lean2/library/algebra/category/basic.lean
Floris van Doorn d8a616fa70 refactor(library): major changes in the library
I made some major changes in the library. I wanted to wait with pushing
until I had finished the formalization of the slice functor, but for
some reason that is very hard to formalize, requiring a lot of casts and
manipulation of casts. So I've not finished that yet.

Changes:

- in multiple files make more use of variables

- move dependent congr_arg theorems to logic.cast and proof them using heq (which doesn't involve nested inductions and fewer casts).

- prove some more theorems involving heq, e.g. hcongr_arg3 (which do not
  require piext)

- in theorems where casts are used in the statement use eq.rec_on
  instead of eq.drec_on

- in category split basic into basic, functor and natural_transformation

- change the definition of functor to use fully bundled
categories. @avigad: this means that the file semisimplicial.lean will
also need changes (but I'm quite sure nothing major).  You want to
define the fully bundled category Delta, and use only fully bundled
categories (type and ᵒᵖ are notations for the fully bundled
Type_category and Opposite if you open namespace category.ops). If you
want I can make the changes.

- lots of minor changes
2014-11-03 18:45:12 -08:00

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-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
-- Released under Apache 2.0 license as described in the file LICENSE.
-- Author: Floris van Doorn
import logic.axioms.funext
open eq eq.ops
inductive category [class] (ob : Type) : Type :=
mk : Π (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),
(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) →
(Π ⦃a b : ob⦄ {f : hom a b}, comp id f = f) →
(Π ⦃a b : ob⦄ {f : hom a b}, comp f id = f) →
category ob
namespace category
variables {ob : Type} [C : category ob]
variables {a b c d : ob}
include C
definition hom [reducible] : ob → ob → Type := rec (λ hom compose id assoc idr idl, hom) C
-- note: needs to be reducible to typecheck composition in opposite category
definition compose [reducible] : Π {a b c : ob}, hom b c → hom a b → hom a c :=
rec (λ hom compose id assoc idr idl, compose) C
definition id [reducible] : Π {a : ob}, hom a a := rec (λ hom compose id assoc idr idl, id) C
definition ID [reducible] (a : ob) : hom a a := id
infixr `∘`:60 := compose
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}
theorem assoc : Π ⦃a b c d : ob⦄ (h : hom c d) (g : hom b c) (f : hom a b),
h ∘ (g ∘ f) = (h ∘ g) ∘ f :=
rec (λ hom comp id assoc idr idl, assoc) C
theorem id_left : Π ⦃a b : ob⦄ (f : hom a b), id ∘ f = f :=
rec (λ hom comp id assoc idl idr, idl) C
theorem id_right : Π ⦃a b : ob⦄ (f : hom a b), f ∘ id = f :=
rec (λ hom comp id assoc idl idr, idr) C
--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 left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
calc i = i ∘ id : id_right
... = id : H
theorem right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
calc i = id ∘ i : id_left
... = id : H
end category
inductive Category : Type := mk : Π (ob : Type), category ob → Category
namespace category
definition Mk {ob} (C) : Category := Category.mk ob C
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] (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
open category
theorem Category.equal (C : Category) : Category.mk C C = C :=
Category.rec (λ ob c, rfl) C