150 lines
5.1 KiB
Text
150 lines
5.1 KiB
Text
-- Copyright (c) 2014 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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-- Author: Floris van Doorn
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-- category
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import logic.core.eq logic.core.connectives
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import data.unit data.sigma data.prod
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import struc.function
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inductive category [class] (ob : Type) (mor : ob → ob → Type) : Type :=
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mk : Π (comp : Π⦃A B C : ob⦄, mor B C → mor A B → mor A C)
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(id : Π {A : ob}, mor A A),
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(Π {A B C D : ob} {f : mor A B} {g : mor B C} {h : mor C D},
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comp h (comp g f) = comp (comp h g) f) →
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(Π {A B : ob} {f : mor A B}, comp f id = f) →
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(Π {A B : ob} {f : mor A B}, comp id f = f) →
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category ob mor
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namespace category
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precedence `∘` : 60
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section
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parameters {ob : Type} {mor : ob → ob → Type} {Cat : category ob mor}
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abbreviation compose := rec (λ comp id assoc idr idl, comp) Cat
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abbreviation id := rec (λ comp id assoc idr idl, id) Cat
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abbreviation ID (A : ob) := @id A
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end
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infixr `∘` := compose
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section
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parameters {ob : Type} {mor : ob → ob → Type} {Cat : category ob mor}
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theorem assoc : Π {A B C D : ob} {f : mor A B} {g : mor B C} {h : mor C D},
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h ∘ (g ∘ f) = (h ∘ g) ∘ f :=
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rec (λ comp id assoc idr idl, assoc) Cat
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theorem id_right : Π {A B : ob} {f : mor A B}, f ∘ id = f :=
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rec (λ comp id assoc idr idl, idr) Cat
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theorem id_left : Π {A B : ob} {f : mor A B}, id ∘ f = f :=
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rec (λ comp id assoc idr idl, idl) Cat
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theorem left_id_unique {A : ob} (i : mor A A) (H : Π{B} {f : mor B A}, i ∘ f = f) : i = id :=
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calc
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i = i ∘ id : eq.symm id_right
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... = id : H
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theorem right_id_unique {A : ob} (i : mor A A) (H : Π{B} {f : mor A B}, f ∘ i = f) : i = id :=
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calc
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i = id ∘ i : eq.symm id_left
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... = id : H
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definition has_left_inverse {A B : ob} (f : mor A B) : Type :=
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including Cat, Σ g, g ∘ f = id
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definition left_inverse {A B : ob} (f : mor A B) (H : has_left_inverse f) : mor B A :=
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sigma.dpr1 H
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definition has_right_inverse {A B : ob} (f : mor A B) : Type :=
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including Cat, Σ g, f ∘ g = id
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definition right_inverse {A B : ob} (f : mor A B) (H : has_right_inverse f) : mor B A :=
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sigma.dpr1 H
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definition iso {A B : ob} (f : mor A B) : Type :=
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including Cat, Σ g, f ∘ g = id ∧ g ∘ f = id
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definition inverse {A B : ob} (f : mor A B) (H : iso f) : mor B A :=
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sigma.dpr1 H
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theorem iso_imp_left_inverse {A B : ob} (f : mor A B) (H : iso f) : has_left_inverse f :=
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sorry
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theorem iso_imp_right_inverse {A B : ob} (f : mor A B) (H : iso f) : has_left_inverse f :=
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sorry
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theorem left_right_inverse_imp_iso {A B : ob} (f : mor A B)
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(Hl : has_left_inverse f) (Hr : has_right_inverse f) : iso f :=
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sorry
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postfix `⁻¹` := inverse
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set_option pp.implicit true
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-- theorem foo {A B : ob} {f : mor A B} (H : iso f) : true :=
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-- including Cat, (λx (y : iso f),x) _ H
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theorem compose_inverse {A B : ob} {f : mor A B} (H : iso f) : f ∘ f⁻¹ H = id :=
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and.elim_left (sigma.dpr2 H)
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theorem inverse_compose {A B : ob} {f : mor A B} (H : iso f) : f⁻¹ H ∘ f = id :=
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and.elim_right (sigma.dpr2 H)
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theorem inverse_unique {A B : ob} {f : mor A B} (H H' : iso f) : f⁻¹ H = f⁻¹ H' :=
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sorry
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-- calc
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-- inverse f H = f⁻¹ H ∘ id : symm id.right
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-- ... = f⁻¹ H ∘ f ∘ f⁻¹ H' : {symm (compose_inverse H')}
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-- ... = (f⁻¹ H ∘ f) ∘ f⁻¹ H' : assoc
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-- ... = id ∘ f⁻¹ H' : {inverse_compose H}
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-- ... = f⁻¹ H' : id.left
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definition mono {A B : ob} (f : mor A B) : Prop :=
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including Cat, ∀⦃C⦄ {g h : mor C A}, f ∘ g = f ∘ h → g = h
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definition epi {A B : ob} (f : mor A B) : Prop :=
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including Cat, ∀⦃C⦄ {g h : mor B C}, g ∘ f = h ∘ f → g = h
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end
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postfix `⁻¹` := inverse
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section
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parameters {obC obD : Type} {morC : obC → obC → Type} {morD : obD → obD → Type}
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parameters (C : category obC morC)
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parameters (D : category obD morD)
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definition tst (a b c : obC) (m1 : morC a b) (m2 : morC b c) :=
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(λx y, x) (compose m2 m1) (including C, false)
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definition tst2 (C : category obC morC) (a b c : obC) (m1 : morC a b) (m2 : morC b c) :=
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compose m2 m1
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parameter a : obC
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parameter f : morC a a
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-- inductive foo : Type :=
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-- mk : including C, foo
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-- inductive functor : Type :=
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-- functor.mk : including C D,
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-- Π (obF : obC → obD) (morF : Π{A B}, morC A B → morD (obF A) (obF B)),
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-- (Π {A : obC}, morF (ID A) = ID (obF A)) →
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-- (Π {A B C : obC} {f : morC A B} {g : morC B C}, morF (g ∘ f) = morF g ∘ morF f) →
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-- functor
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end
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section
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open unit
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definition one [instance] : category unit (λa b, unit) :=
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category.mk (λ a b c f g, star) (λ a, star) (λ a b c d f g h, unit.equal _ _)
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(λ a b f, unit.equal _ _) (λ a b f, unit.equal _ _)
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end
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section
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--need extensionality
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definition type_cat : category Type (λA B, A → B) :=
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mk (λ a b c f g, function.compose f g) (λ a, function.id) (λ a b c d f g h, sorry)
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(λ a b f, sorry) (λ a b f, sorry)
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end
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end category
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