feat(hott): some more renaming in category library
This commit is contained in:
Floris van Doorn
parent
326eaffafb
commit
1559e0e58c
13 changed files with 391 additions and 115 deletions
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@ -8,7 +8,7 @@ Author: Jakob von Raumer
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import algebra.precategory.iso
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open morphism is_equiv eq is_trunc
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open iso is_equiv eq is_trunc
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-- A category is a precategory extended by a witness
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-- that the function from paths to isomorphisms,
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@ -45,7 +45,7 @@ namespace category
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fapply is_trunc_is_equiv_closed,
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exact (@eq_of_iso _ _ a b),
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apply is_equiv_inv,
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apply is_hset_isomorphic,
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apply is_hset_iso,
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end
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end basic
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@ -8,23 +8,51 @@ Authors: Floris van Doorn
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import .basic algebra.precategory.constructions
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open eq prod eq eq.ops equiv is_trunc funext pi category.ops morphism category
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--open eq eq.ops equiv category.ops iso category is_trunc
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open eq category equiv iso is_equiv category.ops is_trunc iso.iso
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--TODO: MOVE THIS
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namespace equiv
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variables {A B : Type}
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protected definition eq_mk' {f f' : A → B} [H : is_equiv f] [H' : is_equiv f'] (p : f = f')
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: equiv.mk f H = equiv.mk f' H' :=
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apD011 equiv.mk p sorry --!is_hprop.elim
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protected definition eq_mk {f f' : A ≃ B} (p : to_fun f = to_fun f') : f = f' :=
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by (cases f; cases f'; apply (equiv.eq_mk' p))
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end equiv
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namespace category
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section hset
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definition is_univalent_hset (a b : Precategory_hset) : is_equiv (@iso_of_eq _ _ a b) :=
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namespace set
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definition equiv_equiv_iso (A B : Precategory_hset) : (A ≃ B) ≃ (A ≅ B) :=
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equiv.MK (λf, iso.MK (to_fun f)
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(equiv.to_inv f)
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(eq_of_homotopy (sect (to_fun f)))
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(eq_of_homotopy (retr (to_fun f))))
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(λf, equiv.MK (to_hom f)
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(iso.to_inv f)
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(ap10 (right_inverse (to_hom f)))
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(ap10 (left_inverse (to_hom f))))
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(λf, iso.eq_mk idp)
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(λf, equiv.eq_mk idp)
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definition equiv_eq_iso (A B : Precategory_hset) : (A ≃ B) = (A ≅ B) :=
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ua !equiv_equiv_iso
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definition is_univalent (A B : Precategory_hset) : is_equiv (@iso_of_eq _ _ A B) :=
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sorry
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end set
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definition category_hset [reducible] [instance] : category hset :=
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category.mk' hset precategory_hset is_univalent_hset
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category.mk' hset precategory_hset set.is_univalent
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definition Category_hset [reducible] : Category :=
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Category.mk hset category_hset
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definition isomorphic_hset_eq_equiv (a b : Category_hset) : (a ≅ b) = (a ≃ b) := sorry
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end hset
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namespace ops
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abbreviation set := Category_hset
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end ops
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@ -8,9 +8,9 @@ Author: Jakob von Raumer
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Ported from Coq HoTT
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-/
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import .precategory.morphism .group
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import .precategory.iso .group
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open eq is_trunc morphism category path_algebra nat unit
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open eq is_trunc iso category path_algebra nat unit
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namespace category
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@ -48,8 +48,8 @@ namespace category
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apply (ID (center ob)),
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intro f, apply id_left,
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intro f, apply id_right,
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intro f, exact (morphism.inverse f),
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intro f, exact (morphism.inverse_comp f),
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intro f, exact (iso.inverse f),
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intro f, exact (iso.left_inverse f),
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end
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definition group_of_groupoid_unit [G : groupoid unit] : group (hom ⋆ ⋆) :=
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@ -61,8 +61,8 @@ namespace category
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apply (ID ⋆),
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intro f, apply id_left,
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intro f, apply id_right,
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intro f, exact (morphism.inverse f),
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intro f, exact (morphism.inverse_comp f),
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intro f, exact (iso.inverse f),
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intro f, exact (iso.left_inverse f),
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end
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-- Conversely we can turn each group into a groupoid on the unit type
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@ -91,8 +91,8 @@ namespace category
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apply (ID a),
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intro f, apply id_left,
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intro f, apply id_right,
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intro f, exact (morphism.inverse f),
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intro f, exact (morphism.inverse_comp f),
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intro f, exact (iso.inverse f),
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intro f, exact (iso.left_inverse f),
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end
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-- Bundled version of categories
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@ -152,7 +152,7 @@ namespace category
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Precategory.mk hset precategory_hset
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section precategory_functor
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open morphism functor nat_trans
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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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@ -177,16 +177,16 @@ namespace category
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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 to_fun.comp_inverse_eq_of_eq_comp,
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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 to_fun.eq_inverse_comp_of_comp_eq,
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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 inverse_comp,
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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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@ -194,7 +194,7 @@ namespace category
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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 comp_inverse,
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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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@ -1,72 +1,316 @@
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/-
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Copyright (c) 2014 Jakob von Raumer. All rights reserved.
