lean2/hott/algebra/category/precategory.hlean

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/-
Copyright (c) 2015 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
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Module: algebra.category.precategory
Authors: Floris van Doorn
-/
import types.trunc types.pi arity
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open eq is_trunc pi
namespace category
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/-
Just as in Coq-HoTT we add two redundant fields to precategories: assoc' and id_id.
The first is to make (Cᵒᵖ)ᵒᵖ = C definitionally when C is a constructor.
The second is to ensure that the functor from the terminal category 1 ⇒ Cᵒᵖ is
opposite to the functor 1 ⇒ C.
-/
structure precategory [class] (ob : Type) : Type :=
mk' ::
(hom : ob → ob → Type)
(comp : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c)
(ID : Π (a : ob), hom a a)
(assoc : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
comp h (comp g f) = comp (comp h g) f)
(assoc' : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
comp (comp h g) f = comp h (comp g f))
(id_left : Π ⦃a b : ob⦄ (f : hom a b), comp !ID f = f)
(id_right : Π ⦃a b : ob⦄ (f : hom a b), comp f !ID = f)
(id_id : Π (a : ob), comp !ID !ID = ID a)
(is_hset_hom : Π(a b : ob), is_hset (hom a b))
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attribute precategory [multiple-instances]
attribute precategory.is_hset_hom [instance]
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infixr `∘` := precategory.comp
-- input ⟶ using \--> (this is a different arrow than \-> (→))
infixl [parsing-only] `⟶`:25 := precategory.hom
namespace hom
infixl `⟶`:25 := precategory.hom -- if you open this namespace, hom a b is printed as a ⟶ b
end hom
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abbreviation hom := @precategory.hom
abbreviation comp := @precategory.comp
abbreviation ID := @precategory.ID
abbreviation assoc := @precategory.assoc
abbreviation assoc' := @precategory.assoc'
abbreviation id_left := @precategory.id_left
abbreviation id_right := @precategory.id_right
abbreviation id_id := @precategory.id_id
abbreviation is_hset_hom := @precategory.is_hset_hom
-- the constructor you want to use in practice
protected definition precategory.mk {ob : Type} (hom : ob → ob → Type)
[hset : Π (a b : ob), is_hset (hom a b)]
(comp : Π ⦃a b c : ob⦄, hom b c → hom a b → hom a c) (ID : Π (a : ob), hom a a)
(ass : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
comp h (comp g f) = comp (comp h g) f)
(idl : Π ⦃a b : ob⦄ (f : hom a b), comp (ID b) f = f)
(idr : Π ⦃a b : ob⦄ (f : hom a b), comp f (ID a) = f) : precategory ob :=
precategory.mk' hom comp ID ass (λa b c d h g f, !ass⁻¹) idl idr (λa, !idl) hset
section basic_lemmas
variables {ob : Type} [C : precategory ob]
variables {a b c d : ob} {h : c ⟶ d} {g : hom b c} {f f' : hom a b} {i : a ⟶ a}
include C
definition id [reducible] := ID a
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definition id_comp (a : ob) : ID a ∘ ID a = ID a := !id_left
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definition id_leftright (f : hom a b) : id ∘ f ∘ id = f := !id_left ⬝ !id_right
definition comp_id_eq_id_comp (f : hom a b) : f ∘ id = id ∘ f := !id_right ⬝ !id_left⁻¹
