statement stringlengths 5 1.82k | proof stringlengths 0 5.57k | type stringclasses 4
values | symbolic_name stringlengths 1 58 | library stringclasses 164
values | filename stringclasses 562
values | imports listlengths 0 99 | deps listlengths 0 64 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
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|---|---|---|---|---|---|---|---|---|---|---|
cartesian-weak-monic-cancell
: ∀ {x y z} {f : Hom y z} {g : Hom x y}
→ ∀ {x' y' z'} {f' : Hom[ f ] y' z'} {g' : Hom[ g ] x' y'}
→ is-weak-monic f'
→ is-cartesian (f ∘ g) (f' ∘' g')
→ is-cartesian g g' | cartesian-weak-monic-cancell {f = f} {g = g} {f' = f'} {g' = g'} f-weak-mono fg-cart = g-cart where
module fg = is-cartesian fg-cart
open is-cartesian
g-cart : is-cartesian g g'
g-cart .universal m h' =
fg.universal' (sym (assoc f g m)) (f' ∘' h')
g-cart .commutes m h' = f-weak-mono _ h' refl $ begin[]
... | function | cartesian-weak-monic-cancell | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"is-cartesian",
"is-weak-monic",
"refl",
"sym"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cartesian-cancell
: ∀ {x y z} {f : Hom y z} {g : Hom x y}
→ ∀ {x' y' z'} {f' : Hom[ f ] y' z'} {g' : Hom[ g ] x' y'}
→ is-cartesian f f'
→ is-cartesian (f ∘ g) (f' ∘' g')
→ is-cartesian g g' | cartesian-cancell f-cart fg-cart =
cartesian-weak-monic-cancell (cartesian→weak-monic f-cart) fg-cart | function | cartesian-cancell | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian-weak-monic-cancell",
"cartesian→weak-monic",
"f'",
"is-cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cart-paste
: ∀ {x y z x' y' z'} {f : Hom y z} {g : Hom x y}
→ Cartesian-morphism f y' z'
→ Cartesian-morphism (f ∘ g) x' z'
→ Cartesian-morphism g x' y' | cart-paste {x' = x'} {y' = y'} {f = f} {g = g} f' fg' = g' where
open Cartesian-morphism
open is-cartesian
module f' = is-cartesian (f' .cartesian)
module fg' = is-cartesian (fg' .cartesian)
g' : Cartesian-morphism g x' y'
g' .hom' = f'.universal g (fg' .hom')
g' .cartesian = cartesian-cancell (f' .... | function | cart-paste | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-morphism",
"cartesian-cancell",
"f'",
"is-cartesian",
"refl",
"subst-is-cartesian",
"sym"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
vertical+cartesian→invertible
: ∀ {x} {x' x'' : Ob[ x ]} {f' : Hom[ id ] x' x''}
→ is-cartesian id f'
→ is-invertible↓ f' | vertical+cartesian→invertible {x' = x'} {x'' = x''} {f' = f'} f-cart =
make-invertible↓ f⁻¹' f'-invl f'-invr where
open is-cartesian f-cart
f⁻¹' : Hom[ id ] x'' x'
f⁻¹' = universal' (idl _) id'
f'-invl : f' ∘' f⁻¹' ≡[ idl _ ] id'
f'-invl = commutesp _ id'
path : f' ∘' f⁻¹' ∘' f' ≡[ elimr (... | function | vertical+cartesian→invertible | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"id",
"is-cartesian",
"is-invertible↓",
"make-invertible↓"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
postcompose-equiv→cartesian
: ∀ {x y x' y'} {f : Hom x y}
→ (f' : Hom[ f ] x' y')
→ (∀ {w w'} {g : Hom w x} → is-equiv {A = Hom[ g ] w' x'} (f' ∘'_))
→ is-cartesian f f' | postcompose-equiv→cartesian f' eqv = record where
universal m h' = equiv→inverse eqv h'
commutes m h' = equiv→counit eqv h'
unique m' p = sym (equiv→unit eqv m') ∙ ap (equiv→inverse eqv) p | function | postcompose-equiv→cartesian | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"ap",
"equiv→counit",
"equiv→inverse",
"equiv→unit",
"f'",
"is-cartesian",
"is-equiv",
"sym"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cartesian→postcompose-equiv
: ∀ {x y z x' y' z'} {f : Hom y z} {g : Hom x y} {f' : Hom[ f ] y' z'}
→ is-cartesian f f'
→ is-equiv {A = Hom[ g ] x' y'} (f' ∘'_) | cartesian→postcompose-equiv cart = is-iso→is-equiv record where
open is-cartesian cart
from g = universal _ g
rinv g = commutes _ g
linv g = sym (unique g refl) | function | cartesian→postcompose-equiv | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"is-cartesian",
"is-equiv",
"linv",
"refl",
"rinv",
"sym"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Cartesian-fibration : Type _ | Cartesian-fibration = ∀ {x y} (f : Hom x y) (y' : Ob[ y ]) → Cartesian-lift f y' | function | Cartesian-fibration | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-lift"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
f' ∘cart g' = fg' where
open Cartesian-morphism
fg' : Cartesian-morphism _ _ _
fg' .hom' = f' .hom' ∘' g' .hom'
fg' .cartesian = cartesian-∘ (f' .cartesian) (g' .cartesian) | function | f' | Cat.Displayed | src/Cat/Displayed/Cartesian.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-morphism",
"cartesian-∘"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
is-cocartesian
{a b a' b'} (f : Hom a b) (f' : Hom[ f ] a' b')
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality | record | is-cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Cocartesian-morphism
{x y : Ob} (f : Hom x y) (x' : Ob[ x ]) (y' : Ob[ y ])
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
field
hom' : Hom[ f ] x' y'
cocartesian : is-cocartesian f hom'
open is-cocartesian cocartesian public | record | Cocartesian-morphism | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Cocartesian-lift {x y} (f : Hom x y) (x' : Ob[ x ]) : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
{y'} : Ob[ y ]
lifting : Hom[ f ] x' y'
cocartesian : is-cocartesian f lifting
open is-cocartesian cocartesian public | record | Cocartesian-lift | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
co-cartesian→cocartesian
: ∀ {x y} {f : Hom x y} {x' y'} {f' : Hom[ f ] x' y'}
→ is-cartesian (ℰ ^total-op) f f'
→ is-cocartesian f f' | co-cartesian→cocartesian cart^op .is-cocartesian.universal m h' =
is-cartesian.universal cart^op m h'
