lean2/hott/algebra/precategory/constructions.lean

-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Authors: Floris van Doorn, Jakob von Raumer

-- This file contains basic constructions on precategories, including common precategories


import .natural_transformation hott.path
import data.unit data.sigma data.prod data.empty data.bool hott.types.prod hott.types.sigma

open path prod eq eq.ops

namespace precategory
  namespace opposite
  section
  definition opposite {ob : Type} (C : precategory ob) : precategory ob :=
  mk (λ a b, hom b a)
     (λ b a, !homH)
     (λ a b c f g, g ∘ f)
     (λ a, id)
     (λ a b c d f g h, !assoc⁻¹)
     (λ a b f, !id_right)
     (λ a b f, !id_left)

  definition Opposite (C : Precategory) : Precategory := Mk (opposite C)

  infixr `∘op`:60 := @compose _ (opposite _) _ _ _

  variables {C : Precategory} {a b c : C}

  theorem compose_op {f : hom a b} {g : hom b c} : f ∘op g ≈ g ∘ f := idp

  -- TODO: Decide whether just to use funext for this theorem or
  --       take the trick they use in Coq-HoTT, and introduce a further
  --       axiom in the definition of precategories that provides thee
  --       symmetric associativity proof.
  theorem op_op' {ob : Type} (C : precategory ob) : opposite (opposite C) ≈ C :=
  sorry

  theorem op_op : Opposite (Opposite C) ≈ C :=
  (ap (Precategory.mk C) (op_op' C)) ⬝ !Precategory.equal

  end
  end opposite

  /-definition type_category : precategory Type :=
  mk (λa b, a → b)
     (λ a b c, function.compose)
     (λ a, function.id)
     (λ a b c d h g f, symm (function.compose_assoc h g f))
     (λ a b f, function.compose_id_left f)
     (λ a b f, function.compose_id_right f)

  definition Type_category : Category := Mk type_category-/

  -- Note: Discrete precategory doesn't really make sense in HoTT,
  -- We'll define a discrete _category_ later.
  /-section
  open decidable unit empty
  variables {A : Type} [H : decidable_eq A]
  include H
  definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
  theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
  definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
  decidable.rec_on
    (H b c)
    (λ Hbc g, decidable.rec_on
      (H a b)
      (λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
      (λh f, empty.rec _ f) f)
    (λh (g : empty), empty.rec _ g) g
  omit H
  definition discrete_precategory (A : Type) [H : decidable_eq A] : precategory A :=
  mk (λa b, set_hom a b)
     (λ a b c g f, set_compose g f)
     (λ a, decidable.rec_on_true rfl ⋆)
     (λ a b c d h g f, @subsingleton.elim (set_hom a d) _ _ _)
     (λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
     (λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
  definition Discrete_category (A : Type) [H : decidable_eq A] := Mk (discrete_category A)
  end
  section
  open unit bool
  definition category_one := discrete_category unit
  definition Category_one := Mk category_one
  definition category_two := discrete_category bool
  definition Category_two := Mk category_two
  end-/

  namespace product
  section
  open prod truncation

  definition prod_precategory {obC obD : Type} (C : precategory obC) (D : precategory obD)
      : precategory (obC × obD) :=
  mk (λ a b, hom (pr1 a) (pr1 b) × hom (pr2 a) (pr2 b))
     (λ a b, !trunc_prod)
     (λ a b c g f, (pr1 g ∘ pr1 f , pr2 g ∘ pr2 f) )
     (λ a, (id, id))
     (λ a b c d h g f, pair_path  !assoc    !assoc   )
     (λ a b f,         prod.path  !id_left  !id_left )
     (λ a b f,         prod.path  !id_right !id_right)

  definition Prod_precategory (C D : Precategory) : Precategory := Mk (prod_precategory C D)

  end
  end product

  namespace ops

    --notation `type`:max := Type_category
    --notation 1 := Category_one --it was confusing for me (Floris) that no ``s are needed here
    --notation 2 := Category_two
    postfix `ᵒᵖ`:max := opposite.Opposite
    infixr `×c`:30 := product.Prod_precategory
    --instance [persistent] type_category category_one
    --                      category_two product.prod_category
    instance [persistent] product.prod_precategory

  end ops

  open ops
  namespace opposite
  section
  open ops functor
  set_option pp.universes true

  definition opposite_functor {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-/ sorry