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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.to_fun
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Authors: Floris van Doorn, Jakob von Raumer
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Module: algebra.category.morphism
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Author: Floris van Doorn, Jakob von Raumer
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-/
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import .basic .morphism types.sigma
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import algebra.precategory.basic types.sigma
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open eq category sigma sigma.ops equiv is_equiv function is_trunc
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open prod
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open eq category prod equiv is_equiv sigma sigma.ops is_trunc
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namespace morphism
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variables {ob : Type} [C : precategory ob] include C
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namespace iso
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structure split_mono [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
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{retraction_of : b ⟶ a}
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(retraction_comp : retraction_of ∘ f = id)
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structure split_epi [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
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{section_of : b ⟶ a}
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(comp_section : f ∘ section_of = id)
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structure is_iso [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
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{inverse : b ⟶ a}
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(left_inverse : inverse ∘ f = id)
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(right_inverse : f ∘ inverse = id)
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attribute is_iso [multiple-instances]
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open split_mono split_epi is_iso
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definition retraction_of [reducible] := @split_mono.retraction_of
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definition retraction_comp [reducible] := @split_mono.retraction_comp
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definition section_of [reducible] := @split_epi.section_of
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definition comp_section [reducible] := @split_epi.comp_section
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definition inverse [reducible] := @is_iso.inverse
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definition left_inverse [reducible] := @is_iso.left_inverse
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definition right_inverse [reducible] := @is_iso.right_inverse
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postfix `⁻¹` := inverse
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--a second notation for the inverse, which is not overloaded
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postfix [parsing-only] `⁻¹ʰ`:std.prec.max_plus := inverse
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variables {ob : Type} [C : precategory ob]
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variables {a b c : ob} {g : b ⟶ c} {f : a ⟶ b} {h : b ⟶ a}
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include C
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definition split_mono_of_is_iso [instance] [priority 300] [reducible]
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(f : a ⟶ b) [H : is_iso f] : split_mono f :=
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split_mono.mk !left_inverse
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definition split_epi_of_is_iso [instance] [priority 300] [reducible]
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(f : a ⟶ b) [H : is_iso f] : split_epi f :=
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split_epi.mk !right_inverse
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definition is_iso_id [instance] [priority 500] (a : ob) : is_iso (ID a) :=
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is_iso.mk !id_comp !id_comp
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definition is_iso_inverse [instance] [priority 200] (f : a ⟶ b) [H : is_iso f] : is_iso f⁻¹ :=
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is_iso.mk !right_inverse !left_inverse
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definition left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
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(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
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by rewrite [-(id_right g), -Hr, assoc, Hl, id_left]
