f513538631
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
88 lines
3 KiB
Text
88 lines
3 KiB
Text
/-
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Copyright (c) 2015 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.basic
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Authors: Floris van Doorn
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-/
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open eq is_trunc
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namespace category
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structure precategory [class] (ob : Type) : Type :=
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(hom : ob → ob → Type)
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(homH : Π(a b : ob), is_hset (hom a b))
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(comp : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c)
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(ID : Π (a : ob), hom a a)
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(assoc : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
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comp h (comp g f) = comp (comp h g) f)
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(id_left : Π ⦃a b : ob⦄ (f : hom a b), comp !ID f = f)
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(id_right : Π ⦃a b : ob⦄ (f : hom a b), comp f !ID = f)
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attribute precategory [multiple-instances]
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attribute precategory.homH [instance]
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infixr `∘` := precategory.comp
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-- input ⟶ using \--> (this is a different arrow than \-> (→))
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infixl [parsing-only] `⟶`:25 := precategory.hom
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namespace hom
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infixl `⟶` := precategory.hom -- if you open this namespace, hom a b is printed as a ⟶ b
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end hom
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abbreviation hom := @precategory.hom
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abbreviation homH := @precategory.homH
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abbreviation comp := @precategory.comp
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abbreviation ID := @precategory.ID
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abbreviation assoc := @precategory.assoc
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abbreviation id_left := @precategory.id_left
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abbreviation id_right := @precategory.id_right
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section basic_lemmas
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variables {ob : Type} [C : precategory ob]
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variables {a b c d : ob} {h : c ⟶ d} {g : hom b c} {f f' : hom a b} {i : a ⟶ a}
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include C
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definition id [reducible] := ID a
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definition id_compose (a : ob) : ID a ∘ ID a = ID a := !id_left
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definition left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
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calc i = i ∘ id : id_right
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... = id : H
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definition right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
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calc i = id ∘ i : id_left
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... = id : H
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definition homset [reducible] (x y : ob) : hset :=
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hset.mk (hom x y) _
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definition is_hprop_eq_hom [instance] : is_hprop (f = f') :=
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!is_trunc_eq
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end basic_lemmas
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structure Precategory : Type :=
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(carrier : Type)
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(struct : precategory carrier)
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definition precategory.Mk [reducible] {ob} (C) : Precategory := Precategory.mk ob C
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definition precategory.MK [reducible] (a b c d e f g h) : Precategory :=
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Precategory.mk a (precategory.mk b c d e f g h)
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abbreviation carrier := @Precategory.carrier
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attribute Precategory.carrier [coercion]
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attribute Precategory.struct [instance] [priority 10000] [coercion]
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-- definition precategory.carrier [coercion] [reducible] := Precategory.carrier
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-- definition precategory.struct [instance] [coercion] [reducible] := Precategory.struct
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notation g `∘⁅` C `⁆` f := @comp (Precategory.carrier C) (Precategory.struct C) _ _ _ g f
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-- TODO: make this left associative
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-- TODO: change this notation?
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definition Precategory.eta (C : Precategory) : Precategory.mk C C = C :=
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Precategory.rec (λob c, idp) C
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end category
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open category
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