285 lines
12 KiB
Text
285 lines
12 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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Authors: Floris van Doorn
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-/
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import types.trunc types.pi arity
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open eq is_trunc pi equiv equiv.ops
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namespace category
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/-
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Just as in Coq-HoTT we add two redundant fields to precategories: assoc' and id_id.
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The first is to make (Cᵒᵖ)ᵒᵖ = C definitionally when C is a constructor.
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The second is to ensure that the functor from the terminal category 1 ⇒ Cᵒᵖ is
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opposite to the functor 1 ⇒ C.
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-/
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structure precategory [class] (ob : Type) : Type :=
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mk' ::
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(hom : ob → ob → Type)
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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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(assoc' : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
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comp (comp h g) f = comp h (comp 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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(id_id : Π (a : ob), comp !ID !ID = ID a)
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(is_set_hom : Π(a b : ob), is_set (hom a b))
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-- attribute precategory [multiple-instances] --this is not used anywhere
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attribute precategory.is_set_hom [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] ` ⟶ `:60 := precategory.hom
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namespace hom
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infixl ` ⟶ `:60 := 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 [unfold 2] := @precategory.hom
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abbreviation comp [unfold 2] := @precategory.comp
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abbreviation ID [unfold 2] := @precategory.ID
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abbreviation assoc [unfold 2] := @precategory.assoc
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abbreviation assoc' [unfold 2] := @precategory.assoc'
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abbreviation id_left [unfold 2] := @precategory.id_left
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abbreviation id_right [unfold 2] := @precategory.id_right
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abbreviation id_id [unfold 2] := @precategory.id_id
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abbreviation is_set_hom [unfold 2] := @precategory.is_set_hom
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definition is_prop_hom_eq {ob : Type} [C : precategory ob] {x y : ob} (f g : x ⟶ y)
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: is_prop (f = g) :=
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_
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-- the constructor you want to use in practice
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protected definition precategory.mk [constructor] {ob : Type} (hom : ob → ob → Type)
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[set : Π (a b : ob), is_set (hom a b)]
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(comp : Π ⦃a b c : ob⦄, hom b c → hom a b → hom a c) (ID : Π (a : ob), hom a a)
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(ass : 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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(idl : Π ⦃a b : ob⦄ (f : hom a b), comp (ID b) f = f)
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(idr : Π ⦃a b : ob⦄ (f : hom a b), comp f (ID a) = f) : precategory ob :=
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precategory.mk' hom comp ID ass (λa b c d h g f, !ass⁻¹) idl idr (λa, !idl) set
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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] [unfold 2] := ID a
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definition id_leftright (f : hom a b) : id ∘ f ∘ id = f := !id_left ⬝ !id_right
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definition comp_id_eq_id_comp (f : hom a b) : f ∘ id = id ∘ f := !id_right ⬝ !id_left⁻¹
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definition id_comp_eq_comp_id (f : hom a b) : id ∘ f = f ∘ id := !id_left ⬝ !id_right⁻¹
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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 : by rewrite id_right
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... = id : by rewrite 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 : by rewrite id_left
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... = id : by rewrite H
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definition homset [reducible] [constructor] (x y : ob) : Set :=
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Set.mk (hom x y) _
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end basic_lemmas
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section squares
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parameters {ob : Type} [C : precategory ob]
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local infixl ` ⟶ `:25 := @precategory.hom ob C
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local infixr ∘ := @precategory.comp ob C _ _ _
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definition compose_squares {xa xb xc ya yb yc : ob}
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{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb}
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{wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
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: wc ∘ (xg ∘ xf) = (yg ∘ yf) ∘ wa :=
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calc
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wc ∘ (xg ∘ xf) = (wc ∘ xg) ∘ xf : by rewrite assoc
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... = (yg ∘ wb) ∘ xf : by rewrite xybc
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... = yg ∘ (wb ∘ xf) : by rewrite assoc
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... = yg ∘ (yf ∘ wa) : by rewrite xyab
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... = (yg ∘ yf) ∘ wa : by rewrite assoc