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definition retraction_eq [H : split_mono f] (H2 : f ∘ h = id) : retraction_of f = h :=
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left_inverse_eq_right_inverse !retraction_comp H2
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definition section_eq [H : split_epi f] (H2 : h ∘ f = id) : section_of f = h :=
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(left_inverse_eq_right_inverse H2 !comp_section)⁻¹
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definition inverse_eq_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
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left_inverse_eq_right_inverse !left_inverse H2
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definition inverse_eq_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
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(left_inverse_eq_right_inverse H2 !right_inverse)⁻¹
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definition retraction_eq_section (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f] :
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retraction_of f = section_of f :=
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retraction_eq !comp_section
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definition is_iso_of_split_epi_of_split_mono (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f]
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: is_iso f :=
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is_iso.mk ((retraction_eq_section f) ▹ (retraction_comp f)) (comp_section f)
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definition inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
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inverse_eq_left !left_inverse
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definition inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ = f :=
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inverse_eq_right !left_inverse
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definition retraction_id (a : ob) : retraction_of (ID a) = id :=
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retraction_eq !id_comp
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definition section_id (a : ob) : section_of (ID a) = id :=
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section_eq !id_comp
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definition id_inverse (a : ob) [H : is_iso (ID a)] : (ID a)⁻¹ = id :=
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inverse_eq_left !id_comp
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definition split_mono_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
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[Hf : split_mono f] [Hg : split_mono g] : split_mono (g ∘ f) :=
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split_mono.mk
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(show (retraction_of f ∘ retraction_of g) ∘ g ∘ f = id,
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by rewrite [-assoc, assoc _ g f, retraction_comp, id_left, retraction_comp])
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definition split_epi_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
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[Hf : split_epi f] [Hg : split_epi g] : split_epi (g ∘ f) :=
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split_epi.mk
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(show (g ∘ f) ∘ section_of f ∘ section_of g = id,
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by rewrite [-assoc, {f ∘ _}assoc, comp_section, id_left, comp_section])
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definition is_iso_comp [instance] [priority 150] (g : b ⟶ c) (f : a ⟶ b)
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[Hf : is_iso f] [Hg : is_iso g] : is_iso (g ∘ f) :=
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!is_iso_of_split_epi_of_split_mono
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-- "is_iso f" is equivalent to a certain sigma type
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protected definition sigma_char (f : hom a b) :
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(Σ (g : hom b a), (g ∘ f = id) × (f ∘ g = id)) ≃ is_iso f :=
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-- definition is_iso.sigma_char (f : hom a b) :
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-- (Σ (g : hom b a), (g ∘ f = id) × (f ∘ g = id)) ≃ is_iso f :=
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-- begin
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-- fapply equiv.MK,
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-- {intro S, apply is_iso.mk,
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-- exact (pr₁ S.2),
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-- exact (pr₂ S.2)},
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-- {intro H, cases H with (g, η, ε),
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-- exact (sigma.mk g (pair η ε))},
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-- {intro H, cases H, apply idp},
