lean2/library/hott/algebra/precategory/basic.lean

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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 hott.axioms.funext hott.trunc hott.equiv
open path truncation
structure precategory [class] (ob : Type) : Type :=
(hom : ob → ob → Type)
(homH : 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)
(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)
(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)
namespace precategory
variables {ob : Type} [C : precategory ob]
variables {a b c d : ob}
include C
instance [persistent] homH
definition compose := comp
definition id [reducible] {a : ob} : hom a a := ID a
infixr `∘` := 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}
--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 precategory
inductive Precategory : Type := mk : Π (ob : Type), precategory ob → Precategory
namespace precategory
definition Mk {ob} (C) : Precategory := Precategory.mk ob C
definition MK (a b c d e f g h) : Precategory := Precategory.mk a (precategory.mk b c d e f g h)
definition objects [coercion] [reducible] (C : Precategory) : Type
:= Precategory.rec (fun c s, c) C
definition category_instance [instance] [coercion] [reducible] (C : Precategory) : precategory (objects C)
:= Precategory.rec (fun c s, s) C
end precategory
open precategory
theorem Precategory.equal (C : Precategory) : Precategory.mk C C ≈ C :=
Precategory.rec (λ ob c, idp) C