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definition compose_squares_2x2 {xa xb xc ya yb yc za zb zc : ob}
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{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb} {zg : zb ⟶ zc} {zf : za ⟶ zb}
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{va : ya ⟶ za} {vb : yb ⟶ zb} {vc : yc ⟶ zc} {wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
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(yzab : vb ∘ yf = zf ∘ va) (yzbc : vc ∘ yg = zg ∘ vb)
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: (vc ∘ wc) ∘ (xg ∘ xf) = (zg ∘ zf) ∘ (va ∘ wa) :=
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calc
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(vc ∘ wc) ∘ (xg ∘ xf) = vc ∘ (wc ∘ (xg ∘ xf)) : by rewrite (assoc vc wc _)
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... = vc ∘ ((yg ∘ yf) ∘ wa) : by rewrite (compose_squares xyab xybc)
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... = (vc ∘ (yg ∘ yf)) ∘ wa : by rewrite assoc
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... = ((zg ∘ zf) ∘ va) ∘ wa : by rewrite (compose_squares yzab yzbc)
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... = (zg ∘ zf) ∘ (va ∘ wa) : by rewrite assoc
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definition square_precompose {xa xb xc yb yc : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (xf : xa ⟶ xb) : wc ∘ xg ∘ xf = yg ∘ wb ∘ xf :=
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calc
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wc ∘ xg ∘ xf = (wc ∘ xg) ∘ xf : by rewrite assoc
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... = (yg ∘ wb) ∘ xf : by rewrite H
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... = yg ∘ wb ∘ xf : by rewrite assoc
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definition square_postcompose {xb xc yb yc yd : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) : (yh ∘ wc) ∘ xg = (yh ∘ yg) ∘ wb :=
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calc
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(yh ∘ wc) ∘ xg = yh ∘ wc ∘ xg : by rewrite assoc
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... = yh ∘ yg ∘ wb : by rewrite H
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... = (yh ∘ yg) ∘ wb : by rewrite assoc
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definition square_prepostcompose {xa xb xc yb yc yd : ob}
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{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
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(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) (xf : xa ⟶ xb)
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: (yh ∘ wc) ∘ (xg ∘ xf) = (yh ∘ yg) ∘ (wb ∘ xf) :=
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square_precompose (square_postcompose H yh) xf
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end squares
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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] [constructor] {ob} (C) : Precategory := Precategory.mk ob C
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definition precategory.MK [reducible] [constructor] (a b c d e f g h) : Precategory :=
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Precategory.mk a (@precategory.mk a b c d e f g h)
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abbreviation carrier [unfold 1] := @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 ` ∘[`:60 C:0 `] `:0 f:60 :=
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@comp (Precategory.carrier C) (Precategory.struct C) _ _ _ g f
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-- TODO: make this left associative
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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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/-Characterization of paths between precategories-/
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-- introduction tule for paths between precategories
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definition precategory_eq {ob : Type}
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{C D : precategory ob}
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(p : Π{a b}, @hom ob C a b = @hom ob D a b)
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(q : Πa b c g f, cast p (@comp ob C a b c g f) = @comp ob D a b c (cast p g) (cast p f))
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: C = D :=
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begin
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induction C with hom1 comp1 ID1 a b il ir, induction D with hom2 comp2 ID2 a' b' il' ir',
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esimp at *,
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revert q, eapply homotopy2.rec_on @p, esimp, clear p, intro p q, induction p,
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esimp at *,
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assert H : comp1 = comp2,
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{ apply eq_of_homotopy3, intros, apply eq_of_homotopy2, intros, apply q},
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induction H,
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assert K : ID1 = ID2,
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{ apply eq_of_homotopy, intro a, exact !ir'⁻¹ ⬝ !il},
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induction K,
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apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_prop.elim
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end
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definition precategory_eq_of_equiv {ob : Type}
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{C D : precategory ob}
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(p : Π⦃a b⦄, @hom ob C a b ≃ @hom ob D a b)
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(q : Π{a b c} g f, p (@comp ob C a b c g f) = @comp ob D a b c (p g) (p f))
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: C = D :=
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begin
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fapply precategory_eq,
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{ intro a b, exact ua !@p},
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{ intros, refine !cast_ua ⬝ !q ⬝ _, apply ap011 !@comp !cast_ua⁻¹ !cast_ua⁻¹},
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end
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/- if we need to prove properties about precategory_eq, it might be easier with the following proof:
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begin
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induction C with hom1 comp1 ID1, induction D with hom2 comp2 ID2, esimp at *,
