fec45abda5
Jakob accidentally undid some of my changes in commit aad4592
, reverted that;
made style changes in multiple files;
in types/sigma: finished porting Coq-HoTT, and finished unfinished proof;
in axioms/funext: rename path_forall, make arguments implicit and make instance visible
368 lines
15 KiB
Text
368 lines
15 KiB
Text
-- 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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-- 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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import .natural_transformation hott.path
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import data.unit data.sigma data.prod data.empty data.bool hott.types.prod hott.types.sigma
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open path prod eq eq.ops
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namespace precategory
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namespace opposite
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section
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definition opposite {ob : Type} (C : precategory ob) : precategory ob :=
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mk (λ a b, hom b a)
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(λ b a, !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 (C : Precategory) : Precategory := Mk (opposite C)
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infixr `∘op`:60 := @compose _ (opposite _) _ _ _
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variables {C : Precategory} {a b c : C}
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theorem 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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theorem op_op' {ob : Type} (C : precategory ob) : opposite (opposite C) ≈ C :=
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sorry
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theorem op_op : Opposite (Opposite C) ≈ C :=
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(ap (Precategory.mk C) (op_op' C)) ⬝ !Precategory.equal
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end
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end opposite
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/-definition type_category : precategory Type :=
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mk (λa b, a → b)
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(λ a b c, function.compose)
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(λ a, function.id)
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(λ a b c d h g f, symm (function.compose_assoc h g f))
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(λ a b f, function.compose_id_left f)
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(λ a b f, function.compose_id_right f)
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definition Type_category : Category := Mk type_category-/
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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 truncation
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definition prod_precategory {obC obD : Type} (C : precategory obC) (D : precategory obD)
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: precategory (obC × obD) :=
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mk (λ a b, hom (pr1 a) (pr1 b) × hom (pr2 a) (pr2 b))
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(λ a b, !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_path !assoc !assoc )
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(λ a b f, prod.path !id_left !id_left )
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(λ a b f, prod.path !id_right !id_right)
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definition Prod_precategory (C D : Precategory) : Precategory := Mk (prod_precategory C D)
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end
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end product
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namespace ops
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--notation `type`:max := Type_category
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--notation 1 := Category_one --it was confusing for me (Floris) that no ``s are needed here
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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.Prod_precategory
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--instance [persistent] type_category category_one
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-- category_two product.prod_category
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instance [persistent] product.prod_precategory
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end ops
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open ops
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namespace opposite
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section
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open ops functor
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set_option pp.universes true
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definition opposite_functor {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-/ sorry
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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 {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_path !respect_id !respect_id)
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(λ a b c g f, pair_path !respect_comp !respect_comp)
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end
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end product
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namespace ops
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infixr `×f`:30 := product.prod_functor
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infixr `ᵒᵖᶠ`:max := opposite.opposite_functor
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end ops
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section functor_category
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variables (C D : Precategory)
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definition functor_category [fx : funext] : precategory (functor C D) :=
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mk (λa b, natural_transformation a b)
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sorry --TODO: Prove that the nat trafos between two functors are an hset
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(λ a b c g f, natural_transformation.compose g f)
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(λ a, natural_transformation.id)
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(λ a b c d h g f, !natural_transformation.assoc)
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(λ a b f, !natural_transformation.id_left)
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(λ a b f, !natural_transformation.id_right)
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end functor_category
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namespace slice
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open sigma function
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variables {ob : Type} {C : precategory ob} {c : ob}
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protected definition slice_obs (C : precategory ob) (c : ob) := Σ(b : ob), hom b c
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variables {a b : slice_obs C c}
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protected definition to_ob (a : slice_obs C c) : ob := dpr1 a
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protected definition to_ob_def (a : slice_obs C c) : to_ob a = dpr1 a := rfl
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protected definition ob_hom (a : slice_obs C c) : hom (to_ob a) c := dpr2 a
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-- protected theorem slice_obs_equal (H₁ : to_ob a = to_ob b)
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-- (H₂ : eq.drec_on H₁ (ob_hom a) = ob_hom b) : a = b :=
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-- sigma.equal H₁ H₂
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protected definition slice_hom (a b : slice_obs C c) : Type :=
