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chore(library/algebra): change category to be a structure

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Jakob von Raumer 2014-12-02 13:33:15 -05:00 committed by Leonardo de Moura
parent b37a77d25e
commit cda828bfe8
1 changed files with 19 additions and 13 deletions
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32
library/algebra/category/basic.lean
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@ -8,23 +8,26 @@ open eq eq.ops
structure category [class] (ob : Type) : Type := structure category [class] (ob : Type) : Type :=
(hom : ob → ob → Type) (hom : ob → ob → Type)
(compose : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c) (comp : Π⦃a b c : ob⦄, hom b c → hom a b → hom a c)
(ID : Π (a : ob), hom a a) (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), (assoc : Π ⦃a b c d : ob⦄ {h : hom c d} {g : hom b c} {f : hom a b},
compose h (compose g f) = compose (compose h g) f) comp h (comp g f) = comp (comp h g) f)
(id_left : Π ⦃a b : ob⦄ (f : hom a b), compose !ID f = 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), compose f !ID = f) (id_right : Π ⦃a b : ob⦄ {f : hom a b}, comp f !ID = f)
namespace category namespace category
variables {ob : Type} [C : category ob] variables {ob : Type} [C : category ob]
variables {a b c d : ob} {h : hom c d} {g : hom b c} {f : hom a b} {i : hom a a} variables {a b c d : ob}
include C include C
definition id [reducible] {a : ob} : hom a a := ID a definition id [reducible] {a : ob} : hom a a := ID a
infixr `∘` := compose infixr `∘` := comp
infixl `⟶`:25 := hom -- input ⟶ using \--> (this is a different arrow than \-> (→)) infixl `⟶`:25 := hom -- input ⟶ using \--> (this is a different arrow than \-> (→))
variables {h : hom c d} {g : hom b c} {f : hom a b} {i : hom a a}
--the following is the only theorem for which "include C" is necessary if C is a variable (why?)
theorem id_compose (a : ob) : (ID a) ∘ id = id := !id_left theorem id_compose (a : ob) : (ID a) ∘ id = id := !id_left
theorem left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id := theorem left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
@ -36,15 +39,18 @@ namespace category
... = id : H ... = id : H
end category end category
structure Category : Type := inductive Category : Type := mk : Π (ob : Type), category ob → Category
(objects : Type)
(category_instance : category objects)
namespace category namespace category
definition Mk {ob} (C) : Category := Category.mk ob C definition Mk {ob} (C) : Category := Category.mk ob C
definition MK (o h c i a l r) : Category := Category.mk o (category.mk h c i a l r) definition MK (a b c d e f g) : Category := Category.mk a (category.mk b c d e f g)
definition objects [coercion] [reducible] := Category.objects
definition category_instance [instance] [coercion] [reducible] := Category.category_instance definition objects [coercion] [reducible] (C : Category) : Type
:= Category.rec (fun c s, c) C
definition category_instance [instance] [coercion] [reducible] (C : Category) : category (objects C)
:= Category.rec (fun c s, s) C
end category end category
open category open category