co-cartesian→cocartesian cart^op .is-cocartesian.commutes m h' =
is-cartesian.commutes cart^op m h'
co-cartesian→cocartesian cart^op .is-cocartesian.unique m' p =
is-cartesian.unique cart^op m' p | function | co-cartesian→cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"is-cartesian",
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian→co-cartesian
: ∀ {x y} {f : Hom x y} {x' y'} {f' : Hom[ f ] x' y'}
→ is-cocartesian f f'
→ is-cartesian (ℰ ^total-op) f f' | cocartesian→co-cartesian cocart .is-cartesian.universal m h' =
is-cocartesian.universal cocart m h'
cocartesian→co-cartesian cocart .is-cartesian.commutes m h' =
is-cocartesian.commutes cocart m h'
cocartesian→co-cartesian cocart .is-cartesian.unique m' p =
is-cocartesian.unique cocart m' p | function | cocartesian→co-cartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"is-cartesian",
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
co-cartesian→cocartesian-is-equiv
: ∀ {x y} {f : Hom x y} {x' y'} {f' : Hom[ f ] x' y'}
→ is-equiv (co-cartesian→cocartesian {f' = f'}) | co-cartesian→cocartesian-is-equiv {f' = f'} =
is-iso→is-equiv $ iso cocartesian→co-cartesian cocart-invl cocart-invr
where
cocart-invl
: ∀ f
→ co-cartesian→cocartesian {f' = f'} (cocartesian→co-cartesian f) ≡ f
cocart-invl f i .is-cocartesian.universal m h' = is-cocartesian.universal f m h'
... | function | co-cartesian→cocartesian-is-equiv | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"co-cartesian→cocartesian",
"cocartesian→co-cartesian",
"f'",
"is-equiv"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
co-cartesian≡cocartesian
: ∀ {x y} {f : Hom x y} {x' y'} {f' : Hom[ f ] x' y'}
→ is-cartesian (ℰ ^total-op) f f' ≡ is-cocartesian f f' | co-cartesian≡cocartesian =
ua (co-cartesian→cocartesian , co-cartesian→cocartesian-is-equiv) | function | co-cartesian≡cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"co-cartesian→cocartesian",
"co-cartesian→cocartesian-is-equiv",
"f'",
"is-cartesian",
"is-cocartesian",
"ua"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-∘
: ∀ {x y z} {f : Hom y z} {g : Hom x y}
→ ∀ {x' y' z'} {f' : Hom[ f ] y' z'} {g' : Hom[ g ] x' y'}
→ is-cocartesian f f' → is-cocartesian g g'
→ is-cocartesian (f ∘ g) (f' ∘' g') | cocartesian-∘ f-cocart g-cocart =
co-cartesian→cocartesian $
cartesian-∘ _
(cocartesian→co-cartesian g-cocart)
(cocartesian→co-cartesian f-cocart) | function | cocartesian-∘ | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian-∘",
"co-cartesian→cocartesian",
"cocartesian→co-cartesian",
"f'",
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-id : ∀ {x x'} → is-cocartesian id (id' {x} {x'}) | cocartesian-id = co-cartesian→cocartesian (cartesian-id _) | function | cocartesian-id | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian-id",
"co-cartesian→cocartesian",
"id",
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
invertible→cocartesian
: ∀ {x y} {f : Hom x y} {x' y'} {f' : Hom[ f ] x' y'}
→ (f-inv : is-invertible f)
→ is-invertible[ f-inv ] f'
→ is-cocartesian f f' | invertible→cocartesian f-inv f'-inv =
co-cartesian→cocartesian $
invertible→cartesian _ _ (invertible[]→co-invertible[] f'-inv) | function | invertible→cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"co-cartesian→cocartesian",
"f'",
"invertible[]→co-invertible[]",
"invertible→cartesian",
"is-cocartesian",
"is-invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian→weak-epic
: ∀ {x y} {f : Hom x y}
→ ∀ {x' y'} {f' : Hom[ f ] x' y'}
→ is-cocartesian f f'
→ is-weak-epic f' | cocartesian→weak-epic cocart =
cartesian→weak-monic (ℰ ^total-op) (cocartesian→co-cartesian cocart) | function | cocartesian→weak-epic | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian→weak-monic",
"cocartesian→co-cartesian",
"f'",
"is-cocartesian",
"is-weak-epic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-codomain-unique
: ∀ {x y} {f : Hom x y}
→ ∀ {x' y' y''} {f' : Hom[ f ] x' y'} {f'' : Hom[ f ] x' y''}
→ is-cocartesian f f'
→ is-cocartesian f f''
→ y' ≅↓ y'' | cocartesian-codomain-unique f'-cocart f''-cocart =
vertical-co-iso→vertical-iso $
cartesian-domain-unique (ℰ ^total-op)
(cocartesian→co-cartesian f''-cocart)
(cocartesian→co-cartesian f'-cocart) | function | cocartesian-codomain-unique | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian-domain-unique",
"cocartesian→co-cartesian",
"f'",
"is-cocartesian",
"vertical-co-iso→vertical-iso"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-vertical-section-stable
: ∀ {x y} {f : Hom x y}
→ ∀ {x' y' y''} {f' : Hom[ f ] x' y'} {f'' : Hom[ f ] x' y''} {ϕ : Hom[ id ] y'' y'}
→ is-cocartesian f f'
→ has-retract↓ ϕ
→ ϕ ∘' f'' ≡[ idl _ ] f'
→ is-cocartesian f f'' | cocartesian-vertical-section-stable cocart ret factor =
co-cartesian→cocartesian $
cartesian-vertical-retraction-stable (ℰ ^total-op)
(cocartesian→co-cartesian cocart)
(vertical-retract→vertical-co-section ret)
factor | function | cocartesian-vertical-section-stable | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"co-cartesian→cocartesian",
"cocartesian→co-cartesian",
"f'",
"factor",
"has-retract↓",
"id",
"is-cocartesian",
"vertical-retract→vertical-co-section"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-cancelr
: ∀ {x y z} {f : Hom y z} {g : Hom x y}
→ ∀ {x' y' z'} {f' : Hom[ f ] y' z'} {g' : Hom[ g ] x' y'}
→ is-cocartesian g g'
→ is-cocartesian (f ∘ g) (f' ∘' g')
→ is-cocartesian f f' | cocartesian-cancelr g-cocart fg-cocart =
co-cartesian→cocartesian $
cartesian-cancell (ℰ ^total-op)
(cocartesian→co-cartesian g-cocart)
(cocartesian→co-cartesian fg-cocart) | function | cocartesian-cancelr | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian-cancell",
"co-cartesian→cocartesian",