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assert H : Σ(s : hom1 = hom2), (λa b, equiv_of_eq (apd100 s a b)) = p,
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{ fconstructor,
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{ apply eq_of_homotopy2, intros, apply ua, apply p},
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{ apply eq_of_homotopy2, intros, rewrite [to_right_inv !eq_equiv_homotopy2, equiv_of_eq_ua]}},
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induction H with H1 H2, induction H1, esimp at H2,
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assert K : (λa b, equiv.refl) = p,
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{ refine _ ⬝ H2, apply eq_of_homotopy2, intros, exact !equiv_of_eq_refl⁻¹},
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induction K, clear H2,
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esimp at *,
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assert H : comp1 = comp2,
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{ apply eq_of_homotopy3, intros, apply eq_of_homotopy2, intros, apply q},
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assert K : ID1 = ID2,
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{ apply eq_of_homotopy, intros, apply r},
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induction H, induction K,
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apply ap0111111 (precategory.mk' hom1 comp1 ID1): apply is_prop.elim
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end
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-/
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definition Precategory_eq {C D : Precategory}
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(p : carrier C = carrier D)
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(q : Π{a b : C}, a ⟶ b = cast p a ⟶ cast p b)
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(r : Π{a b c : C} (g : b ⟶ c) (f : a ⟶ b), cast q (g ∘ f) = cast q g ∘ cast q f)
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: C = D :=
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begin
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induction C with X C, induction D with Y D, esimp at *, induction p,
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esimp at *,
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apply ap (Precategory.mk X),
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apply precategory_eq @q @r
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end
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definition Precategory_eq_of_equiv {C D : Precategory}
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(p : carrier C ≃ carrier D)
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(q : Π⦃a b : C⦄, a ⟶ b ≃ p a ⟶ p b)
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(r : Π{a b c : C} (g : b ⟶ c) (f : a ⟶ b), q (g ∘ f) = q g ∘ q f)
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: C = D :=
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begin
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induction C with X C, induction D with Y D, esimp at *,
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revert q r, eapply equiv.rec_on_ua p, clear p, intro p, induction p, esimp,
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intros,
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apply ap (Precategory.mk X),
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apply precategory_eq_of_equiv @q @r
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end
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-- elimination rules for paths between precategories.
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-- The first elimination rule is "ap carrier"
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definition Precategory_eq_hom [unfold 3] {C D : Precategory} (p : C = D) (a b : C)
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: hom a b = hom (cast (ap carrier p) a) (cast (ap carrier p) b) :=
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by induction p; reflexivity
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--(ap10 (ap10 (apd (λx, @hom (carrier x) (Precategory.struct x)) p⁻¹ᵖ) a) b)⁻¹ᵖ ⬝ _
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-- beta/eta rules
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definition ap_Precategory_eq' {C D : Precategory}
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(p : carrier C = carrier D)
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(q : Π{a b : C}, a ⟶ b = cast p a ⟶ cast p b)
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(r : Π{a b c : C} (g : b ⟶ c) (f : a ⟶ b), cast q (g ∘ f) = cast q g ∘ cast q f)
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(s : Πa, cast q (ID a) = ID (cast p a)) : ap carrier (Precategory_eq p @q @r) = p :=
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begin
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induction C with X C, induction D with Y D, esimp at *, induction p,
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rewrite [↑Precategory_eq, -ap_compose,↑function.compose,ap_constant]
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end
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/-
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theorem Precategory_eq'_eta {C D : Precategory} (p : C = D) :
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Precategory_eq
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(ap carrier p)
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(Precategory_eq_hom p)
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(by induction p; intros; reflexivity) = p :=
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begin
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induction p, induction C with X C, unfold Precategory_eq,
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induction C, unfold precategory_eq, exact sorry
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end
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-/
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/-
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theorem Precategory_eq2 {C D : Precategory} (p q : C = D)
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(r : ap carrier p = ap carrier q)
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(s : Precategory_eq_hom p =[r] Precategory_eq_hom q)
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: p = q :=
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begin
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end
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-/
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end category
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