83 lines
2.7 KiB
Text
83 lines
2.7 KiB
Text
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/-
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Copyright (c) 2015 Jakob von Raumer. 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.strict
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Authors: Floris van Doorn, Jakob von Raumer
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-/
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import .precategory .functor
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open is_trunc eq
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namespace category
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structure strict_precategory [class] (ob : Type) extends precategory ob :=
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mk' :: (is_hset_ob : is_hset ob)
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attribute strict_precategory.is_hset_ob [instance]
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definition strict_precategory.mk [reducible] {ob : Type} (C : precategory ob)
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(H : is_hset ob) : strict_precategory ob :=
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precategory.rec_on C strict_precategory.mk' H
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structure Strict_precategory : Type :=
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(carrier : Type)
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(struct : strict_precategory carrier)
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attribute Strict_precategory.struct [instance] [coercion]
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definition Strict_precategory.to_Precategory [coercion] [reducible]
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(C : Strict_precategory) : Precategory :=
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Precategory.mk (Strict_precategory.carrier C) C
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open functor
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definition precat_strict_precat : precategory Strict_precategory :=
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precategory.mk (λ a b, functor a b)
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(λ a b c g f, functor.compose g f)
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(λ a, functor.id)
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(λ a b c d h g f, !functor.assoc)
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(λ a b f, !functor.id_left)
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(λ a b f, !functor.id_right)
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definition Precat_of_strict_precats := precategory.Mk precat_strict_precat
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namespace ops
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abbreviation SPreCat := Precat_of_strict_precats
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--attribute precat_strict_precat [instance]
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end ops
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end category
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/-section
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open decidable unit empty
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variables {A : Type} [H : decidable_eq A]
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include H
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definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
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theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
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definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
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decidable.rec_on
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(H b c)
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(λ Hbc g, decidable.rec_on
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(H a b)
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(λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
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(λh f, empty.rec _ f) f)
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(λh (g : empty), empty.rec _ g) g
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omit H
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definition discrete_precategory (A : Type) [H : decidable_eq A] : precategory A :=
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mk (λa b, set_hom a b)
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(λ a b c g f, set_compose g f)
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(λ a, decidable.rec_on_true rfl ⋆)
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(λ a b c d h g f, @subsingleton.elim (set_hom a d) _ _ _)
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(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
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(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
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definition Discrete_category (A : Type) [H : decidable_eq A] := Mk (discrete_category A)
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end
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section
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open unit bool
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definition category_one := discrete_category unit
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definition Category_one := Mk category_one
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definition category_two := discrete_category bool
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definition Category_two := Mk category_two
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end-/
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