"cocartesian→co-cartesian",
"f'",
"is-cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
vertical+cocartesian→invertible
: ∀ {x} {x' x'' : Ob[ x ]} {f' : Hom[ id ] x' x''}
→ is-cocartesian id f'
→ is-invertible↓ f' | vertical+cocartesian→invertible cocart =
vertical-co-invertible→vertical-invertible $
vertical+cartesian→invertible (ℰ ^total-op)
(cocartesian→co-cartesian cocart) | function | vertical+cocartesian→invertible | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cocartesian→co-cartesian",
"f'",
"id",
"is-cocartesian",
"is-invertible↓",
"vertical+cartesian→invertible",
"vertical-co-invertible→vertical-invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso→cocartesian
: ∀ {x y x' y'} {f : x ≅ y}
→ (f' : x' ≅[ f ] y')
→ is-cocartesian (f .to) (f' .to') | iso→cocartesian {f = f} f' =
invertible→cocartesian (iso→invertible f) (iso[]→invertible[] f') | function | iso→cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"f'",
"invertible→cocartesian",
"is-cocartesian",
"iso[]→invertible[]",
"iso→invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
precompose-equiv→cocartesian
: ∀ {x y x' y'} {f : Hom x y}
→ (f' : Hom[ f ] x' y')
→ (∀ {z z'} {g : Hom y z} → is-equiv {A = Hom[ g ] y' z'} (_∘' f'))
→ is-cocartesian f f' | precompose-equiv→cocartesian f' cocart =
co-cartesian→cocartesian $
postcompose-equiv→cartesian (ℰ ^total-op) f' cocart | function | precompose-equiv→cocartesian | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"co-cartesian→cocartesian",
"f'",
"is-cocartesian",
"is-equiv",
"postcompose-equiv→cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian→precompose-equiv
: ∀ {x y z x' y' z'} {g : Hom y z} {f : Hom x y} {f' : Hom[ f ] x' y'}
→ is-cocartesian f f'
→ is-equiv {A = Hom[ g ] y' z'} (_∘' f') | cocartesian→precompose-equiv cocart =
cartesian→postcompose-equiv (ℰ ^total-op) $
cocartesian→co-cartesian cocart | function | cocartesian→precompose-equiv | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"cartesian→postcompose-equiv",
"cocartesian→co-cartesian",
"f'",
"is-cocartesian",
"is-equiv"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
co-cartesian-lift→cocartesian-lift
: ∀ {x y} {f : Hom x y} {x' : Ob[ x ]}
→ Cartesian-lift (ℰ ^total-op) f x'
→ Cocartesian-lift f x' | co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.y' =
Cartesian-lift.x' cart
co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.lifting =
Cartesian-lift.lifting cart
co-cartesian-lift→cocartesian-lift cart .Cocartesian-lift.cocartesian =
co-cartesian→cocartesian (Cartesian-lift.cartesian cart) | function | co-cartesian-lift→cocartesian-lift | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-lift",
"Cocartesian-lift",
"co-cartesian→cocartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
cocartesian-lift→co-cartesian-lift
: ∀ {x y} {f : Hom x y} {x' : Ob[ x ]}
→ Cocartesian-lift f x'
→ Cartesian-lift (ℰ ^total-op) f x' | cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.x' =
Cocartesian-lift.y' cocart
cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.lifting =
Cocartesian-lift.lifting cocart
cocartesian-lift→co-cartesian-lift cocart .Cartesian-lift.cartesian =
cocartesian→co-cartesian (Cocartesian-lift.cocartesian... | function | cocartesian-lift→co-cartesian-lift | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-lift",
"Cocartesian-lift",
"cocartesian→co-cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Cocartesian-fibration : Type _ | Cocartesian-fibration = ∀ {x y} (f : Hom x y) (x' : Ob[ x ]) → Cocartesian-lift f x' | function | Cocartesian-fibration | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cocartesian-lift"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
op-fibration→opfibration : Cartesian-fibration (ℰ ^total-op) → Cocartesian-fibration | op-fibration→opfibration fib f x' =
co-cartesian-lift→cocartesian-lift (fib f x') | function | op-fibration→opfibration | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-fibration",
"Cocartesian-fibration",
"co-cartesian-lift→cocartesian-lift"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
opfibration→op-fibration : Cocartesian-fibration → Cartesian-fibration (ℰ ^total-op) | opfibration→op-fibration opfib f y' =
cocartesian-lift→co-cartesian-lift (opfib f y') | function | opfibration→op-fibration | Cat.Displayed | src/Cat/Displayed/Cocartesian.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Total.Op",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Morphism.Duality",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism",
"Cat.Reasoning"
] | [
"Cartesian-fibration",
"Cocartesian-fibration",
"cocartesian-lift→co-cartesian-lift"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
_D∘_
: ∀ {o ℓ o' ℓ' o'' ℓ''} {ℬ : Precategory o ℓ}
→ (ℰ : Displayed ℬ o' ℓ') (ℱ : Displayed (∫ ℰ) o'' ℓ'')
→ Displayed ℬ (o' ⊔ o'') (ℓ' ⊔ ℓ'') | _D∘_ {ℬ = ℬ} ℰ ℱ = disp where
module ℰ = Displayed ℰ
module ℱ = Displayed ℱ
open Displayed
disp : Displayed ℬ _ _
disp .Ob[_] X = Σ[ X' ∈ ℰ.Ob[ X ] ] ℱ.Ob[ X , X' ]
disp .Hom[_] f (X , X') (Y , Y') =
Σ[ f' ∈ ℰ.Hom[ f ] X Y ] ℱ.Hom[ ∫hom f f' ] X' Y'
disp .Hom[_]-set f x y = hlevel 2
disp .Displayed... | function | _D∘_ | Cat.Displayed | src/Cat/Displayed/Composition.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Displayed.Total",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning"
] | [
"Displayed",
"Precategory",
"ap₂",
"f'",
"hlevel"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
πᵈ : ∀ {o ℓ o' ℓ' o'' ℓ''}
→ {ℬ : Precategory o ℓ}
→ {ℰ : Displayed ℬ o' ℓ'} {ℱ : Displayed (∫ ℰ) o'' ℓ''}
→ Displayed-functor Id (ℰ D∘ ℱ) ℰ | πᵈ .Displayed-functor.F₀' = fst
πᵈ .Displayed-functor.F₁' = fst
πᵈ .Displayed-functor.F-id' = refl