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-- {intro S, cases S with (g, ηε), cases ηε, apply idp},
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-- end
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definition is_hprop_is_iso [instance] (f : hom a b) : is_hprop (is_iso f) :=
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begin
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fapply (equiv.mk),
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{intro S, apply is_iso.mk,
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exact (pr₁ S.2),
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exact (pr₂ S.2)},
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{fapply adjointify,
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{intro H, cases H with (g, η, ε),
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exact (sigma.mk g (pair η ε))},
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{intro H, cases H, apply idp},
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{intro S, cases S with (g, ηε), cases ηε, apply idp}},
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apply is_hprop.mk, intros (H, H'),
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cases H with (g, li, ri), cases H' with (g', li', ri'),
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fapply (apD0111 (@is_iso.mk ob C a b f)),
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apply left_inverse_eq_right_inverse,
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apply li,
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apply ri',
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apply is_hprop.elim,
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apply is_hprop.elim,
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end
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-- The structure for isomorphism can be characterized up to equivalence
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-- by a sigma type.
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definition isomorphic.sigma_char ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
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/- iso objects -/
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structure iso (a b : ob) :=
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(to_hom : hom a b)
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[struct : is_iso to_hom]
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infix `≅`:50 := iso.iso
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attribute iso.struct [instance] [priority 400]
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namespace iso
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attribute to_hom [coercion]
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definition MK (f : a ⟶ b) (g : b ⟶ a) (H1 : g ∘ f = id) (H2 : f ∘ g = id) :=
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@mk _ _ _ _ f (is_iso.mk H1 H2)
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definition to_inv (f : a ≅ b) : b ⟶ a :=
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(to_hom f)⁻¹
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protected definition refl (a : ob) : a ≅ a :=
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mk (ID a)
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protected definition symm ⦃a b : ob⦄ (H : a ≅ b) : b ≅ a :=
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mk (to_hom H)⁻¹
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protected definition trans ⦃a b c : ob⦄ (H1 : a ≅ b) (H2 : b ≅ c) : a ≅ c :=
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mk (to_hom H2 ∘ to_hom H1)
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protected definition eq_mk' {f f' : a ⟶ b} [H : is_iso f] [H' : is_iso f'] (p : f = f')
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: iso.mk f = iso.mk f' :=
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apD011 iso.mk p !is_hprop.elim
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protected definition eq_mk {f f' : a ≅ b} (p : to_hom f = to_hom f') : f = f' :=
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by (cases f; cases f'; apply (iso.eq_mk' p))
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-- The structure for isomorphism can be characterized up to equivalence by a sigma type.
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definition sigma_char ⦃a b : ob⦄ : (Σ (f : hom a b), is_iso f) ≃ (a ≅ b) :=
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begin
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fapply (equiv.mk),
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{intro S, apply isomorphic.mk, apply (S.2)},
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{intro S, apply iso.mk, apply (S.2)},
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{fapply adjointify,
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{intro p, cases p with (f, H), exact (sigma.mk f H)},
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{intro p, cases p, apply idp},
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{intro S, cases S, apply idp}},
