feat(hott) prove that rezk functor is a weak equivalence
This commit is contained in:
Jakob von Raumer
Leonardo de Moura
parent
57bf0a09dd
commit
82a8d137da
1 changed files with 46 additions and 21 deletions
|
|
@ -1,4 +1,13 @@
|
|||
import algebra.category hit.two_quotient types.trunc types.arrow
|
||||
/-
|
||||
Copyright (c) 2016 Jakob von Raumer. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
|
||||
Authors: Jakob von Raumer
|
||||
|
||||
The Rezk completion
|
||||
-/
|
||||
|
||||
import algebra.category hit.two_quotient types.trunc types.arrow algebra.category.functor.attributes
|
||||
|
||||
open eq category equiv trunc_two_quotient is_trunc iso relation e_closure function pi
|
||||
|
||||
|
|
@ -141,7 +150,7 @@ namespace rezk_completion
|
|||
{ clear b, intro b b' b'' f g x, apply !assoc⁻¹ }
|
||||
end
|
||||
|
||||
private definition transport_rezk_hom_left_pt_eq_comp {a b c : A} (f : hom a b) (g : b ≅ c) :
|
||||
private definition pathover_rezk_hom_left_pt {a b c : A} (f : hom a b) (g : b ≅ c) :
|
||||
pathover (rezk_hom_left_pt a) f (pth g) ((to_hom g) ∘ f) :=
|
||||
begin
|
||||
apply pathover_of_tr_eq, apply @homotopy_of_eq _ _ _ (λ f, (to_hom g) ∘ f),
|
||||
|
|
@ -159,9 +168,9 @@ namespace rezk_completion
|
|||
refine @rezk_carrier.rec _ _ _ (rezk_hom_left_pth_1_trunc a a' f) _ _ _ b,
|
||||
intro b, apply equiv_precompose (to_hom f⁻¹ⁱ), --how do i unfold properly at this point?
|
||||
{ intro b b' g, apply equiv_pathover, intro g' g'' H,
|
||||
refine !transport_rezk_hom_left_pt_eq_comp ⬝op _,
|
||||
refine !pathover_rezk_hom_left_pt ⬝op _,
|
||||
refine !assoc ⬝ ap (λ x, x ∘ _) _, refine eq_of_parallel_po_right _ H,
|
||||
apply transport_rezk_hom_left_pt_eq_comp },
|
||||
apply pathover_rezk_hom_left_pt },
|
||||
intro b b' b'' g g', apply @is_prop.elim, apply is_trunc_pathover, apply is_trunc_equiv
|
||||
end
|
||||
|
||||
|
|
@ -174,17 +183,17 @@ namespace rezk_completion
|
|||
apply assoc, apply is_prop.elimo, apply is_set.elimo }
|
||||
end
|
||||
|
||||
private definition transport_rezk_hom_left_eq_comp {a b c : A} (f : hom a c) (g : a ≅ b) :
|
||||
private definition pathover_rezk_hom_left {a b c : A} (f : hom a c) (g : a ≅ b) :
|
||||
pathover (λ x, rezk_hom x (elt c)) f (pth g) (f ∘ (to_hom g)⁻¹) :=
|
||||
begin
|
||||
apply pathover_of_tr_eq, apply @homotopy_of_eq _ _ _ (λ f, f ∘ (to_hom g)⁻¹),
|
||||
apply rezk_carrier.elim_set_pth,
|
||||
end
|
||||
|
||||
private definition transport_rezk_hom_right_eq_comp {a b c : A} (f : hom a b) (g : b ≅ c) : --todo delete?
|
||||
private definition pathover_rezk_hom_right {a b c : A} (f : hom a b) (g : b ≅ c) : --todo delete?
|
||||
pathover (rezk_hom (elt a)) f (pth g) ((to_hom g) ∘ f) :=
|
||||
begin
|
||||
apply transport_rezk_hom_left_pt_eq_comp,
|
||||
apply pathover_rezk_hom_left_pt,
|
||||
end
|
||||
|
||||
private definition transport_rezk_hom_eq_comp {a c : A} (f : hom a a) (g : a ≅ c) :
|
||||
|
|
@ -192,8 +201,8 @@ namespace rezk_completion
|
|||
begin
|
||||
apply concat, apply tr_diag_eq_tr_tr rezk_hom,
|
||||
apply concat, apply ap (λ x, _ ▸ x),
|
||||
apply tr_eq_of_pathover, apply transport_rezk_hom_left_eq_comp,
|
||||
apply tr_eq_of_pathover, apply transport_rezk_hom_left_pt_eq_comp
|
||||
apply tr_eq_of_pathover, apply pathover_rezk_hom_left,
|
||||
apply tr_eq_of_pathover, apply pathover_rezk_hom_left_pt
|
||||
end
|
||||
|
||||
definition rezk_id (a : @rezk_carrier A C) : rezk_hom a a :=
|
||||
|
|
@ -212,9 +221,9 @@ namespace rezk_completion
|
|||
induction c using rezk_carrier.set_rec with c c c' ic,
|
||||
exact g ∘ f,
|
||||
{ apply arrow_pathover_left, intro d,
|
||||
apply concato !transport_rezk_hom_left_pt_eq_comp, apply pathover_idp_of_eq,
|
||||
apply concato !pathover_rezk_hom_left_pt, apply pathover_idp_of_eq,
|
||||
apply concat, apply assoc, apply ap (λ x, x ∘ f),
|
||||
apply inverse, apply tr_eq_of_pathover, apply transport_rezk_hom_left_pt_eq_comp },
|
||||
apply inverse, apply tr_eq_of_pathover, apply pathover_rezk_hom_left_pt },
|
||||