πᵈ .Displayed-functor.F-∘' = refl | function | πᵈ | Cat.Displayed | src/Cat/Displayed/Composition.lagda.md | [
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Displayed.Total",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning"
] | [
"Displayed",
"Id",
"Precategory",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Comprehension-structure : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
field
Comprehend : Vertical-functor E (Slices B)
Comprehend-is-fibred : is-fibred-functor Comprehend | record | Comprehension-structure | Cat.Displayed | src/Cat/Displayed/Comprehension.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Instances.Slice",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Diagram.Pullback",
"Cat.Diagram.Comonad",
"Cat.Displayed.Total",
"Cat.Instances.Slice",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
... | [] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Comprehension-comonad : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
field
comprehend : Functor (∫ E) (∫ E)
comonad : Comonad-on comprehend
open Comonad-on comonad public
field
counit-cartesian
: ∀ {Γ x} → is-cartesian E (counit.ε (Γ , x) .fst) (counit.ε (Γ , x) .snd)
cartesian-pullba... | record | Comprehension-comonad | Cat.Displayed | src/Cat/Displayed/Comprehension.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Instances.Slice",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Diagram.Pullback",
"Cat.Diagram.Comonad",
"Cat.Displayed.Total",
"Cat.Instances.Slice",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
... | [
"Functor",
"is-cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Comprehension→comonad
: Cartesian-fibration E
→ Comprehension-structure
→ Comprehension-comonad | Comprehension→comonad fib P = comp-comonad where
open Cartesian-fibration E fib
open Comprehension fib P
open Comonad-on | function | Comprehension→comonad | Cat.Displayed | src/Cat/Displayed/Comprehension.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Instances.Slice",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Diagram.Pullback",
"Cat.Diagram.Comonad",
"Cat.Displayed.Total",
"Cat.Instances.Slice",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
... | [
"Cartesian-fibration",
"Comprehension-comonad",
"Comprehension-structure"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Comonad→comprehension
: Cartesian-fibration E → Comprehension-comonad → Comprehension-structure | Comonad→comprehension fib comp-comonad = comprehension where
open Comprehension-comonad comp-comonad
open Comprehension-structure
open is-fibred-functor
open Vertical-functor
open is-pullback
vert : Vertical-functor E (Slices B)
vert .F₀' {Γ} x = cut (counit.ε (Γ , x) .fst)
vert .F₁' {f = σ} f = record... | function | Comonad→comprehension | Cat.Displayed | src/Cat/Displayed/Comprehension.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Instances.Slice",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Functor",
"Cat.Diagram.Pullback",
"Cat.Diagram.Comonad",
"Cat.Displayed.Total",
"Cat.Instances.Slice",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
... | [
"Cartesian-fibration",
"Comprehension-comonad",
"Comprehension-structure",
"Slice-path",
"ap",
"map"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Regular-hyperdoctrine o' ℓ' : Type (o ⊔ ℓ ⊔ lsuc (o' ⊔ ℓ')) where | record | Regular-hyperdoctrine | Cat.Displayed | src/Cat/Displayed/Doctrine.lagda.md | [
"Cat.Displayed.Cocartesian",
"Cat.Diagram.Limit.Finite",
"Cat.Displayed.Cartesian",
"Cat.Diagram.Pullback",
"Cat.Diagram.Terminal",
"Cat.Diagram.Product",
"Cat.Displayed.Fibre",
"Cat.Displayed.Base",
"Cat.Prelude",
"Order.Base",
"Order.Cat",
"Cat.Displayed.Reasoning",
"Cat.Reasoning",
"Ord... | [
"lsuc"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Fibre : (X : Ob) → Precategory _ _ | Fibre X .Precategory.Ob = Ob[ X ]
Fibre X .Precategory.Hom = Hom[ id ]
Fibre X .Precategory.Hom-set = Hom[ id ]-set
Fibre X .Precategory.id = id'
Fibre X .Precategory._∘_ f g = hom[ idl id ] (f ∘' g)
Fibre X .Precategory.idr f =
hom[ idl id ] (f ∘' id') ≡⟨ Ds.disp! E ⟩
f ∎
Fibre X .Precategor... | function | Fibre | Cat.Displayed | src/Cat/Displayed/Fibre.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Solver"
] | [
"Precategory",
"hom[",
"id"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Generic-object {t} (t' : Ob[ t ]) : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
field
classify : ∀ {x} → (x' : Ob[ x ]) → Hom x t
classify' : ∀ {x} → (x' : Ob[ x ]) → Hom[ classify x' ] x' t'
classify-cartesian
: ∀ {x} (x' : Ob[ x ]) → is-cartesian E (classify x') (classify' x')
module classi... | record | Generic-object | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"classify",
"is-cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Globally-small : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
field
{U} : Ob
Gen : Ob[ U ]
has-generic-ob : Generic-object Gen
open Generic-object has-generic-ob public | record | Globally-small | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"Generic-object"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
is-gaunt-generic-object
{t} {t' : Ob[ t ]}
(gobj : Generic-object t')
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ') where
no-eta-equality
open Generic-object gobj | record | is-gaunt-generic-object | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"Generic-object"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
is-skeletal-generic-object : ∀ {t} {t' : Ob[ t ]} → Generic-object t' → Type _ | is-skeletal-generic-object {t} {t'} gobj =
∀ {x} {x' : Ob[ x ]} {u : Hom x t} {f' : Hom[ u ] x' t'}
→ is-cartesian E u f' → u ≡ classify x'
