54 lines
1.6 KiB
Text
54 lines
1.6 KiB
Text
|
/-
|
||
|
|
Copyright (c) 2015 Floris van Doorn. All rights reserved.
|
||
|
|
Released under Apache 2.0 license as described in the file LICENSE.
|
||
|
|
|
||
|
|
Module: algebra.category.constructions.opposite
|
||
|
|
Authors: Floris van Doorn, Jakob von Raumer
|
||
|
|
|
||
|
|
Opposite precategory and (TODO) category
|
||
|
|
-/
|
||
|
|
|
||
|
|
import ..functor ..category
|
||
|
|
|
||
|
|
open eq functor
|
||
|
|
|
||
|
|
namespace category
|
||
|
|
|
||
|
|
definition opposite [reducible] {ob : Type} (C : precategory ob) : precategory ob :=
|
||
|
|
precategory.mk' (λ a b, hom b a)
|
||
|
|
(λ a b c f g, g ∘ f)
|
||
|
|
(λ a, id)
|
||
|
|
(λ a b c d f g h, !assoc')
|
||
|
|
(λ a b c d f g h, !assoc)
|
||
|
|
(λ a b f, !id_right)
|
||
|
|
(λ a b f, !id_left)
|
||
|
|
(λ a, !id_id)
|
||
|
|
(λ a b, !is_hset_hom)
|
||
|
|
|
||
|
|
definition Opposite [reducible] (C : Precategory) : Precategory := precategory.Mk (opposite C)
|
||
|
|
|
||
|
|
infixr `∘op`:60 := @comp _ (opposite _) _ _ _
|
||
|
|
|
||
|
|
variables {C : Precategory} {a b c : C}
|
||
|
|
|
||
|
|
definition compose_op {f : hom a b} {g : hom b c} : f ∘op g = g ∘ f := idp
|
||
|
|
|
||
|
|
definition opposite_opposite' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
|
||
|
|
by cases C; apply idp
|
||
|
|
|
||
|
|
definition opposite_opposite : Opposite (Opposite C) = C :=
|
||
|
|
(ap (Precategory.mk C) (opposite_opposite' C)) ⬝ !Precategory.eta
|
||
|
|
|
||
|
|
postfix `ᵒᵖ`:(max+1) := Opposite
|
||
|
|
|
||
|
|
definition opposite_functor [reducible] {C D : Precategory} (F : C ⇒ D) : Cᵒᵖ ⇒ Dᵒᵖ :=
|
||
|
|
begin
|
||
|
|
apply (@functor.mk (Cᵒᵖ) (Dᵒᵖ)),
|
||
|
|
intro a, apply (respect_id F),
|
||
|
|
intros, apply (@respect_comp C D)
|
||
|
|
end
|
||
|
|
|
||
|
|
infixr `ᵒᵖᶠ`:(max+1) := opposite_functor
|
||
|
|
|
||
|
|
end category
|