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end
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set_option apply.class_instance false -- disable class instance resolution in the apply tactic
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-- The statement "f is an isomorphism" is a mere proposition
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definition is_hset_iso : is_hset (is_iso f) :=
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begin
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apply is_trunc_is_equiv_closed,
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apply (equiv.to_is_equiv (!sigma_char)),
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apply is_trunc_sigma,
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apply (!homH),
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{intro g, apply is_trunc_prod,
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repeat (apply is_trunc_eq; apply is_trunc_succ; apply (!homH))},
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end
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end iso
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-- The type of isomorphisms between two objects is a set
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definition is_hset_isomorphic : is_hset (a ≅ b) :=
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definition is_hset_iso [instance] : is_hset (a ≅ b) :=
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begin
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apply is_trunc_is_equiv_closed,
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apply (equiv.to_is_equiv (!isomorphic.sigma_char)),
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apply is_trunc_sigma,
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apply homH,
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{intro f, apply is_hset_iso},
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apply (equiv.to_is_equiv (!iso.sigma_char)),
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end
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-- In a precategory, equal objects are isomorphic
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definition iso_of_eq (p : a = b) : a ≅ b :=
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eq.rec_on p (isomorphic.mk id)
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eq.rec_on p (iso.mk id)
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end morphism
|
||||
structure mono [class] (f : a ⟶ b) :=
|
||||
(elim : ∀c (g h : hom c a), f ∘ g = f ∘ h → g = h)
|
||||
structure epi [class] (f : a ⟶ b) :=
|
||||
(elim : ∀c (g h : hom b c), g ∘ f = h ∘ f → g = h)
|
||||
|
||||
definition mono_of_split_mono [instance] (f : a ⟶ b) [H : split_mono f] : mono f :=
|
||||
mono.mk
|
||||
(λ c g h H,
|
||||
calc
|
||||
g = id ∘ g : by rewrite id_left
|
||||
... = (retraction_of f ∘ f) ∘ g : by rewrite -retraction_comp
|
||||
... = (retraction_of f ∘ f) ∘ h : by rewrite [-assoc, H, -assoc]
|
||||
... = id ∘ h : by rewrite retraction_comp
|
||||
... = h : by rewrite id_left)
|
||||
|
||||
definition epi_of_split_epi [instance] (f : a ⟶ b) [H : split_epi f] : epi f :=
|
||||
epi.mk
|
||||
(λ c g h H,
|
||||
calc
|
||||
g = g ∘ id : by rewrite id_right
|
||||
... = g ∘ f ∘ section_of f : by rewrite -comp_section
|
||||
... = h ∘ f ∘ section_of f : by rewrite [assoc, H, -assoc]
|
||||
... = h ∘ id : by rewrite comp_section
|
||||
... = h : by rewrite id_right)
|
||||
|
||||
definition mono_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : mono f] [Hg : mono g]
|
||||
: mono (g ∘ f) :=
|
||||
mono.mk
|
||||
(λ d h₁ h₂ H,
|
||||
have H2 : g ∘ (f ∘ h₁) = g ∘ (f ∘ h₂),
|
||||
begin
|
||||
rewrite *assoc, exact H
|
||||
end,
|
||||
!mono.elim (!mono.elim H2))
|
||||
|
||||
definition epi_comp [instance] (g : b ⟶ c) (f : a ⟶ b) [Hf : epi f] [Hg : epi g]
|
||||
: epi (g ∘ f) :=
|
||||
epi.mk
|
||||
(λ d h₁ h₂ H,
|
||||
have H2 : (h₁ ∘ g) ∘ f = (h₂ ∘ g) ∘ f,
|
||||
begin
|
||||
rewrite -*assoc, exact H
|
||||
end,
|
||||
!epi.elim (!epi.elim H2))
|
||||
|
||||
end iso
|
||||
|
||||
namespace iso
|
||||
/-
|
||||
rewrite lemmas for inverses, modified from
|
||||
https://github.com/JasonGross/HoTT-categories/blob/master/theories/Categories/Category/Morphisms.v
|
||||
-/
|
||||
section
|
||||
variables {ob : Type} [C : precategory ob] include C
|
||||
variables {a b c d : ob} (f : b ⟶ a)
|
||||
(r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
|
||||
(g : d ⟶ c)
|
||||
variable [Hq : is_iso q] include Hq
|
||||
definition comp.right_inverse : q ∘ q⁻¹ = id := !right_inverse
|
||||
definition comp.left_inverse : q⁻¹ ∘ q = id := !left_inverse
|
||||
definition inverse_comp_cancel_left : q⁻¹ ∘ (q ∘ p) = p :=
|
||||
by rewrite [assoc, left_inverse, id_left]
|
||||
definition comp_inverse_cancel_left : q ∘ (q⁻¹ ∘ g) = g :=
|
||||
by rewrite [assoc, right_inverse, id_left]
|
||||
definition comp_inverse_cancel_right : (r ∘ q) ∘ q⁻¹ = r :=
|
||||
by rewrite [-assoc, right_inverse, id_right]
|
||||
definition inverse_comp_cancel_right : (f ∘ q⁻¹) ∘ q = f :=
|
||||
by rewrite [-assoc, left_inverse, id_right]