where open Generic-object gobj | function | is-skeletal-generic-object | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"Generic-object",
"classify",
"f'",
"is-cartesian"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-skeletal-generic-object-is-prop
: ∀ {t} {t' : Ob[ t ]} {gobj : Generic-object t'}
→ is-prop (is-skeletal-generic-object gobj) | is-skeletal-generic-object-is-prop = hlevel 1 | function | is-skeletal-generic-object-is-prop | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"Generic-object",
"hlevel",
"is-prop",
"is-skeletal-generic-object"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
gaunt-generic-object→skeletal-generic-object
: ∀ {t} {t' : Ob[ t ]} {gobj : Generic-object t'}
→ is-gaunt-generic-object gobj → is-skeletal-generic-object gobj | gaunt-generic-object→skeletal-generic-object =
is-gaunt-generic-object.classify-unique | function | gaunt-generic-object→skeletal-generic-object | Cat.Displayed | src/Cat/Displayed/GenericObject.lagda.md | [
"Cat.Displayed.Cartesian.Indexing",
"Cat.Displayed.Cartesian",
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Displayed.Morphism"
] | [
"Generic-object",
"is-gaunt-generic-object",
"is-skeletal-generic-object"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Internal-sum : Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
∐F : Vertical-functor (Disp-family E) E
∐F-fibred : is-fibred-functor ∐F
∐F⊣ConstFam : ∐F ⊣↓ ConstDispFam E | record | Internal-sum | Cat.Displayed | src/Cat/Displayed/InternalSum.lagda.md | [
"Cat.Displayed.Instances.DisplayedFamilies",
"Cat.Displayed.Instances.Slice",
"Cat.Displayed.Adjoint",
"Cat.Displayed.Functor",
"Cat.Displayed.Base",
"Cat.Prelude"
] | [
"ConstDispFam",
"Disp-family"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
_↪[_]_
{a b} (a' : Ob[ a ]) (f : a ↪ b) (b' : Ob[ b ])
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
mor' : Hom[ f .mor ] a' b'
monic' : is-monic[ f .monic ] mor' | record | _↪[_]_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
weak-mono-over
{a b} (f : Hom a b) (a' : Ob[ a ]) (b' : Ob[ b ])
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
mor' : Hom[ f ] a' b'
weak-monic : is-weak-monic mor' | record | weak-mono-over | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"is-weak-monic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
_↠[_]_
{a b} (a' : Ob[ a ]) (f : a ↠ b) (b' : Ob[ b ])
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
mor' : Hom[ f .mor ] a' b'
epic' : is-epic[ f .epic ] mor' | record | _↠[_]_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
weak-epi-over
{a b} (f : Hom a b) (a' : Ob[ a ]) (b' : Ob[ b ])
: Type (o ⊔ ℓ ⊔ o' ⊔ ℓ')
where
no-eta-equality
field
mor' : Hom[ f ] a' b'
weak-epic : is-weak-epic mor' | record | weak-epi-over | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"is-weak-epic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
has-section[_]
{x y x' y'} {r : Hom x y} (sect : has-section r) (r' : Hom[ r ] x' y')
: Type ℓ'
where
no-eta-equality
field
section' : Hom[ sect .section ] y' x'
is-section' : section' section-of[ sect .is-section ] r' | record | has-section[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"has-section",
"r'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
has-retract[_]
{x y x' y'} {s : Hom x y} (ret : has-retract s) (s' : Hom[ s ] x' y')
: Type ℓ'
where
no-eta-equality
field
retract' : Hom[ ret .retract ] y' x'
is-retract' : retract' retract-of[ ret .is-retract ] s' | record | has-retract[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"has-retract",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
Inverses[_]
{a b a' b'} {f : Hom a b} {g : Hom b a}
(inv : Inverses f g)
(f' : Hom[ f ] a' b') (g' : Hom[ g ] b' a')
: Type ℓ'
where
no-eta-equality
field
invl' : f' ∘' g' ≡[ Inverses.invl inv ] id'
invr' : g' ∘' f' ≡[ Inverses.invr inv ] id' | record | Inverses[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
is-invertible[_]
{a b a' b'} {f : Hom a b}
(f-inv : is-invertible f)
(f' : Hom[ f ] a' b')
: Type ℓ'
where
no-eta-equality
field
inv' : Hom[ is-invertible.inv f-inv ] b' a'
inverses' : Inverses[ is-invertible.inverses f-inv ] f' inv'
open Inverses[_] inverses' public | record | is-invertible[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses[_]",
"f'",
"is-invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
_≅[_]_
{a b} (a' : Ob[ a ]) (i : a ≅ b) (b' : Ob[ b ])
: Type ℓ'
where
no-eta-equality
field
to' : Hom[ i .to ] a' b'
from' : Hom[ i .from ] b' a'
inverses' : Inverses[ i .inverses ] to' from'
open Inverses[_] inverses' public | record | _≅[_]_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses[_]"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
is-monic[_]
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ is-monic f → Hom[ f ] a' b'
→ Type _ | is-monic[_] {a = a} {a' = a'} {f = f} mono f' =
∀ {c c'} {g h : Hom c a}
→ (g' : Hom[ g ] c' a') (h' : Hom[ h ] c' a')
→ (p : f ∘ g ≡ f ∘ h)
→ f' ∘' g' ≡[ p ] f' ∘' h'
→ g' ≡[ mono g h p ] h' | function | is-monic[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-monic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-monic[]-is-prop
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ (mono : is-monic f) → (f' : Hom[ f ] a' b')
→ is-prop (is-monic[ mono ] f') | is-monic[]-is-prop {a' = a'} mono f' mono[] mono[]' i {c' = c'} g' h' p p' =
is-set→squarep (λ i j → Hom[ mono _ _ p j ]-set c' a')
refl (mono[] g' h' p p') (mono[]' g' h' p p') refl i | function | is-monic[]-is-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-monic",
"is-prop",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-weak-monic
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ Hom[ f ] a' b'
→ Type _ | is-weak-monic {a = a} {a' = a'} {f = f} f' =