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Σ(g : hom (to_ob a) (to_ob b)), ob_hom b ∘ g = ob_hom a
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protected definition hom_hom (f : slice_hom a b) : hom (to_ob a) (to_ob b) := dpr1 f
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protected definition commute (f : slice_hom a b) : ob_hom b ∘ (hom_hom f) = ob_hom a := dpr2 f
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-- protected theorem slice_hom_equal (f g : slice_hom a b) (H : hom_hom f = hom_hom g) : f = g :=
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-- sigma.equal H !proof_irrel
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/- TODO wait for some helping lemmas
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definition slice_category (C : precategory ob) (c : ob) : precategory (slice_obs C c) :=
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mk (λa b, slice_hom a b)
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sorry
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(λ a b c g f, dpair (hom_hom g ∘ hom_hom f)
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(show ob_hom c ∘ (hom_hom g ∘ hom_hom f) ≈ ob_hom a,
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proof
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calc
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ob_hom c ∘ (hom_hom g ∘ hom_hom f) ≈ (ob_hom c ∘ hom_hom g) ∘ hom_hom f : !assoc
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... ≈ ob_hom b ∘ hom_hom f : {commute g}
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... ≈ ob_hom a : {commute f}
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qed))
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(λ a, dpair id !id_right)
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(λ a b c d h g f, dpair_path !assoc sorry)
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(λ a b f, sigma.path !id_left sorry)
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(λ a b f, sigma.path !id_right sorry)
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-/
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-- definition slice_category {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom b c)
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-- :=
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-- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), dpr2 b ∘ g = dpr2 a)
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-- (λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
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-- (show dpr2 c ∘ (dpr1 g ∘ dpr1 f) = dpr2 a,
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-- proof
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-- calc
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-- dpr2 c ∘ (dpr1 g ∘ dpr1 f) = (dpr2 c ∘ dpr1 g) ∘ dpr1 f : !assoc
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-- ... = dpr2 b ∘ dpr1 f : {dpr2 g}
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-- ... = dpr2 a : {dpr2 f}
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-- qed))
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-- (λ a, dpair id !id_right)
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-- (λ a b c d h g f, dpair_eq !assoc !proof_irrel)
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-- (λ a b f, sigma.equal !id_left !proof_irrel)
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-- (λ a b f, sigma.equal !id_right !proof_irrel)
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-- We use !proof_irrel instead of rfl, to give the unifier an easier time
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exit
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definition Slice_category [reducible] (C : Category) (c : C) := Mk (slice_category C c)
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open category.ops
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instance [persistent] slice_category
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variables {D : Category}
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definition forgetful (x : D) : (Slice_category D x) ⇒ D :=
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functor.mk (λ a, to_ob a)
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(λ a b f, hom_hom f)
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(λ a, rfl)
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(λ a b c g f, rfl)
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definition postcomposition_functor {x y : D} (h : x ⟶ y)
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: Slice_category D x ⇒ Slice_category D y :=
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functor.mk (λ a, dpair (to_ob a) (h ∘ ob_hom a))
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(λ a b f, dpair (hom_hom f)
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(calc
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(h ∘ ob_hom b) ∘ hom_hom f = h ∘ (ob_hom b ∘ hom_hom f) : assoc h (ob_hom b) (hom_hom f)⁻¹
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... = h ∘ ob_hom a : congr_arg (λx, h ∘ x) (commute f)))
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(λ a, rfl)
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(λ a b c g f, dpair_eq rfl !proof_irrel)
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-- -- in the following comment I tried to have (A = B) in the type of a == b, but that doesn't solve the problems
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-- definition heq2 {A B : Type} (H : A = B) (a : A) (b : B) := a == b
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-- definition heq2.intro {A B : Type} {a : A} {b : B} (H : a == b) : heq2 (heq.type_eq H) a b := H
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-- definition heq2.elim {A B : Type} {a : A} {b : B} (H : A = B) (H2 : heq2 H a b) : a == b := H2
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-- definition heq2.proof_irrel {A B : Prop} (a : A) (b : B) (H : A = B) : heq2 H a b :=
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-- hproof_irrel H a b
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-- theorem functor.mk_eq2 {C D : Category} {obF obG : C → D} {homF homG idF idG compF compG}
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-- (Hob : ∀x, obF x = obG x)
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-- (Hmor : ∀(a b : C) (f : a ⟶ b), heq2 (congr_arg (λ x, x a ⟶ x b) (funext Hob)) (homF a b f) (homG a b f))
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-- : functor.mk obF homF idF compF = functor.mk obG homG idG compG :=
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-- hddcongr_arg4 functor.mk
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-- (funext Hob)
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-- (hfunext (λ a, hfunext (λ b, hfunext (λ f, !Hmor))))
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-- !proof_irrel
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-- !proof_irrel
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-- set_option pp.implicit true
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-- set_option pp.coercions true
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-- definition slice_functor : D ⇒ Category_of_categories :=
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-- functor.mk (λ a, Category.mk (slice_obs D a) (slice_category D a))
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-- (λ a b f, postcomposition_functor f)
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-- (λ a, functor.mk_heq
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-- (λx, sigma.equal rfl !id_left)
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-- (λb c f, sigma.hequal sorry !heq.refl (hproof_irrel sorry _ _)))
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-- (λ a b c g f, functor.mk_heq
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-- (λx, sigma.equal (sorry ⬝ refl (dpr1 x)) sorry)
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-- (λb c f, sorry))
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--the error message generated here is really confusing: the type of the above refl should be
