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lean2/hott/algebra/category/groupoid.hlean

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feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
/-
Copyright (c) 2014 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
fix(hott): notation spacing and markdown files
2015-10-01 19:52:28 +00:00
Authors: Jakob von Raumer, Floris van Doorn
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
Ported from Coq HoTT
-/
refactor(category): merge precategory/ and category/, organize construction files differently.
2015-04-25 04:20:59 +00:00
import .iso ..group
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
feat(hott): port parts of natural numbers and integers from standard library to HoTT This also involves: - adding definitions about logic and natural numbers existing in the standard library to init - porting the current algebraic hierarchy
2015-05-05 21:25:35 +00:00
open eq is_trunc iso category algebra nat unit
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
namespace category
structure groupoid [class] (ob : Type) extends parent : precategory ob :=
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
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mk' :: (all_iso : Π ⦃a b : ob⦄ (f : hom a b), @is_iso ob parent a b f)
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
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abbreviation all_iso := @groupoid.all_iso
feat(categories): prove introduction rule for equivalences
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attribute groupoid.all_iso [instance] [priority 3000]
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
2015-04-25 03:04:24 +00:00
definition groupoid.mk [reducible] {ob : Type} (C : precategory ob)
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
(H : Π (a b : ob) (f : a ⟶ b), is_iso f) : groupoid ob :=
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
2015-04-25 03:04:24 +00:00
precategory.rec_on C groupoid.mk' H
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
feat(category): add limits in a category
2015-09-24 02:44:36 +00:00
-- We can turn each group into a groupoid on the unit type
feat(datatypes): let the type of unit be the lowest non-Prop universe The definitional package (brec_on and cases_on) now use poly_unit instead of unit closes #698
2015-06-24 21:59:17 +00:00
definition groupoid_of_group.{l} (A : Type.{l}) [G : group A] : groupoid.{0 l} unit :=
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
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begin
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
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fapply groupoid.mk, fapply precategory.mk,
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
intros, exact A,
feat(hott): port parts of natural numbers and integers from standard library to HoTT This also involves: - adding definitions about logic and natural numbers existing in the standard library to init - porting the current algebraic hierarchy
2015-05-05 21:25:35 +00:00
intros, apply (@group.is_hset_carrier A G),
feat(frontends/lean): uniform notation for lists in tactics closes #504
2015-03-28 00:26:06 +00:00
intros [a, b, c, g, h], exact (@group.mul A G g h),
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
intro a, exact (@group.one A G),
style(hott): let inverse notation have higher binding power than application
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intros, exact (@group.mul_assoc A G h g f)⁻¹,
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
intros, exact (@group.one_mul A G f),
intros, exact (@group.mul_one A G f),
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
2015-04-25 03:04:24 +00:00
intros, esimp [precategory.mk], apply is_iso.mk,
feat(hott): port parts of natural numbers and integers from standard library to HoTT This also involves: - adding definitions about logic and natural numbers existing in the standard library to init - porting the current algebraic hierarchy
2015-05-05 21:25:35 +00:00
apply mul.left_inv,
apply mul.right_inv,
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
end
feat(category): add limits in a category
2015-09-24 02:44:36 +00:00
definition hom_group {A : Type} [G : groupoid A] (a : A) :
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
group (hom a a) :=
begin
fapply group.mk,
feat(frontends/lean): nicer syntax for 'intros' 'reverts' and 'clears'
2015-04-30 18:00:39 +00:00
intro f g, apply (comp f g),
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
2015-04-25 03:04:24 +00:00
apply is_hset_hom,
feat(frontends/lean): nicer syntax for 'intros' 'reverts' and 'clears'
2015-04-30 18:00:39 +00:00
intros f g h, apply (assoc f g h)⁻¹,
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
apply (ID a),
intro f, apply id_left,
intro f, apply id_right,
feat(hott): some more renaming in category library
2015-02-28 06:16:20 +00:00
intro f, exact (iso.inverse f),
intro f, exact (iso.left_inverse f),
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
end
feat(category): add limits in a category
2015-09-24 02:44:36 +00:00
definition group_of_is_contr_groupoid {ob : Type} [H : is_contr ob]
[G : groupoid ob] : group (hom (center ob) (center ob)) := !hom_group
definition group_of_groupoid_unit [G : groupoid unit] : group (hom ⋆ ⋆) := !hom_group
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
-- Bundled version of categories
-- we don't use Groupoid.carrier explicitly, but rather use Groupoid.carrier (to_Precategory C)
structure Groupoid : Type :=
(carrier : Type)
(struct : groupoid carrier)
attribute Groupoid.struct [instance] [coercion]
definition Groupoid.to_Precategory [coercion] [reducible] (C : Groupoid) : Precategory :=
refactor(library,hott): remove coercions between algebraic structures They are classes, and mixing coercion with type class resolution is a recipe for disaster (aka counterintuitive behavior).
2015-11-11 19:32:05 +00:00
Precategory.mk (Groupoid.carrier C) _
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
definition groupoid.Mk [reducible] := Groupoid.mk
fix(hott/algebra) correct the name of a groupoid constructor
2015-03-13 22:33:53 +00:00
definition groupoid.MK [reducible] (C : Precategory) (H : Π (a b : C) (f : a ⟶ b), is_iso f)
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
: Groupoid :=
feat(precategory): add two redundant fields to precategory. Also some cleanup. In particular, all instances of "set_option apply.class_instance false" are removed
2015-04-25 03:04:24 +00:00
Groupoid.mk C (groupoid.mk C H)
feat(hott): more cleanup of HoTT library remove funext class, remove a couple of sorry's, add characterization of equality in trunctypes, use Jeremy's format for headers everywhere in the HoTT library, continue working on Yoneda embedding
2015-02-26 18:19:54 +00:00
definition Groupoid.eta (C : Groupoid) : Groupoid.mk C C = C :=
Groupoid.rec (λob c, idp) C
end category
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