∀ {c c'} {g h : Hom c a}
→ (g' : Hom[ g ] c' a') (h' : Hom[ h ] c' a')
→ (p : g ≡ h)
→ f' ∘' g' ≡[ ap (f ∘_) p ] f' ∘' h'
→ g' ≡[ p ] h' | function | is-weak-monic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"ap",
"f'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-weak-monic-is-prop
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ (f' : Hom[ f ] a' b')
→ is-prop (is-weak-monic f') | is-weak-monic-is-prop f' = hlevel 1 | function | is-weak-monic-is-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"hlevel",
"is-prop",
"is-weak-monic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
∘-is-weak-monic
: is-weak-monic f'
→ is-weak-monic g'
→ is-weak-monic (f' ∘' g') | ∘-is-weak-monic {f' = f'} {g' = g'} f'-weak-monic g'-weak-monic h' k' p p' =
g'-weak-monic h' k' p $
f'-weak-monic (g' ∘' h') (g' ∘' k') (ap₂ _∘_ refl p) $ begin[]
f' ∘' g' ∘' h' ≡[]⟨ assoc' f' g' h' ⟩
(f' ∘' g') ∘' h' ≡[]⟨ p' ⟩
(f' ∘' g') ∘' k' ≡[]˘⟨ assoc' f' g' k' ⟩
f' ∘' g' ∘' k' ∎[] | function | ∘-is-weak-monic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"_∘_",
"ap₂",
"f'",
"is-weak-monic",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-monic[]→is-weak-monic
: {f-monic : is-monic f}
→ is-monic[ f-monic ] f'
→ is-weak-monic f' | is-monic[]→is-weak-monic f'-monic g' h' p p' =
cast[] $ f'-monic g' h' (ap₂ _∘_ refl p) p' | function | is-monic[]→is-weak-monic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"_∘_",
"ap₂",
"cast[]",
"f'",
"is-monic",
"is-weak-monic",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
weak-monic-cancell
: is-weak-monic (f' ∘' g')
→ is-weak-monic g' | weak-monic-cancell {f' = f'} {g' = g'} fg-weak-monic h' k' p p' =
fg-weak-monic h' k' p (extendr' _ p') | function | weak-monic-cancell | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-weak-monic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
weak-monic-postcomp-embedding
: {f : Hom b c} {g : Hom a b}
→ {f' : Hom[ f ] b' c'}
→ is-weak-monic f'
→ is-embedding {A = Hom[ g ] a' b'} (f' ∘'_) | weak-monic-postcomp-embedding {f' = f'} f'-weak-monic =
injective→is-embedding (hlevel 2) (f' ∘'_) λ {g'} {h'} → f'-weak-monic g' h' refl | function | weak-monic-postcomp-embedding | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"hlevel",
"injective→is-embedding",
"is-embedding",
"is-weak-monic",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-jointly-weak-monic
: {fᵢ : (ix : Ix) → Hom a (bᵢ ix)}
→ (fᵢ' : (ix : Ix) → Hom[ fᵢ ix ] a' (bᵢ' ix))
→ Type _ | is-jointly-weak-monic {a = a} {a' = a'} {fᵢ = fᵢ} fᵢ' =
∀ {x x'} {g h : Hom x a}
→ (g' : Hom[ g ] x' a') (h' : Hom[ h ] x' a')
→ (p : g ≡ h)
→ (∀ ix → fᵢ' ix ∘' g' ≡[ ap (fᵢ ix ∘_) p ] fᵢ' ix ∘' h')
→ g' ≡[ p ] h' | function | is-jointly-weak-monic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"ap"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
∘-is-jointly-weak-monic
: {fᵢ : (ix : Ix) → Hom a (bᵢ ix)}
→ {fᵢ' : (ix : Ix) → Hom[ fᵢ ix ] a' (bᵢ' ix)}
→ is-jointly-weak-monic fᵢ'
→ is-weak-monic g'
→ is-jointly-weak-monic (λ ix → fᵢ' ix ∘' g') | ∘-is-jointly-weak-monic {g' = g'} {fᵢ' = fᵢ'} fᵢ'-joint-mono g'-joint-mono h' h'' p p' =
g'-joint-mono h' h'' p $
fᵢ'-joint-mono (g' ∘' h') (g' ∘' h'') (ap₂ _∘_ refl p) λ ix → begin[]
fᵢ' ix ∘' g' ∘' h' ≡[]⟨ assoc' (fᵢ' ix) g' h' ⟩
(fᵢ' ix ∘' g') ∘' h' ≡[]⟨ p' ix ⟩
(fᵢ' ix ∘' g') ∘' h'' ≡[]˘⟨ assoc'... | function | ∘-is-jointly-weak-monic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"_∘_",
"ap₂",
"is-jointly-weak-monic",
"is-weak-monic",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
jointly-weak-monic-cancell
: {fᵢ : (ix : Ix) → Hom a (bᵢ ix)}
→ {fᵢ' : (ix : Ix) → Hom[ fᵢ ix ] a' (bᵢ' ix)}
→ is-jointly-weak-monic (λ ix → fᵢ' ix ∘' g')
→ is-weak-monic g' | jointly-weak-monic-cancell fᵢ'-joint-mono h' h'' p p' =
fᵢ'-joint-mono h' h'' p λ _ → extendr' (ap₂ _∘_ refl p) p' | function | jointly-weak-monic-cancell | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"_∘_",
"ap₂",
"is-jointly-weak-monic",
"is-weak-monic",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-epic[_]
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ is-epic f → Hom[ f ] a' b'
→ Type _ | is-epic[_] {b = b} {b' = b'} {f = f} epi f' =
∀ {c} {c'} {g h : Hom b c}
→ (g' : Hom[ g ] b' c') (h' : Hom[ h ] b' c')
→ (p : g ∘ f ≡ h ∘ f)
→ g' ∘' f' ≡[ p ] h' ∘' f'
→ g' ≡[ epi g h p ] h' | function | is-epic[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-epic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-epic[]-is-prop
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ (epi : is-epic f) → (f' : Hom[ f ] a' b')
→ is-prop (is-epic[ epi ] f') | is-epic[]-is-prop {b' = b'} epi f' epi[] epi[]' i {c' = c'} g' h' p p' =
is-set→squarep (λ i j → Hom[ epi _ _ p j ]-set b' c')
refl (epi[] g' h' p p') (epi[]' g' h' p p') refl i | function | is-epic[]-is-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-epic",
"is-prop",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-weak-epic
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ Hom[ f ] a' b'
→ Type _ | is-weak-epic {b = b} {b' = b'} {f = f} f' =
∀ {c c'} {g h : Hom b c}
→ (g' : Hom[ g ] b' c') (h' : Hom[ h ] b' c')
→ (p : g ≡ h)
→ g' ∘' f' ≡[ ap (_∘ f) p ] h' ∘' f'
→ g' ≡[ p ] h' | function | is-weak-epic | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"ap",
"f'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-weak-epic-is-prop
: ∀ {a' : Ob[ a ]} {b' : Ob[ b ]} {f : Hom a b}
→ (f' : Hom[ f ] a' b')
→ is-prop (is-weak-epic f') | is-weak-epic-is-prop f' = hlevel 1 | function | is-weak-epic-is-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"hlevel",
"is-prop",
"is-weak-epic"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
_section-of[_]_
: ∀ {x y} {s : Hom y x} {r : Hom x y}