|
||||
|
||||
definition comp_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
|
||||
inverse_eq_left
|
||||
(show (p⁻¹ʰ ∘ q⁻¹ʰ) ∘ q ∘ p = id, from
|
||||
by rewrite [-assoc, inverse_comp_cancel_left, left_inverse])
|
||||
|
||||
definition inverse_comp_inverse_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q :=
|
||||
inverse_involutive q ▹ comp_inverse q⁻¹ g
|
||||
|
||||
definition inverse_comp_inverse_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ :=
|
||||
inverse_involutive f ▹ comp_inverse q f⁻¹
|
||||
|
||||
definition inverse_comp_inverse_inverse [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q :=
|
||||
inverse_involutive r ▹ inverse_comp_inverse_left q r⁻¹
|
||||
end
|
||||
|
||||
section
|
||||
variables {ob : Type} {C : precategory ob} include C
|
||||
variables {d c b a : ob}
|
||||
{i : b ⟶ c} {f : b ⟶ a}
|
||||
{r : c ⟶ d} {q : b ⟶ c} {p : a ⟶ b}
|
||||
{g : d ⟶ c} {h : c ⟶ b}
|
||||
{x : b ⟶ d} {z : a ⟶ c}
|
||||
{y : d ⟶ b} {w : c ⟶ a}
|
||||
variable [Hq : is_iso q] include Hq
|
||||
|
||||
definition comp_eq_of_eq_inverse_comp (H : y = q⁻¹ ∘ g) : q ∘ y = g :=
|
||||
H⁻¹ ▹ comp_inverse_cancel_left q g
|
||||
definition comp_eq_of_eq_comp_inverse (H : w = f ∘ q⁻¹) : w ∘ q = f :=
|
||||
H⁻¹ ▹ inverse_comp_cancel_right f q
|
||||
definition inverse_comp_eq_of_eq_comp (H : z = q ∘ p) : q⁻¹ ∘ z = p :=
|
||||
H⁻¹ ▹ inverse_comp_cancel_left q p
|
||||
definition comp_inverse_eq_of_eq_comp (H : x = r ∘ q) : x ∘ q⁻¹ = r :=
|
||||
H⁻¹ ▹ comp_inverse_cancel_right r q
|
||||
definition eq_comp_of_inverse_comp_eq (H : q⁻¹ ∘ g = y) : g = q ∘ y :=
|
||||
(comp_eq_of_eq_inverse_comp H⁻¹)⁻¹
|
||||
definition eq_comp_of_comp_inverse_eq (H : f ∘ q⁻¹ = w) : f = w ∘ q :=
|
||||
(comp_eq_of_eq_comp_inverse H⁻¹)⁻¹
|
||||
definition eq_inverse_comp_of_comp_eq (H : q ∘ p = z) : p = q⁻¹ ∘ z :=
|
||||
(inverse_comp_eq_of_eq_comp H⁻¹)⁻¹
|
||||
definition eq_comp_inverse_of_comp_eq (H : r ∘ q = x) : r = x ∘ q⁻¹ :=
|
||||
(comp_inverse_eq_of_eq_comp H⁻¹)⁻¹
|
||||
definition eq_inverse_of_comp_eq_id' (H : h ∘ q = id) : h = q⁻¹ := (inverse_eq_left H)⁻¹
|
||||
definition eq_inverse_of_comp_eq_id (H : q ∘ h = id) : h = q⁻¹ := (inverse_eq_right H)⁻¹
|
||||
definition eq_of_comp_inverse_eq_id (H : i ∘ q⁻¹ = id) : i = q :=
|
||||
eq_inverse_of_comp_eq_id' H ⬝ inverse_involutive q
|
||||
definition eq_of_inverse_comp_eq_id (H : q⁻¹ ∘ i = id) : i = q :=
|
||||
eq_inverse_of_comp_eq_id H ⬝ inverse_involutive q
|
||||
definition eq_of_id_eq_comp_inverse (H : id = i ∘ q⁻¹) : q = i := (eq_of_comp_inverse_eq_id H⁻¹)⁻¹
|
||||
definition eq_of_id_eq_inverse_comp (H : id = q⁻¹ ∘ i) : q = i := (eq_of_inverse_comp_eq_id H⁻¹)⁻¹
|
||||
definition inverse_eq_of_id_eq_comp (H : id = h ∘ q) : q⁻¹ = h :=
|
||||
(eq_inverse_of_comp_eq_id' H⁻¹)⁻¹
|
||||
definition inverse_eq_of_id_eq_comp' (H : id = q ∘ h) : q⁻¹ = h :=
|
||||
(eq_inverse_of_comp_eq_id H⁻¹)⁻¹
|
||||
end
|
||||
end iso
|
||||
|
|
|
|||
|
|
@ -19,8 +19,8 @@ namespace morphism
|
|||
(comp_section : f ∘ section_of = id)
|
||||
structure is_iso [class] {ob : Type} [C : precategory ob] {a b : ob} (f : a ⟶ b) :=
|
||||
{inverse : b ⟶ a}
|
||||
(inverse_comp : inverse ∘ f = id)
|
||||
(comp_inverse : f ∘ inverse = id)
|
||||
(left_inverse : inverse ∘ f = id)
|
||||
(right_inverse : f ∘ inverse = id)
|
||||
|
||||
attribute is_iso [multiple-instances]
|
||||
open split_mono split_epi is_iso
|
||||
|
|
@ -29,8 +29,8 @@ namespace morphism
|
|||
definition section_of [reducible] := @split_epi.section_of
|
||||
definition comp_section [reducible] := @split_epi.comp_section
|
||||
definition inverse [reducible] := @is_iso.inverse
|
||||
definition inverse_comp [reducible] := @is_iso.inverse_comp
|
||||
definition comp_inverse [reducible] := @is_iso.comp_inverse
|
||||
definition left_inverse [reducible] := @is_iso.left_inverse
|
||||
definition right_inverse [reducible] := @is_iso.right_inverse
|
||||
postfix `⁻¹` := inverse
|
||||
--a second notation for the inverse, which is not overloaded
|
||||
postfix [parsing-only] `⁻¹ʰ`:std.prec.max_plus := inverse
|
||||
|
|
@ -41,17 +41,17 @@ namespace morphism
|
|||
|
||||
definition iso_imp_retraction [instance] [priority 300] [reducible]
|
||||
(f : a ⟶ b) [H : is_iso f] : split_mono f :=
|
||||
split_mono.mk !inverse_comp
|
||||
split_mono.mk !left_inverse
|
||||
|
||||
definition iso_imp_section [instance] [priority 300] [reducible]
|
||||
(f : a ⟶ b) [H : is_iso f] : split_epi f :=
|
||||
split_epi.mk !comp_inverse
|
||||
split_epi.mk !right_inverse
|
||||
|
||||
definition is_iso_id [instance] [priority 500] (a : ob) : is_iso (ID a) :=
|
||||
is_iso.mk !id_comp !id_comp
|
||||
|
||||
definition is_iso_inverse [instance] [priority 200] (f : a ⟶ b) [H : is_iso f] : is_iso f⁻¹ :=
|
||||
is_iso.mk !comp_inverse !inverse_comp
|
||||
is_iso.mk !right_inverse !left_inverse
|
||||
|
||||
definition left_inverse_eq_right_inverse {f : a ⟶ b} {g g' : hom b a}
|
||||
(Hl : g ∘ f = id) (Hr : f ∘ g' = id) : g = g' :=
|
||||
|
|
@ -64,10 +64,10 @@ namespace morphism
|
|||
(left_inverse_eq_right_inverse H2 !comp_section)⁻¹
|
||||
|
||||