definition left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
calc i = i ∘ id : by rewrite id_right
... = id : by rewrite H
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definition right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
calc i = id ∘ i : by rewrite id_left
... = id : by rewrite H
definition homset [reducible] (x y : ob) : hset :=
hset.mk (hom x y) _
end basic_lemmas
section squares
parameters {ob : Type} [C : precategory ob]
local infixl `⟶`:25 := @precategory.hom ob C
local infixr `∘` := @precategory.comp ob C _ _ _
definition compose_squares {xa xb xc ya yb yc : ob}
{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb}
{wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
: wc ∘ (xg ∘ xf) = (yg ∘ yf) ∘ wa :=
calc
wc ∘ (xg ∘ xf) = (wc ∘ xg) ∘ xf : by rewrite assoc
... = (yg ∘ wb) ∘ xf : by rewrite xybc
... = yg ∘ (wb ∘ xf) : by rewrite assoc
... = yg ∘ (yf ∘ wa) : by rewrite xyab
... = (yg ∘ yf) ∘ wa : by rewrite assoc
definition compose_squares_2x2 {xa xb xc ya yb yc za zb zc : ob}
{xg : xb ⟶ xc} {xf : xa ⟶ xb} {yg : yb ⟶ yc} {yf : ya ⟶ yb} {zg : zb ⟶ zc} {zf : za ⟶ zb}
{va : ya ⟶ za} {vb : yb ⟶ zb} {vc : yc ⟶ zc} {wa : xa ⟶ ya} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
(xyab : wb ∘ xf = yf ∘ wa) (xybc : wc ∘ xg = yg ∘ wb)
(yzab : vb ∘ yf = zf ∘ va) (yzbc : vc ∘ yg = zg ∘ vb)
: (vc ∘ wc) ∘ (xg ∘ xf) = (zg ∘ zf) ∘ (va ∘ wa) :=
calc
(vc ∘ wc) ∘ (xg ∘ xf) = vc ∘ (wc ∘ (xg ∘ xf)) : by rewrite (assoc vc wc _)
... = vc ∘ ((yg ∘ yf) ∘ wa) : by rewrite (compose_squares xyab xybc)
... = (vc ∘ (yg ∘ yf)) ∘ wa : by rewrite assoc
... = ((zg ∘ zf) ∘ va) ∘ wa : by rewrite (compose_squares yzab yzbc)
... = (zg ∘ zf) ∘ (va ∘ wa) : by rewrite assoc
definition square_precompose {xa xb xc yb yc : ob}
{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
(H : wc ∘ xg = yg ∘ wb) (xf : xa ⟶ xb) : wc ∘ xg ∘ xf = yg ∘ wb ∘ xf :=
calc
wc ∘ xg ∘ xf = (wc ∘ xg) ∘ xf : by rewrite assoc
... = (yg ∘ wb) ∘ xf : by rewrite H
... = yg ∘ wb ∘ xf : by rewrite assoc
definition square_postcompose {xb xc yb yc yd : ob}
{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) : (yh ∘ wc) ∘ xg = (yh ∘ yg) ∘ wb :=
calc
(yh ∘ wc) ∘ xg = yh ∘ wc ∘ xg : by rewrite assoc
... = yh ∘ yg ∘ wb : by rewrite H
... = (yh ∘ yg) ∘ wb : by rewrite assoc
definition square_prepostcompose {xa xb xc yb yc yd : ob}
{xg : xb ⟶ xc} {yg : yb ⟶ yc} {wb : xb ⟶ yb} {wc : xc ⟶ yc}
(H : wc ∘ xg = yg ∘ wb) (yh : yc ⟶ yd) (xf : xa ⟶ xb)
: (yh ∘ wc) ∘ (xg ∘ xf) = (yh ∘ yg) ∘ (wb ∘ xf) :=
square_precompose (square_postcompose H yh) xf
end squares
structure Precategory : Type :=
(carrier : Type)
(struct : precategory carrier)
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definition precategory.Mk [reducible] {ob} (C) : Precategory := Precategory.mk ob C
definition precategory.MK [reducible] (a b c d e f g h) : Precategory :=
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]
attribute Precategory.struct [instance] [priority 10000] [coercion]
-- definition precategory.carrier [coercion] [reducible] := Precategory.carrier
-- definition precategory.struct [instance] [coercion] [reducible] := Precategory.struct
notation g `∘⁅`:60 C:0 `⁆`:0 f:60 :=
@comp (Precategory.carrier C) (Precategory.struct C) _ _ _ g f
-- TODO: make this left associative
-- TODO: change this notation?