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-- "@dpr1 D (λ (a_1 : D), a_1 ⟶ a) x = @dpr1 D (λ (a_1 : D), a_1 ⟶ c) x", but the second dpr1 is not even well-typed
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end slice
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-- section coslice
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-- open sigma
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-- definition coslice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom c b) :=
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-- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), g ∘ dpr2 a = dpr2 b)
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-- (λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
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-- (show (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr2 c,
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-- proof
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-- calc
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-- (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr1 g ∘ (dpr1 f ∘ dpr2 a): symm !assoc
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-- ... = dpr1 g ∘ dpr2 b : {dpr2 f}
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-- ... = dpr2 c : {dpr2 g}
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-- qed))
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-- (λ a, dpair id !id_left)
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-- (λ a b c d h g f, dpair_eq !assoc !proof_irrel)
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-- (λ a b f, sigma.equal !id_left !proof_irrel)
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-- (λ a b f, sigma.equal !id_right !proof_irrel)
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-- -- theorem slice_coslice_opp {ob : Type} (C : category ob) (c : ob) :
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-- -- coslice C c = opposite (slice (opposite C) c) :=
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-- -- sorry
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-- end coslice
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section arrow
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open sigma eq.ops
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-- theorem concat_commutative_squares {ob : Type} {C : category ob} {a1 a2 a3 b1 b2 b3 : ob}
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-- {f1 : a1 => b1} {f2 : a2 => b2} {f3 : a3 => b3} {g2 : a2 => a3} {g1 : a1 => a2}
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-- {h2 : b2 => b3} {h1 : b1 => b2} (H1 : f2 ∘ g1 = h1 ∘ f1) (H2 : f3 ∘ g2 = h2 ∘ f2)
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-- : f3 ∘ (g2 ∘ g1) = (h2 ∘ h1) ∘ f1 :=
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-- calc
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-- f3 ∘ (g2 ∘ g1) = (f3 ∘ g2) ∘ g1 : assoc
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-- ... = (h2 ∘ f2) ∘ g1 : {H2}
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-- ... = h2 ∘ (f2 ∘ g1) : symm assoc
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-- ... = h2 ∘ (h1 ∘ f1) : {H1}
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-- ... = (h2 ∘ h1) ∘ f1 : assoc
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-- definition arrow {ob : Type} (C : category ob) : category (Σ(a b : ob), hom a b) :=
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-- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)) (h : hom (dpr2' a) (dpr2' b)),
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-- dpr3 b ∘ g = h ∘ dpr3 a)
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-- (λ a b c g f, dpair (dpr1 g ∘ dpr1 f) (dpair (dpr2' g ∘ dpr2' f) (concat_commutative_squares (dpr3 f) (dpr3 g))))
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-- (λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
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-- (λ a b c d h g f, dtrip_eq2 assoc assoc !proof_irrel)
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-- (λ a b f, trip.equal2 id_left id_left !proof_irrel)
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-- (λ a b f, trip.equal2 id_right id_right !proof_irrel)
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-- make these definitions private?
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variables {ob : Type} {C : category ob}
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protected definition arrow_obs (ob : Type) (C : category ob) := Σ(a b : ob), hom a b
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variables {a b : arrow_obs ob C}
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protected definition src (a : arrow_obs ob C) : ob := dpr1 a
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protected definition dst (a : arrow_obs ob C) : ob := dpr2' a
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protected definition to_hom (a : arrow_obs ob C) : hom (src a) (dst a) := dpr3 a
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protected definition arrow_hom (a b : arrow_obs ob C) : Type :=
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Σ (g : hom (src a) (src b)) (h : hom (dst a) (dst b)), to_hom b ∘ g = h ∘ to_hom a
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protected definition hom_src (m : arrow_hom a b) : hom (src a) (src b) := dpr1 m
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protected definition hom_dst (m : arrow_hom a b) : hom (dst a) (dst b) := dpr2' m
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protected definition commute (m : arrow_hom a b) : to_hom b ∘ (hom_src m) = (hom_dst m) ∘ to_hom a
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:= dpr3 m
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definition arrow (ob : Type) (C : category ob) : category (arrow_obs ob C) :=
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mk (λa b, arrow_hom a b)
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(λ a b c g f, dpair (hom_src g ∘ hom_src f) (dpair (hom_dst g ∘ hom_dst f)
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(show to_hom c ∘ (hom_src g ∘ hom_src f) = (hom_dst g ∘ hom_dst f) ∘ to_hom a,
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proof
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calc
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to_hom c ∘ (hom_src g ∘ hom_src f) = (to_hom c ∘ hom_src g) ∘ hom_src f : !assoc
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... = (hom_dst g ∘ to_hom b) ∘ hom_src f : {commute g}
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... = hom_dst g ∘ (to_hom b ∘ hom_src f) : symm !assoc
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... = hom_dst g ∘ (hom_dst f ∘ to_hom a) : {commute f}
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... = (hom_dst g ∘ hom_dst f) ∘ to_hom a : !assoc
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qed)
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))
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(λ a, dpair id (dpair id (!id_right ⬝ (symm !id_left))))
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(λ a b c d h g f, ndtrip_eq !assoc !assoc !proof_irrel)
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(λ a b f, ndtrip_equal !id_left !id_left !proof_irrel)
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(λ a b f, ndtrip_equal !id_right !id_right !proof_irrel)
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end arrow
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end category
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-- definition foo
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-- : category (sorry) :=
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-- mk (λa b, sorry)
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-- (λ a b c g f, sorry)
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-- (λ a, sorry)
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-- (λ a b c d h g f, sorry)
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-- (λ a b f, sorry)
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-- (λ a b f, sorry)
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