definition inverse_eq_right [H : is_iso f] (H2 : f ∘ h = id) : f⁻¹ = h :=
|
||||
left_inverse_eq_right_inverse !inverse_comp H2
|
||||
left_inverse_eq_right_inverse !left_inverse H2
|
||||
|
||||
definition inverse_eq_left [H : is_iso f] (H2 : h ∘ f = id) : f⁻¹ = h :=
|
||||
(left_inverse_eq_right_inverse H2 !comp_inverse)⁻¹
|
||||
(left_inverse_eq_right_inverse H2 !right_inverse)⁻¹
|
||||
|
||||
definition retraction_eq_section (f : a ⟶ b) [Hl : split_mono f] [Hr : split_epi f] :
|
||||
retraction_of f = section_of f :=
|
||||
|
|
@ -78,10 +78,10 @@ namespace morphism
|
|||
is_iso.mk ((retraction_eq_section f) ▹ (retraction_comp f)) (comp_section f)
|
||||
|
||||
definition inverse_unique (H H' : is_iso f) : @inverse _ _ _ _ f H = @inverse _ _ _ _ f H' :=
|
||||
inverse_eq_left !inverse_comp
|
||||
inverse_eq_left !left_inverse
|
||||
|
||||
definition inverse_involutive (f : a ⟶ b) [H : is_iso f] : (f⁻¹)⁻¹ = f :=
|
||||
inverse_eq_right !inverse_comp
|
||||
inverse_eq_right !left_inverse
|
||||
|
||||
definition retraction_id (a : ob) : retraction_of (ID a) = id :=
|
||||
retraction_eq !id_comp
|
||||
|
|
@ -188,27 +188,27 @@ namespace morphism
|
|||
(r : c ⟶ d) (q : b ⟶ c) (p : a ⟶ b)
|
||||
(g : d ⟶ c)
|
||||
variable [Hq : is_iso q] include Hq
|
||||
definition comp.right_inverse : q ∘ q⁻¹ = id := !comp_inverse
|
||||
definition comp.left_inverse : q⁻¹ ∘ q = id := !inverse_comp
|
||||
definition comp.right_inverse : q ∘ q⁻¹ = id := !right_inverse
|
||||
definition comp.left_inverse : q⁻¹ ∘ q = id := !left_inverse
|
||||
definition inverse_comp_cancel_left : q⁻¹ ∘ (q ∘ p) = p :=
|
||||
by rewrite [assoc, inverse_comp, id_left]
|
||||
by rewrite [assoc, left_inverse, id_left]
|
||||
definition comp_inverse_cancel_left : q ∘ (q⁻¹ ∘ g) = g :=
|
||||
by rewrite [assoc, comp_inverse, id_left]
|
||||
by rewrite [assoc, right_inverse, id_left]
|
||||
definition comp_inverse_cancel_right : (r ∘ q) ∘ q⁻¹ = r :=
|
||||
by rewrite [-assoc, comp_inverse, id_right]
|
||||
by rewrite [-assoc, right_inverse, id_right]
|
||||
definition inverse_comp_cancel_right : (f ∘ q⁻¹) ∘ q = f :=
|
||||
by rewrite [-assoc, inverse_comp, id_right]
|
||||
by rewrite [-assoc, left_inverse, id_right]
|
||||
|
||||
definition comp_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
|
||||
definition right_inverse [Hp : is_iso p] [Hpq : is_iso (q ∘ p)] : (q ∘ p)⁻¹ʰ = p⁻¹ʰ ∘ q⁻¹ʰ :=
|
||||
inverse_eq_left
|
||||
(show (p⁻¹ʰ ∘ q⁻¹ʰ) ∘ q ∘ p = id, from
|
||||
by rewrite [-assoc, inverse_comp_cancel_left, inverse_comp])
|
||||
by rewrite [-assoc, inverse_comp_cancel_left, left_inverse])
|
||||
|
||||
definition inverse_comp_inverse_left [H' : is_iso g] : (q⁻¹ ∘ g)⁻¹ = g⁻¹ ∘ q :=
|
||||
inverse_involutive q ▹ comp_inverse q⁻¹ g
|
||||
inverse_involutive q ▹ right_inverse q⁻¹ g
|
||||
|
||||
definition inverse_comp_inverse_right [H' : is_iso f] : (q ∘ f⁻¹)⁻¹ = f ∘ q⁻¹ :=
|
||||
inverse_involutive f ▹ comp_inverse q f⁻¹
|
||||
inverse_involutive f ▹ right_inverse q f⁻¹
|
||||
|
||||
definition inverse_comp_inverse_inverse [H' : is_iso r] : (q⁻¹ ∘ r⁻¹)⁻¹ = r ∘ q :=
|
||||
inverse_involutive r ▹ inverse_comp_inverse_left q r⁻¹
|
||||
|
|
|
|||
|
|
@ -6,7 +6,7 @@ Module: algebra.precategory.nat_trans
|
|||
Author: Floris van Doorn, Jakob von Raumer
|
||||
-/
|
||||
|
||||
import .functor .morphism
|
||||
import .functor .iso
|
||||
open eq category functor is_trunc equiv sigma.ops sigma is_equiv function pi funext
|
||||
|
||||
structure nat_trans {C D : Precategory} (F G : C ⇒ D) :=
|
||||
|
|
@ -39,7 +39,6 @@ namespace nat_trans
|
|||
protected definition ID [reducible] {C D : Precategory} (F : functor C D) : nat_trans F F :=
|
||||
id
|
||||
|
||||
local attribute is_hprop_eq_hom [instance]
|
||||
definition nat_trans_eq_mk' {η₁ η₂ : Π (a : C), hom (F a) (G a)}
|
||||
(nat₁ : Π (a b : C) (f : hom a b), G f ∘ η₁ a = η₁ b ∘ F f)
|
||||
(nat₂ : Π (a b : C) (f : hom a b), G f ∘ η₂ a = η₂ b ∘ F f)
|
||||
|
|
|
|||
|
|
@ -29,8 +29,9 @@ namespace yoneda
|
|||
definition hom_functor (C : Precategory) : Cᵒᵖ ×c C ⇒ set :=
|
||||
functor.mk (λ(x : Cᵒᵖ ×c C), homset x.1 x.2)
|
||||
(λ(x y : Cᵒᵖ ×c C) (f : _) (h : homset x.1 x.2), f.2 ∘⁅ C ⁆ (h ∘⁅ C ⁆ f.1))
|
||||
proof (λ(x : Cᵒᵖ ×c C), eq_of_homotopy (λ(h : homset x.1 x.2), !id_left ⬝ !id_right)) qed
|
||||
-- (λ(x y z : Cᵒᵖ ×c C) (g : y ⟶ z) (f : x ⟶ y), eq_of_homotopy (λ(h : hom x.1 x.2), representable_functor_assoc g.2 f.2 h f.1 g.1))
|
||||
proof
|
||||
(λ(x : Cᵒᵖ ×c C), eq_of_homotopy (λ(h : homset x.1 x.2), !id_left ⬝ !id_right))
|
||||
qed
|
||||
begin
|
||||
intros (x, y, z, g, f), apply eq_of_homotopy, intro h,
|
||||
exact (representable_functor_assoc g.2 f.2 h f.1 g.1),
|
||||
|
|
@ -68,10 +69,10 @@ namespace functor
|
|||
definition functor_curry_hom_def ⦃c c' : C⦄ (f : c ⟶ c') (d : D) :
|
||||
(Fhom F f) d = to_fun_hom F (f, id) := idp
|
||||
|
||||
definition functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
|
||||
theorem functor_curry_id (c : C) : Fhom F (ID c) = nat_trans.id :=
|
||||
nat_trans_eq_mk (λd, respect_id F _)
|
||||
|
||||
definition functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
|
||||
