit-u : ( Γ : Functor I A ) → NTrans I A ( K I A (limit-c Γ )) Γ limit-u Γ = t0 ( climit Γ) open Equalizer equalizer-p : {a b : Obj A} (f g : Hom A a b) → Obj A equalizer-p f g = equalizer-c (cequalizer f g ) equalizer-e : {a b : Obj A} (f g : Hom A a b) → Hom A (equalizer-p f g) a equalizer-e f g = equalizer (cequalizer f g ) -- initial object record HasInitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) (i : Obj A) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where field initial : ∀( a : Obj A ) → Hom A i a uniqueness : { a : Obj A } → ( f : Hom A i a ) → A [ f ≈ initial a ] record InitialObject {c₁ c₂ ℓ : Level} (A : Category c₁ c₂ ℓ) : Set (suc ℓ ⊔ (suc c₁ ⊔ suc c₂)) where field initialObject : Obj A hasInitialObject : HasInitialObject A initialObject open import Category.Sets Yoneda : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( ≡←≈ : {a b : Obj A } { x y : Hom A a b } → (x≈y : A [ x ≈ y ]) → x ≡ y ) (a : Obj A) → Functor (Category.op A) (Sets {c₂}) Yoneda {c₁} {c₂} {ℓ} A ≡←≈ a = record { FObj = λ b → Hom (Category.op A) a b ; FMap = λ {x} {y} (f : Hom A y x ) → λ ( g : Hom A x a ) → (Category.op A) [ f o g ] -- f : Hom A x y → Hom Sets (Hom A a x ) (Hom A a y) ; isFunctor = record { identity = λ {b} x → ≡←≈ idL ; distr = λ {a} {b} {c} {f} {g} x → ≡←≈ ( sym assoc ) ; ≈-cong = λ eq x → ≡←≈ ( resp refl-hom eq ) } } where open ≈-Reasoning (Category.op A) -- -- better than ≡←≈ -- Cong≈ : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → Set (c₁ ⊔ c₂ ⊔ ℓ) Cong≈ A a = {c b : Obj A} (f : Hom A a c → Hom A a b ) {x y : Hom A a c} → A [ x ≈ y ] → A [ f x ≈ f y ] where open ≈-Reasoning A Sets≈ : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → Cong≈ A a → Category c₁ c₂ (c₂ ⊔ ℓ) Sets≈ A a ≈-cong = record { Obj = Obj A ; Hom = λ b c → Hom (Category.op A) b a → Hom (Category.op A) c a ; _o_ = λ f g x → f (g x ) ; _≈_ = λ {b} f g → (x : Hom (Category.op A) b a) → f x ≈ g x ; isCategory = record { isEquivalence = record {refl = λ x → refl-hom ; trans = λ i=j j=k x → trans-hom (i=j x) (j=k x) ; sym = λ eq x → sym-hom (eq x)} ; identityL = λ x → refl-hom ; identityR = λ x → refl-hom ; o-resp-≈ = λ {c} {b} {a} {f} {g} {h} {i} f=g h=i x → trans-hom (h=i (f x)) ( ≈-cong i (f=g x) ) ; associative = λ x → refl-hom } } where open ≈-Reasoning A Yoneda≈ : { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) (a : Obj A) → (≈-cong : Cong≈ (Category.op A) a ) → Functor (Category.op A) (Sets≈ (Category.op A) a ≈-cong) Yoneda≈ {c₁} {c₂} {ℓ} A a ≈-cong = record { FObj = λ b → b ; FMap = λ {x} {y} (f : Hom A y x ) ( g : Hom A x a ) → A [ g o f ] ; isFunctor = record { identity = λ {b} x → idR ; distr = λ {a} {b} {c} {f} {g} x → assoc ; ≈-cong = λ eq x → resp eq refl-hom } } where open ≈-Reasoning A -- -- Representable U ≈ Hom(A,-) record Representable { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( ≡←≈ : {a b : Obj A } { x y : Hom A b a } → (x≈y : (Category.op A) [ x ≈ y ]) → x ≡ y ) ( U : Functor (Category.op A) (Sets {c₂}) ) (a : Obj A) : Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁ )) where field -- FObj U x : A → Set -- FMap U f = Set → Set (locally small) -- λ b → Hom (a,b) : A → Set -- λ f → Hom (a,-) = λ b → Hom a b repr→ : NTrans (Category.op A) (Sets {c₂}) U (Yoneda A ≡←≈ a ) repr← : NTrans (Category.op A) (Sets {c₂}) (Yoneda A ≡←≈ a) U reprId→ : {x : Obj A} → Sets [ Sets [ TMap repr→ x o TMap repr← x ] ≈ id1 (Sets {c₂}) (FObj (Yoneda A ≡←≈ a) x )] reprId← : {x : Obj A} → Sets [ Sets [ TMap repr← x o TMap repr→ x ] ≈ id1 (Sets {c₂}) (FObj U x)] record Representable≈ { c₁ c₂ ℓ : Level} ( A : Category c₁ c₂ ℓ ) ( ≡←≈ : {a b : Obj A } { x y : Hom A b a } → (x≈y : (Category.op A) [ x ≈ y ]) → x ≡ y ) (≈-cong : (a : Obj A) → Cong≈ (Category.op A) a ) (a : Obj A) ( U : Functor (Category.op A) (Sets≈ (Category.op A) a (≈-cong a)) ) : Set (suc ℓ ⊔ (suc (suc c₂) ⊔ suc c₁ )) where field repr→ : NTrans (Category.op A) (Sets≈ (Category.op A) a (≈-cong a)) U (Yoneda≈ A a (≈-cong a)) repr← : NTrans (Category.op A) (Sets≈ (Category.op A) a (≈-cong a)) (Yoneda≈ A a (≈-cong a)) U reprId→ : {x : Obj A} → Sets [ Sets [ TMap repr→ x o TMap repr← x ] ≈ id1 (Sets≈ (Category.op A) a (≈-cong a)) (FObj (Yoneda≈ A a (≈-cong a)) x )] reprId← : {x : Obj A} → Sets [ Sets [ TMap repr← x o TMap repr→ x ] ≈ id1 (Sets≈ (Category.op A) a (≈-cong a)) (FObj U x)]