60 lines
1.8 KiB
Text
60 lines
1.8 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. 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.category.constructions
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Authors: Floris van Doorn
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-/
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import .basic algebra.precategory.constructions
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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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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 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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namespace ops
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abbreviation set := Category_hset
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end ops
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end category
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