  end
  end opposite

  namespace product
  section
  open ops functor
  definition prod_functor {C C' D D' : Precategory} (F : C ⇒ D) (G : C' ⇒ D') : C ×c C' ⇒ D ×c D' :=
  functor.mk (λ a, pair (F (pr1 a)) (G (pr2 a)))
             (λ a b f, pair (F (pr1 f)) (G (pr2 f)))
             (λ a, pair_path !respect_id !respect_id)
             (λ a b c g f, pair_path !respect_comp !respect_comp)
  end
  end product

  namespace ops
  infixr `×f`:30 := product.prod_functor
  infixr `ᵒᵖᶠ`:max := opposite.opposite_functor
  end ops

  section functor_category
  variables (C D : Precategory)
  definition functor_category [fx : funext] : precategory (functor C D) :=
  mk (λa b, natural_transformation a b)
     sorry --TODO: Prove that the nat trafos between two functors are an hset
     (λ a b c g f, natural_transformation.compose g f)
     (λ a, natural_transformation.id)
     (λ a b c d h g f, !natural_transformation.assoc)
     (λ a b f, !natural_transformation.id_left)
     (λ a b f, !natural_transformation.id_right)
  end functor_category

  namespace slice
  open sigma function
  variables {ob : Type} {C : precategory ob} {c : ob}
  protected definition slice_obs (C : precategory ob) (c : ob) := Σ(b : ob), hom b c
  variables {a b : slice_obs C c}
  protected definition to_ob  (a : slice_obs C c) : ob              := dpr1 a
  protected definition to_ob_def (a : slice_obs C c) : to_ob a = dpr1 a := rfl
  protected definition ob_hom (a : slice_obs C c) : hom (to_ob a) c := dpr2 a
  -- protected theorem slice_obs_equal (H₁ : to_ob a = to_ob b)
  --     (H₂ : eq.drec_on H₁ (ob_hom a) = ob_hom b) : a = b :=
  -- sigma.equal H₁ H₂


  protected definition slice_hom (a b : slice_obs C c) : Type :=
  Σ(g : hom (to_ob a) (to_ob b)), ob_hom b ∘ g = ob_hom a

  protected definition hom_hom  (f : slice_hom a b) : hom (to_ob a) (to_ob b)          := dpr1 f
  protected definition commute (f : slice_hom a b) : ob_hom b ∘ (hom_hom f) = ob_hom a := dpr2 f
  -- protected theorem slice_hom_equal (f g : slice_hom a b) (H : hom_hom f = hom_hom g) : f = g :=
  -- sigma.equal H !proof_irrel

  /- TODO wait for some helping lemmas
  definition slice_category (C : precategory ob) (c : ob) : precategory (slice_obs C c) :=
  mk (λa b, slice_hom a b)
     sorry
     (λ a b c g f, dpair (hom_hom g ∘ hom_hom f)
       (show ob_hom c ∘ (hom_hom g ∘ hom_hom f) ≈ ob_hom a,
         proof
         calc
           ob_hom c ∘ (hom_hom g ∘ hom_hom f) ≈ (ob_hom c ∘ hom_hom g) ∘ hom_hom f : !assoc
             ... ≈ ob_hom b ∘ hom_hom f : {commute g}
             ... ≈ ob_hom a : {commute f}
         qed))
     (λ a, dpair id !id_right)
     (λ a b c d h g f, dpair_path  !assoc    sorry)
     (λ a b f,         sigma.path !id_left  sorry)
     (λ a b f,         sigma.path !id_right sorry)
  -/


  -- definition slice_category {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom b c)
  -- :=
  -- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), dpr2 b ∘ g = dpr2 a)
  --    (λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
  --      (show dpr2 c ∘ (dpr1 g ∘ dpr1 f) = dpr2 a,
  --        proof
  --        calc
  --          dpr2 c ∘ (dpr1 g ∘ dpr1 f) = (dpr2 c ∘ dpr1 g) ∘ dpr1 f : !assoc
  --            ... = dpr2 b ∘ dpr1 f : {dpr2 g}
  --            ... = dpr2 a : {dpr2 f}
  --        qed))
  --    (λ a, dpair id !id_right)
  --    (λ a b c d h g f, dpair_eq    !assoc    !proof_irrel)
  --    (λ a b f,         sigma.equal !id_left  !proof_irrel)
  --    (λ a b f,         sigma.equal !id_right !proof_irrel)
  -- We use !proof_irrel instead of rfl, to give the unifier an easier time

