lean2/hott/algebra/category/constructions/opposite.hlean

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/-
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