214 lines
7.7 KiB
Text
214 lines
7.7 KiB
Text
/-
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Copyright (c) 2014 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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Module: algebra.precategory.constructions
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Authors: Floris van Doorn, Jakob von Raumer
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This file contains basic constructions on precategories, including common precategories
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-/
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import .nat_trans
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import types.prod types.sigma types.pi
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open eq prod eq eq.ops equiv is_trunc
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namespace category
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namespace opposite
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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, !homH)
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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⁻¹)
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(λ a b f, !id_right)
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(λ a b f, !id_left)
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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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set_option apply.class_instance false -- disable class instance resolution in the apply tactic
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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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-- TODO: Decide whether just to use funext for this theorem or
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-- take the trick they use in Coq-HoTT, and introduce a further
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-- axiom in the definition of precategories that provides thee
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-- symmetric associativity proof.
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definition opposite_opposite' {ob : Type} (C : precategory ob) : opposite (opposite C) = C :=
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begin
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apply (precategory.rec_on C), intros (hom', homH', comp', ID', assoc', id_left', id_right'),
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apply (ap (λassoc'', precategory.mk hom' @homH' comp' ID' assoc'' id_left' id_right')),
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repeat (apply eq_of_homotopy ; intros ),
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apply ap,
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apply (@is_hset.elim), apply !homH',
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end
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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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end opposite
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-- Note: Discrete precategory doesn't really make sense in HoTT,
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-- We'll define a discrete _category_ later.
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/-section
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open decidable unit empty
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variables {A : Type} [H : decidable_eq A]
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include H
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definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
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theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
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definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
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decidable.rec_on
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(H b c)
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(λ Hbc g, decidable.rec_on
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(H a b)
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(λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
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(λh f, empty.rec _ f) f)
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(λh (g : empty), empty.rec _ g) g
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omit H
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definition discrete_precategory (A : Type) [H : decidable_eq A] : precategory A :=
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mk (λa b, set_hom a b)
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(λ a b c g f, set_compose g f)
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(λ a, decidable.rec_on_true rfl ⋆)
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(λ a b c d h g f, @subsingleton.elim (set_hom a d) _ _ _)
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(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
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(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
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definition Discrete_category (A : Type) [H : decidable_eq A] := Mk (discrete_category A)
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end
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section
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open unit bool
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definition category_one := discrete_category unit
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definition Category_one := Mk category_one
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definition category_two := discrete_category bool
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definition Category_two := Mk category_two
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end-/
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namespace product
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section
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open prod is_trunc
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definition precategory_prod [reducible] {obC obD : Type} (C : precategory obC) (D : precategory obD)
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: precategory (obC × obD) :=
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precategory.mk (λ a b, hom (pr1 a) (pr1 b) × hom (pr2 a) (pr2 b))
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(λ a b, !is_trunc_prod)
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(λ a b c g f, (pr1 g ∘ pr1 f , pr2 g ∘ pr2 f))
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(λ a, (id, id))
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(λ a b c d h g f, pair_eq !assoc !assoc )
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(λ a b f, prod_eq !id_left !id_left )
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(λ a b f, prod_eq !id_right !id_right)
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definition Precategory_prod [reducible] (C D : Precategory) : Precategory :=
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precategory.Mk (precategory_prod C D)
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end
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end product
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namespace ops
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--notation 1 := Category_one
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--notation 2 := Category_two
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postfix `ᵒᵖ`:max := opposite.Opposite
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infixr `×c`:30 := product.Precategory_prod
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--instance [persistent] type_category category_one
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-- category_two product.prod_category
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end ops
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open ops
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namespace opposite
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open ops functor
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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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end opposite
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namespace product
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section
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open ops functor
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definition prod_functor [reducible] {C C' D D' : Precategory} (F : C ⇒ D) (G : C' ⇒ D') : C ×c C' ⇒ D ×c D' :=
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functor.mk (λ a, pair (F (pr1 a)) (G (pr2 a)))
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(λ a b f, pair (F (pr1 f)) (G (pr2 f)))
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(λ a, pair_eq !respect_id !respect_id)
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(λ a b c g f, pair_eq !respect_comp !respect_comp)
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end
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end product
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definition precategory_hset [reducible] : precategory hset :=
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precategory.mk (λx y : hset, x → y)
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_
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(λx y z g f a, g (f a))
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(λx a, a)
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(λx y z w h g f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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(λx y f, eq_of_homotopy (λa, idp))
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definition Precategory_hset [reducible] : Precategory :=
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Precategory.mk hset precategory_hset
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section precategory_functor
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open iso functor nat_trans
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definition precategory_functor [instance] [reducible] (D C : Precategory)
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: precategory (functor C D) :=
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precategory.mk (λa b, nat_trans a b)
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(λ a b, @is_hset_nat_trans C D a b)
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(λ a b c g f, nat_trans.compose g f)
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(λ a, nat_trans.id)
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(λ a b c d h g f, !nat_trans.assoc)
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(λ a b f, !nat_trans.id_left)
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(λ a b f, !nat_trans.id_right)
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definition Precategory_functor [reducible] (D C : Precategory) : Precategory :=
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precategory.Mk (precategory_functor D C)
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-- definition Precategory_functor_rev [reducible] (C D : Precategory) : Precategory :=
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-- Precategory_functor D C
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/- we prove that if a natural transformation is pointwise an to_fun, then it is an to_fun -/
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variables {C D : Precategory} {F G : C ⇒ D} (η : F ⟹ G) [iso : Π(a : C), is_iso (η a)]
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include iso
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definition nat_trans_inverse : G ⟹ F :=
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nat_trans.mk
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(λc, (η c)⁻¹)
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(λc d f,
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begin
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apply comp_inverse_eq_of_eq_comp,
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apply concat, rotate_left 1, apply assoc,
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apply eq_inverse_comp_of_comp_eq,
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apply inverse,
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apply naturality,
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end)
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definition nat_trans_left_inverse : nat_trans_inverse η ∘n η = nat_trans.id :=
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begin
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fapply (apD011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply left_inverse,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
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apply is_hset.elim
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end
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definition nat_trans_right_inverse : η ∘n nat_trans_inverse η = nat_trans.id :=
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begin
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fapply (apD011 nat_trans.mk),
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apply eq_of_homotopy, intro c, apply right_inverse,
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apply eq_of_homotopy, intros, apply eq_of_homotopy, intros, apply eq_of_homotopy, intros,
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apply is_hset.elim
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end
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definition nat_trans_iso.mk : is_iso η :=
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is_iso.mk (nat_trans_left_inverse η) (nat_trans_right_inverse η)
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end precategory_functor
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namespace ops
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infixr `^c`:35 := Precategory_functor
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infixr `×f`:30 := product.prod_functor
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infixr `ᵒᵖᶠ`:(max+1) := opposite.opposite_functor
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end ops
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end category
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