end
|
||||
|
||||
definition rezk_comp_pt_pth [reducible] {c : rezk_carrier} {a b b' : A} {ib : iso b b'} :
|
||||
|
|
@ -226,9 +235,9 @@ namespace rezk_completion
|
|||
induction c using rezk_carrier.set_rec with c c c' ic,
|
||||
{ apply pathover_of_eq, apply inverse,
|
||||
apply concat, apply ap (λ x, rezk_comp_pt_pt x _), apply tr_eq_of_pathover,
|
||||
apply transport_rezk_hom_left_eq_comp,
|
||||
apply pathover_rezk_hom_left,
|
||||
apply concat, apply ap (rezk_comp_pt_pt _), apply tr_eq_of_pathover,
|
||||
apply transport_rezk_hom_left_pt_eq_comp,
|
||||
apply pathover_rezk_hom_left_pt,
|
||||
refine !assoc ⬝ ap (λ x, x ∘ y) _,
|
||||
refine !assoc⁻¹ ⬝ _,
|
||||
refine ap (λ y, x ∘ y) !iso.left_inverse ⬝ _,
|
||||
|
|
@ -245,9 +254,9 @@ namespace rezk_completion
|
|||
{ induction b using rezk_carrier.set_rec with b b b' ib,
|
||||
apply arrow_pathover_left, intro f,
|
||||
induction c using rezk_carrier.set_rec with c c c' ic,
|
||||
{ apply concato, apply transport_rezk_hom_left_eq_comp,
|
||||
{ apply concato, apply pathover_rezk_hom_left,
|
||||
apply pathover_idp_of_eq, refine !assoc⁻¹ ⬝ ap (λ x, g ∘ x) _⁻¹,
|
||||
apply tr_eq_of_pathover, apply transport_rezk_hom_left_eq_comp },
|
||||
apply tr_eq_of_pathover, apply pathover_rezk_hom_left },
|
||||
apply is_prop.elimo,
|
||||
apply is_prop.elimo }
|
||||
end
|
||||
|
|
@ -293,22 +302,20 @@ namespace rezk_completion
|
|||
apply rezk_precategory _ _,
|
||||
end
|
||||
|
||||
definition rezk_embedding (C : Precategory) : functor C (to_rezk_Precategory C) :=
|
||||
definition rezk_functor [constructor] (C : Precategory) : functor C (to_rezk_Precategory C) :=
|
||||
begin
|
||||
fapply functor.mk, apply elt,
|
||||
{ intro a b f, exact f },
|
||||
do 2 (intros; reflexivity)
|
||||
end
|
||||
|
||||
--TODO prove that rezk_embedding is a weak equivalence
|
||||
|
||||
section
|
||||
parameters {A : Type} [C : precategory A]
|
||||
include C
|
||||
|
||||
protected definition elt_iso_of_iso [reducible] {a b : A} (f : a ≅ b) : elt a ≅ elt b :=
|
||||
begin
|
||||
fapply iso.mk, apply to_hom f, apply functor.preserve_is_iso (rezk_embedding _)
|
||||
fapply iso.mk, apply to_hom f, apply functor.preserve_is_iso (rezk_functor _)
|
||||
end
|
||||
|
||||
protected definition iso_of_elt_iso [reducible] {a b : A} (f : elt a ≅ elt b) : a ≅ b :=
|
||||
|
|
@ -345,7 +352,7 @@ namespace rezk_completion
|
|||
begin
|
||||
apply pathover_of_tr_eq, apply iso_eq,
|
||||
apply concat, apply hom_transport_eq_transport_hom,
|
||||
apply tr_eq_of_pathover, apply transport_rezk_hom_right_eq_comp A C
|
||||
apply tr_eq_of_pathover, apply pathover_rezk_hom_right A C
|
||||
end
|
||||
|
||||
private definition pathover_iso_pth' {a a' b : A} (f : elt a ≅ elt b)
|
||||
|
|
@ -353,7 +360,7 @@ namespace rezk_completion
|
|||
begin
|
||||
apply pathover_of_tr_eq, apply iso_eq,
|
||||
apply concat, apply hom_transport_eq_transport_hom',
|
||||
apply tr_eq_of_pathover, apply transport_rezk_hom_left_eq_comp A C
|
||||
apply tr_eq_of_pathover, apply pathover_rezk_hom_left A C
|
||||
end
|
||||
|
||||
private definition eq_of_iso_pt {a : A} {b : @rezk_carrier A C} :
|
||||
|
|
@ -422,5 +429,23 @@ namespace rezk_completion
|
|||
apply rezk_completion.eq_of_iso_of_eq
|
||||
end
|
||||
|
||||
section
|
||||
variable (C : Precategory)
|
||||
|
||||
definition fully_faithful_rezk_functor : fully_faithful (rezk_functor C) :=
|
||||
by intros; apply is_equiv.is_equiv_id
|
||||
|
||||
open trunc
|
||||
definition essentially_surj_rezk_functor : essentially_surjective (rezk_functor C) :=
|
||||
begin
|
||||
intro a, esimp[to_rezk_Precategory] at *,
|
||||
induction a using rezk_carrier.prop_rec with a, apply tr,
|
||||
constructor, apply iso.refl (elt a),
|
||||
end
|
||||
|
||||
definition is_weak_equiv_rezk_functor : is_weak_equivalence (rezk_functor C) :=
|
||||
prod.mk (fully_faithful_rezk_functor C) (essentially_surj_rezk_functor C)
|
||||
|
||||
end
|
||||
|
||||
end rezk_completion
|
||||
|
|
|
|||
Loading…
Reference in a new issue