→ ∀ {x' y'} (s' : Hom[ s ] y' x') → s section-of r → (r' : Hom[ r ] x' y')
→ Type _ | function | _section-of[_]_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"r'",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
_section-of↓_
: ∀ {x} {x' x'' : Ob[ x ]} (s' : Hom[ id ] x'' x') → (r : Hom[ id ] x' x'')
→ Type _ | function | _section-of↓_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
has-section↓ : ∀ {x} {x' x'' : Ob[ x ]} (r' : Hom[ id ] x' x'') → Type _ | has-section↓ r' = has-section[ id-has-section ] r' | function | has-section↓ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"id-has-section",
"r'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
_retract-of[_]_
: ∀ {x y} {s : Hom y x} {r : Hom x y}
→ ∀ {x' y'} (r' : Hom[ r ] x' y') → r retract-of s → (s' : Hom[ s ] y' x')
→ Type _ | function | _retract-of[_]_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"r'",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
_retract-of↓_
: ∀ {x} {x' x'' : Ob[ x ]} (r' : Hom[ id ] x' x'') → (s : Hom[ id ] x'' x')
→ Type _ | function | _retract-of↓_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"r'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
has-retract↓ : ∀ {x} {x' x'' : Ob[ x ]} (s' : Hom[ id ] x'' x') → Type _ | has-retract↓ s' = has-retract[ id-has-retract ] s' | function | has-retract↓ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"id-has-retract",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
_≅↓_ : {x : Ob} (A B : Ob[ x ]) → Type ℓ' | _≅↓_ = _≅[ id-iso ]_ | function | _≅↓_ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id-iso"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-invertible↓ : {x : Ob} {x' x'' : Ob[ x ]} → Hom[ id ] x' x'' → Type _ | is-invertible↓ = is-invertible[ id-invertible ] | function | is-invertible↓ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"id-invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
make-invertible↓
: ∀ {x} {x' x'' : Ob[ x ]} {f' : Hom[ id ] x' x''}
→ (g' : Hom[ id ] x'' x')
→ f' ∘' g' ≡[ idl _ ] id'
→ g' ∘' f' ≡[ idl _ ] id'
→ is-invertible↓ f' | make-invertible↓ g' p q .is-invertible[_].inv' = g'
make-invertible↓ g' p q .is-invertible[_].inverses' .Inverses[_].invl' = p
make-invertible↓ g' p q .is-invertible[_].inverses' .Inverses[_].invr' = q | function | make-invertible↓ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"id",
"is-invertible↓"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Inverses[]-are-prop
: ∀ {a b a' b'} {f : Hom a b} {g : Hom b a}
→ (inv : Inverses f g)
→ (f' : Hom[ f ] a' b') (g' : Hom[ g ] b' a')
→ is-prop (Inverses[ inv ] f' g') | Inverses[]-are-prop inv f' g' inv[] inv[]' i .Inverses[_].invl' =
is-set→squarep (λ i j → Hom[ Inverses.invl inv j ]-set _ _)
refl (Inverses[_].invl' inv[]) (Inverses[_].invl' inv[]') refl i
Inverses[]-are-prop inv f' g' inv[] inv[]' i .Inverses[_].invr' =
is-set→squarep (λ i j → Hom[ Inverses.invr inv j ]-set ... | function | Inverses[]-are-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv",
"is-prop",
"refl"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
is-invertible[]-is-prop
: ∀ {a b a' b'} {f : Hom a b}
→ (f-inv : is-invertible f)
→ (f' : Hom[ f ] a' b')
→ is-prop (is-invertible[ f-inv ] f') | is-invertible[]-is-prop inv f' p q = path where
module inv = is-invertible inv
module p = is-invertible[_] p
module q = is-invertible[_] q
inv≡inv' : p.inv' ≡ q.inv'
inv≡inv' =
p.inv' ≡⟨ shiftr (insertr inv.invl) (insertr' _ q.invl') ⟩
hom[] ((p.inv' ∘' f') ∘' q.inv') ≡⟨ wea... | function | is-invertible[]-is-prop | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses[]-are-prop",
"f'",
"hom[]",
"inv",
"is-invertible",
"is-invertible[_]",
"is-prop",
"is-prop→pathp",
"liberate",
"refl",
"shiftr",
"weave"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
make-iso[_]
: ∀ {a b a' b'}
→ (iso : a ≅ b)
→ (f' : Hom[ iso .to ] a' b') (g' : Hom[ iso .from ] b' a')
→ f' ∘' g' ≡[ iso .invl ] id'
→ g' ∘' f' ≡[ iso .invr ] id'
→ a' ≅[ iso ] b'
{-# INLINE make-iso[_] #-} | function | make-iso[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
make-invertible[_]
: ∀ {a b a' b'} {f : Hom a b} {f' : Hom[ f ] a' b'}
→ (f-inv : is-invertible f)
→ (f-inv' : Hom[ is-invertible.inv f-inv ] b' a')
→ f' ∘' f-inv' ≡[ is-invertible.invl f-inv ] id'
→ f-inv' ∘' f' ≡[ is-invertible.invr f-inv ] id'
→ is-invertible[ f-inv ] f' | function | make-invertible[_] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"is-invertible"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
make-vertical-iso
: ∀ {x} {x' x'' : Ob[ x ]}
→ (f' : Hom[ id ] x' x'') (g' : Hom[ id ] x'' x')
→ f' ∘' g' ≡[ idl _ ] id'
→ g' ∘' f' ≡[ idl _ ] id'
→ x' ≅↓ x'' | make-vertical-iso = make-iso[ id-iso ] | function | make-vertical-iso | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"id",
"id-iso",
"make-iso["
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
invertible[]→iso[]
: ∀ {a b a' b'} {f : Hom a b} {f' : Hom[ f ] a' b'}
→ {i : is-invertible f}
→ is-invertible[ i ] f'
→ a' ≅[ invertible→iso f i ] b' | invertible[]→iso[] {f' = f'} i = make-iso[ _ ] f'
(is-invertible[_].inv' i)
(is-invertible[_].invl' i)
(is-invertible[_].invr' i) | function | invertible[]→iso[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"invertible→iso",
"is-invertible",
"make-iso["
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso[]→invertible[]
: ∀ {a b a' b'}
→ {i : a ≅ b}
→ (i' : a' ≅[ i ] b')
→ is-invertible[ iso→invertible i ] (i' .to') | iso[]→invertible[] {i = i} i' =
make-invertible[ (iso→invertible i) ] (i' .from') (i' .invl') (i' .invr') | function | iso[]→invertible[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"i'",
"iso→invertible",
"make-invertible["