exit
  definition Slice_category [reducible] (C : Category) (c : C) := Mk (slice_category C c)
  open category.ops
  instance [persistent] slice_category
  variables {D : Category}
  definition forgetful (x : D) : (Slice_category D x) ⇒ D :=
  functor.mk (λ a, to_ob a)
             (λ a b f, hom_hom f)
             (λ a, rfl)
             (λ a b c g f, rfl)

  definition postcomposition_functor {x y : D} (h : x ⟶ y)
      : Slice_category D x ⇒ Slice_category D y :=
  functor.mk (λ a, dpair (to_ob a) (h ∘ ob_hom a))
             (λ a b f, dpair (hom_hom f)
       (calc
         (h ∘ ob_hom b) ∘ hom_hom f = h ∘ (ob_hom b ∘ hom_hom f) : assoc h (ob_hom b) (hom_hom f)⁻¹
           ... = h ∘ ob_hom a : congr_arg (λx, h ∘ x) (commute f)))
             (λ a, rfl)
             (λ a b c g f, dpair_eq rfl !proof_irrel)

  -- -- in the following comment I tried to have (A = B) in the type of a == b, but that doesn't solve the problems
  -- definition heq2 {A B : Type} (H : A = B) (a : A) (b : B) := a == b
  -- definition heq2.intro {A B : Type} {a : A} {b : B} (H : a == b) : heq2 (heq.type_eq H) a b := H
  -- definition heq2.elim {A B : Type} {a : A} {b : B} (H : A = B) (H2 : heq2 H a b) : a == b := H2
  -- definition heq2.proof_irrel {A B : Prop} (a : A) (b : B) (H : A = B) : heq2 H a b :=
  -- hproof_irrel H a b
  -- theorem functor.mk_eq2 {C D : Category} {obF obG : C → D} {homF homG idF idG compF compG}
  --    (Hob : ∀x, obF x = obG x)
  --    (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))
  --      : functor.mk obF homF idF compF = functor.mk obG homG idG compG :=
  -- hddcongr_arg4 functor.mk
  --             (funext Hob)
  --             (hfunext (λ a, hfunext (λ b, hfunext (λ f, !Hmor))))
  --             !proof_irrel
  --             !proof_irrel

--  set_option pp.implicit true
--  set_option pp.coercions true

  -- definition slice_functor : D ⇒ Category_of_categories :=
  -- functor.mk (λ a, Category.mk (slice_obs D a) (slice_category D a))
  --            (λ a b f, postcomposition_functor f)
  --            (λ a, functor.mk_heq
  --              (λx, sigma.equal rfl !id_left)
  --              (λb c f, sigma.hequal sorry !heq.refl (hproof_irrel sorry _ _)))
  --            (λ a b c g f, functor.mk_heq
  --                    (λx, sigma.equal (sorry ⬝ refl (dpr1 x)) sorry)
  --                    (λb c f, sorry))

  --the error message generated here is really confusing: the type of the above refl should be
  -- "@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

  end slice

  -- section coslice
  -- open sigma

  -- definition coslice {ob : Type} (C : category ob) (c : ob) : category (Σ(b : ob), hom c b) :=
  -- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)), g ∘ dpr2 a = dpr2 b)
  --    (λ a b c g f, dpair (dpr1 g ∘ dpr1 f)
  --      (show (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr2 c,
  --        proof
  --        calc
  --          (dpr1 g ∘ dpr1 f) ∘ dpr2 a = dpr1 g ∘ (dpr1 f ∘ dpr2 a): symm !assoc
  --            ... = dpr1 g ∘ dpr2 b : {dpr2 f}
  --            ... = dpr2 c : {dpr2 g}
  --        qed))
  --    (λ a, dpair id !id_left)
  --    (λ a b c d h g f, dpair_eq    !assoc    !proof_irrel)
  --    (λ a b f,         sigma.equal !id_left  !proof_irrel)
  --    (λ a b f,         sigma.equal !id_right !proof_irrel)