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definition Precategory.eta (C : Precategory) : Precategory.mk C C = C :=
Precategory.rec (λob c, idp) C
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/-Characterization of paths between precategories-/
definition precategory_mk'_eq (ob : Type)
(hom1 : ob → ob → Type)
(hom2 : ob → ob → Type)
(homH1 : Π(a b : ob), is_hset (hom1 a b))
(homH2 : Π(a b : ob), is_hset (hom2 a b))
(comp1 : Π⦃a b c : ob⦄, hom1 b c → hom1 a b → hom1 a c)
(comp2 : Π⦃a b c : ob⦄, hom2 b c → hom2 a b → hom2 a c)
(ID1 : Π (a : ob), hom1 a a)
(ID2 : Π (a : ob), hom2 a a)
(assoc1 : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
comp1 h (comp1 g f) = comp1 (comp1 h g) f)
(assoc2 : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
comp2 h (comp2 g f) = comp2 (comp2 h g) f)
(assoc1' : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
comp1 (comp1 h g) f = comp1 h (comp1 g f))
(assoc2' : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
comp2 (comp2 h g) f = comp2 h (comp2 g f))
(id_left1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 !ID1 f = f)
(id_left2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 !ID2 f = f)
(id_right1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 f !ID1 = f)
(id_right2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 f !ID2 = f)
(id_id1 : Π (a : ob), comp1 !ID1 !ID1 = ID1 a)
(id_id2 : Π (a : ob), comp2 !ID2 !ID2 = ID2 a)
(p : hom1 = hom2)
(q : p ▹ comp1 = comp2)
(r : p ▹ ID1 = ID2) :
precategory.mk' hom1 comp1 ID1 assoc1 assoc1' id_left1 id_right1 id_id1 homH1
= precategory.mk' hom2 comp2 ID2 assoc2 assoc2' id_left2 id_right2 id_id2 homH2 :=
begin
cases p, cases q, cases r,
apply (ap0111111 (precategory.mk' hom2 comp2 ID2)),
repeat (apply is_hprop.elim),
end
definition precategory_eq (ob : Type)
(C D : precategory ob)
(p : @hom ob C = @hom ob D)
(q : transport (λ x, Πa b c, x b c → x a b → x a c) p
(@comp ob C) = @comp ob D)
(r : transport (λ x, Πa, x a a) p (@ID ob C) = @ID ob D) : C = D :=
begin
cases C, cases D,
apply precategory_mk'_eq, apply q, apply r,
end
definition precategory_mk_eq (ob : Type)
(hom1 : ob → ob → Type)
(hom2 : ob → ob → Type)
(homH1 : Π(a b : ob), is_hset (hom1 a b))
(homH2 : Π(a b : ob), is_hset (hom2 a b))
(comp1 : Π⦃a b c : ob⦄, hom1 b c → hom1 a b → hom1 a c)
(comp2 : Π⦃a b c : ob⦄, hom2 b c → hom2 a b → hom2 a c)
(ID1 : Π (a : ob), hom1 a a)
(ID2 : Π (a : ob), hom2 a a)
(assoc1 : Π ⦃a b c d : ob⦄ (h : hom1 c d) (g : hom1 b c) (f : hom1 a b),
comp1 h (comp1 g f) = comp1 (comp1 h g) f)
(assoc2 : Π ⦃a b c d : ob⦄ (h : hom2 c d) (g : hom2 b c) (f : hom2 a b),
comp2 h (comp2 g f) = comp2 (comp2 h g) f)
(id_left1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 !ID1 f = f)
(id_left2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 !ID2 f = f)
(id_right1 : Π ⦃a b : ob⦄ (f : hom1 a b), comp1 f !ID1 = f)
(id_right2 : Π ⦃a b : ob⦄ (f : hom2 a b), comp2 f !ID2 = f)
(p : Π (a b : ob), hom1 a b = hom2 a b)
(q : transport (λ x, Π a b c, x b c → x a b → x a c)
(eq_of_homotopy (λ a, eq_of_homotopy (λ b, p a b))) @comp1 = @comp2)
(r : transport (λ x, Π a, x a a)
(eq_of_homotopy (λ (x : ob), eq_of_homotopy (λ (x_1 : ob), p x x_1)))
ID1 = ID2) :
precategory.mk hom1 comp1 ID1 assoc1 id_left1 id_right1
= precategory.mk hom2 comp2 ID2 assoc2 id_left2 id_right2 :=
begin
fapply precategory_eq,
apply eq_of_homotopy, intros,
apply eq_of_homotopy, intros,
exact (p _ _),
exact q,
exact r,
end
definition Precategory_eq (C D : Precategory)
(p : carrier C = carrier D)
(q : p ▹ (Precategory.struct C) = Precategory.struct D) : C = D :=
begin
cases C, cases D,
cases p, cases q,
apply idp,
end
end category