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
≅[]-path
: {x y : Ob} {A : Ob[ x ]} {B : Ob[ y ]} {f : x ≅ y}
{p q : A ≅[ f ] B}
→ p .to' ≡ q .to'
→ p ≡ q | ≅[]-path {f = f} {p = p} {q = q} a = it where
p' : PathP (λ i → is-invertible[ iso→invertible f ] (a i))
(record { inv' = p .from' ; inverses' = p .inverses' })
(record { inv' = q .from' ; inverses' = q .inverses' })
p' = is-prop→pathp (λ i → is-invertible[]-is-prop _ (a i)) _ _
it : p ≡ q
it i .to' ... | function | ≅[]-path | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"is-invertible[]-is-prop",
"is-prop→pathp",
"iso→invertible",
"it"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
Extensional-≅[]
: ∀ {ℓr} {x y : Ob} {x' : Ob[ x ]} {y' : Ob[ y ]} {f : x ≅ y}
→ ⦃ sa : Extensional (Hom[ f .to ] x' y') ℓr ⦄
→ Extensional (x' ≅[ f ] y') ℓr | Extensional-≅[] ⦃ sa ⦄ = injection→extensional! ≅[]-path sa | function | Extensional-≅[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Extensional",
"injection→extensional!",
"≅[]-path"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
_Iso[]⁻¹
: ∀ {a b a' b'} {i : a ≅ b}
→ a' ≅[ i ] b'
→ b' ≅[ i Iso⁻¹ ] a' | function | _Iso[]⁻¹ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | ||
id-iso↓ : ∀ {x} {x' : Ob[ x ]} → x' ≅↓ x' | id-iso↓ = make-iso[ id-iso ] id' id' (idl' id') (idl' id') | function | id-iso↓ | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id-iso",
"make-iso["
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
inverses[]→to-has-section[]
: ∀ {f : Hom a b} {g : Hom b a}
→ ∀ {a' b'} {f' : Hom[ f ] a' b'} {g' : Hom[ g ] b' a'}
→ {inv : Inverses f g} → Inverses[ inv ] f' g'
→ has-section[ inverses→to-has-section inv ] f' | inverses[]→to-has-section[] {g' = g'} inv' .section' = g'
inverses[]→to-has-section[] inv' .is-section' = Inverses[_].invl' inv' | function | inverses[]→to-has-section[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv",
"inverses→to-has-section"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
inverses[]→from-has-section[]
: ∀ {f : Hom a b} {g : Hom b a}
→ ∀ {a' b'} {f' : Hom[ f ] a' b'} {g' : Hom[ g ] b' a'}
→ {inv : Inverses f g} → Inverses[ inv ] f' g'
→ has-section[ inverses→from-has-section inv ] g' | inverses[]→from-has-section[] {f' = f'} inv' .section' = f'
inverses[]→from-has-section[] inv' .is-section' = Inverses[_].invr' inv' | function | inverses[]→from-has-section[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv",
"inverses→from-has-section"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
inverses[]→to-has-retract[]
: ∀ {f : Hom a b} {g : Hom b a}
→ ∀ {a' b'} {f' : Hom[ f ] a' b'} {g' : Hom[ g ] b' a'}
→ {inv : Inverses f g} → Inverses[ inv ] f' g'
→ has-retract[ inverses→to-has-retract inv ] f' | inverses[]→to-has-retract[] {g' = g'} inv' .retract' = g'
inverses[]→to-has-retract[] inv' .is-retract' = Inverses[_].invr' inv' | function | inverses[]→to-has-retract[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv",
"inverses→to-has-retract"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
inverses[]→from-has-retract[]
: ∀ {f : Hom a b} {g : Hom b a}
→ ∀ {a' b'} {f' : Hom[ f ] a' b'} {g' : Hom[ g ] b' a'}
→ {inv : Inverses f g} → Inverses[ inv ] f' g'
→ has-retract[ inverses→from-has-retract inv ] g' | inverses[]→from-has-retract[] {f' = f'} inv' .retract' = f'
inverses[]→from-has-retract[] inv' .is-retract' = Inverses[_].invl' inv' | function | inverses[]→from-has-retract[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"Inverses",
"f'",
"inv",
"inverses→from-has-retract"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso[]→to-has-section[]
: {f : a ≅ b} → (f' : a' ≅[ f ] b')
→ has-section[ iso→to-has-section f ] (f' .to') | iso[]→to-has-section[] f' .section' = f' .from'
iso[]→to-has-section[] f' .is-section' = f' .invl' | function | iso[]→to-has-section[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"iso→to-has-section"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso[]→from-has-section[]
: {f : a ≅ b} → (f' : a' ≅[ f ] b')
→ has-section[ iso→from-has-section f ] (f' .from') | iso[]→from-has-section[] f' .section' = f' .to'
iso[]→from-has-section[] f' .is-section' = f' .invr' | function | iso[]→from-has-section[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"iso→from-has-section"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso[]→to-has-retract[]
: {f : a ≅ b} → (f' : a' ≅[ f ] b')
→ has-retract[ iso→to-has-retract f ] (f' .to') | iso[]→to-has-retract[] f' .retract' = f' .from'
iso[]→to-has-retract[] f' .is-retract' = f' .invr' | function | iso[]→to-has-retract[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"iso→to-has-retract"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
iso[]→from-has-retract[]
: {f : a ≅ b} → (f' : a' ≅[ f ] b')
→ has-retract[ iso→from-has-retract f ] (f' .from') | iso[]→from-has-retract[] f' .retract' = f' .to'
iso[]→from-has-retract[] f' .is-retract' = f' .invl' | function | iso[]→from-has-retract[] | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"f'",
"iso→from-has-retract"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
s' section-of[ p ] r' = r' ∘' s' ≡[ p ] id' | s' section-of↓ r' = s' section-of[ idl id ] r' | function | s' | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"r'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c | |
r' retract-of[ p ] s' = r' ∘' s' ≡[ p ] id' | r' retract-of↓ s' = r' retract-of[ idl id ] s' | function | r' | Cat.Displayed | src/Cat/Displayed/Morphism.lagda.md | [
"Cat.Displayed.Base",
"Cat.Prelude",
"Cat.Displayed.Reasoning",
"Cat.Reasoning"
] | [
"id",
"s'"
] | https://github.com/plt-amy/1lab | e5a99a399a3c58922adef713f38314805810937c |
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