  -- -- theorem slice_coslice_opp {ob : Type} (C : category ob) (c : ob) :
  -- --     coslice C c = opposite (slice (opposite C) c) :=
  -- -- sorry
  -- end coslice

  section arrow
  open sigma eq.ops
  -- theorem concat_commutative_squares {ob : Type} {C : category ob} {a1 a2 a3 b1 b2 b3 : ob}
  --     {f1 : a1 => b1} {f2 : a2 => b2} {f3 : a3 => b3} {g2 : a2 => a3} {g1 : a1 => a2}
  --     {h2 : b2 => b3} {h1 : b1 => b2} (H1 : f2 ∘ g1 = h1 ∘ f1) (H2 : f3 ∘ g2 = h2 ∘ f2)
  --       : f3 ∘ (g2 ∘ g1) = (h2 ∘ h1) ∘ f1 :=
  -- calc
  --   f3 ∘ (g2 ∘ g1) = (f3 ∘ g2) ∘ g1 : assoc
  --     ... = (h2 ∘ f2) ∘ g1 : {H2}
  --     ... = h2 ∘ (f2 ∘ g1) : symm assoc
  --     ... = h2 ∘ (h1 ∘ f1) : {H1}
  --     ... = (h2 ∘ h1) ∘ f1 : assoc

  -- definition arrow {ob : Type} (C : category ob) : category (Σ(a b : ob), hom a b) :=
  -- mk (λa b, Σ(g : hom (dpr1 a) (dpr1 b)) (h : hom (dpr2' a) (dpr2' b)),
  --      dpr3 b ∘ g = h ∘ dpr3 a)
  --    (λ a b c g f, dpair (dpr1 g ∘ dpr1 f) (dpair (dpr2' g ∘ dpr2' f) (concat_commutative_squares (dpr3 f) (dpr3 g))))
  --    (λ a, dpair id (dpair id (id_right ⬝ (symm id_left))))
  --    (λ a b c d h g f, dtrip_eq2   assoc    assoc    !proof_irrel)
  --    (λ a b f,         trip.equal2 id_left  id_left  !proof_irrel)
  --    (λ a b f,         trip.equal2 id_right id_right !proof_irrel)

  -- make these definitions private?
  variables {ob : Type} {C : category ob}
  protected definition arrow_obs (ob : Type) (C : category ob) := Σ(a b : ob), hom a b
  variables {a b : arrow_obs ob C}
  protected definition src    (a : arrow_obs ob C) : ob                  := dpr1 a
  protected definition dst    (a : arrow_obs ob C) : ob                  := dpr2' a
  protected definition to_hom (a : arrow_obs ob C) : hom (src a) (dst a) := dpr3 a

  protected definition arrow_hom (a b : arrow_obs ob C) : Type :=
  Σ (g : hom (src a) (src b)) (h : hom (dst a) (dst b)), to_hom b ∘ g = h ∘ to_hom a

  protected definition hom_src (m : arrow_hom a b) : hom (src a) (src b) := dpr1 m
  protected definition hom_dst (m : arrow_hom a b) : hom (dst a) (dst b) := dpr2' m
  protected definition commute (m : arrow_hom a b) : to_hom b ∘ (hom_src m) = (hom_dst m) ∘ to_hom a
  := dpr3 m

  definition arrow (ob : Type) (C : category ob) : category (arrow_obs ob C) :=
  mk (λa b, arrow_hom a b)
     (λ a b c g f, dpair (hom_src g ∘ hom_src f) (dpair (hom_dst g ∘ hom_dst f)
        (show to_hom c ∘ (hom_src g ∘ hom_src f) = (hom_dst g ∘ hom_dst f) ∘ to_hom a,
         proof
         calc
         to_hom c ∘ (hom_src g ∘ hom_src f) = (to_hom c ∘ hom_src g) ∘ hom_src f : !assoc
           ... = (hom_dst g ∘ to_hom b) ∘ hom_src f  : {commute g}
           ... = hom_dst g ∘ (to_hom b ∘ hom_src f)  : symm !assoc
           ... = hom_dst g ∘ (hom_dst f ∘ to_hom a)  : {commute f}
           ... = (hom_dst g ∘ hom_dst f) ∘ to_hom a  : !assoc
         qed)
       ))
     (λ a, dpair id (dpair id (!id_right ⬝ (symm !id_left))))
     (λ a b c d h g f, ndtrip_eq       !assoc    !assoc    !proof_irrel)
     (λ a b f,         ndtrip_equal !id_left  !id_left  !proof_irrel)
     (λ a b f,         ndtrip_equal !id_right !id_right !proof_irrel)

  end arrow

end category

  -- definition foo
  --     : category (sorry) :=
  -- mk (λa b, sorry)
  --    (λ a b c g f, sorry)
  --    (λ a, sorry)
  --    (λ a b c d h g f, sorry)
  --    (λ a b f, sorry)
  --    (λ a b f, sorry)