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