theorem functor_curry_comp ⦃c c' c'' : C⦄ (f' : c' ⟶ c'') (f : c ⟶ c')
|
||||
: Fhom F (f' ∘ f) = Fhom F f' ∘n Fhom F f :=
|
||||
nat_trans_eq_mk (λd, calc
|
||||
natural_map (Fhom F (f' ∘ f)) d = F (f' ∘ f, id) : functor_curry_hom_def
|
||||
|
|
@ -94,14 +95,14 @@ namespace functor
|
|||
to_fun_hom (to_fun_ob G p'.1) f.2 ∘ natural_map (to_fun_hom G f.1) p.2
|
||||
local abbreviation Ghom := @functor_uncurry_hom
|
||||
|
||||
definition functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
|
||||
theorem functor_uncurry_id (p : C ×c D) : Ghom G (ID p) = id :=
|
||||
calc
|
||||
Ghom G (ID p) = to_fun_hom (to_fun_ob G p.1) id ∘ natural_map (to_fun_hom G id) p.2 : idp
|
||||
... = id ∘ natural_map (to_fun_hom G id) p.2 : ap (λx, x ∘ _) (respect_id (to_fun_ob G p.1) p.2)
|
||||
... = id ∘ natural_map nat_trans.id p.2 : {respect_id G p.1}
|
||||
... = id : id_comp
|
||||
|
||||
definition functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
|
||||
theorem functor_uncurry_comp ⦃p p' p'' : C ×c D⦄ (f' : p' ⟶ p'') (f : p ⟶ p')
|
||||
: Ghom G (f' ∘ f) = Ghom G f' ∘ Ghom G f :=
|
||||
calc
|
||||
Ghom G (f' ∘ f)
|
||||
|
|
|
|||
|
|
@ -228,9 +228,11 @@ namespace equiv
|
|||
|
||||
variables {A B C : Type}
|
||||
|
||||
protected definition MK (f : A → B) (g : B → A) (retr : f ∘ g ∼ id) (sect : g ∘ f ∼ id) : A ≃ B :=
|
||||
protected definition MK [reducible] (f : A → B) (g : B → A)
|
||||
(retr : f ∘ g ∼ id) (sect : g ∘ f ∼ id) : A ≃ B :=
|
||||
equiv.mk f (adjointify f g retr sect)
|
||||
definition to_inv (f : A ≃ B) : B → A :=
|
||||
|
||||
definition to_inv [reducible] (f : A ≃ B) : B → A :=
|
||||
f⁻¹
|
||||
|
||||
protected definition refl : A ≃ A :=
|
||||
|
|
|
|||
|
|
@ -108,14 +108,16 @@ namespace is_trunc
|
|||
have K : ∀ (r : x = y), center_eq x y = r, from (λ r, eq.rec_on r !con.left_inv),
|
||||
(K p)⁻¹ ⬝ K q
|
||||
|
||||
definition is_contr_eq [instance] {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y)
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definition is_contr_eq {A : Type} [H : is_contr A] (x y : A) : is_contr (x = y)
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:=
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is_contr.mk !center_eq (λ p, !hprop_eq)
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local attribute is_contr_eq [instance]
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/- truncation is upward close -/
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-- n-types are also (n+1)-types
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definition is_trunc_succ [instance] (A : Type) (n : trunc_index) [H : is_trunc n A] : is_trunc (n.+1) A :=
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definition is_trunc_succ [instance] [priority 100] (A : Type) (n : trunc_index)
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[H : is_trunc n A] : is_trunc (n.+1) A :=
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trunc_index.rec_on n
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(λ A (H : is_contr A), !is_trunc_succ_intro)
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(λ n IH A (H : is_trunc (n.+1) A), @is_trunc_succ_intro _ _ (λ x y, IH _ !is_trunc_eq))
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@ -190,9 +192,9 @@ namespace is_trunc
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/- truncated universe -/
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|
||||
structure trunctype (n : trunc_index) :=
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||||
(trunctype_type : Type) (is_trunc_trunctype_type : is_trunc n trunctype_type)
|
||||
attribute trunctype.trunctype_type [coercion]
|
||||
attribute trunctype.is_trunc_trunctype_type [instance]
|
||||
(carrier : Type) (struct : is_trunc n carrier)
|
||||
attribute trunctype.carrier [coercion]
|
||||
attribute trunctype.struct [instance]
|
||||
|
||||
notation n `-Type` := trunctype n
|
||||
abbreviation hprop := -1-Type
|
||||
|
|
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|||
|
|
@ -122,7 +122,7 @@ namespace is_trunc
|
|||
(trunctype.{l} n) ≃ (Σ (A : Type.{l}), is_trunc n A) :=
|
||||
begin
|
||||
fapply equiv.MK,
|
||||
/--/ intro A, exact (⟨trunctype_type A, is_trunc_trunctype_type A⟩),
|
||||
/--/ intro A, exact (⟨carrier A, struct A⟩),
|
||||
/--/ intro S, exact (trunctype.mk S.1 S.2),
|
||||
/--/ intro S, apply (sigma.rec_on S), intros (S1, S2), apply idp,
|
||||
intro A, apply (trunctype.rec_on A), intros (A1, A2), apply idp,
|
||||
|
|
@ -130,11 +130,11 @@ namespace is_trunc
|
|||
|
||||
-- set_option pp.notation false
|
||||
protected definition trunctype.eq (n : trunc_index) (A B : n-Type) :
|
||||
(A = B) ≃ (trunctype_type A = trunctype_type B) :=
|
||||
(A = B) ≃ (carrier A = carrier B) :=
|
||||
calc
|
||||
(A = B) ≃ (trunctype.sigma_char n A = trunctype.sigma_char n B) : eq_equiv_fn_eq_of_equiv
|
||||
... ≃ ((trunctype.sigma_char n A).1 = (trunctype.sigma_char n B).1) : equiv.symm (!equiv_subtype)
|
||||
... ≃ (trunctype_type A = trunctype_type B) : equiv.refl
|
||||
... ≃ (carrier A = carrier B) : equiv.refl
|
||||
|
||||
|
||||
end is_trunc
|
||||
|
|
|
|||
Loading…
Reference in a new issue