lean2/hott/algebra/precategory/basic.hlean

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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
open eq is_trunc
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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),
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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)
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attribute precategory [multiple-instances]
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namespace precategory
variables {ob : Type} [C : precategory ob]
variables {a b c d : ob}
include C
attribute homH [instance]
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definition compose [reducible] := comp
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definition id [reducible] {a : ob} : hom a a := ID a
infixr `∘` := comp
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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 id_compose (a : ob) : ID a ∘ ID a = ID a := !id_left
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theorem left_id_unique (H : Π{b} {f : hom b a}, i ∘ f = f) : i = id :=
calc i = i ∘ id : id_right
... = id : H
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theorem right_id_unique (H : Π{b} {f : hom a b}, f ∘ i = f) : i = id :=
calc i = id ∘ i : id_left
... = id : H
definition homset [reducible] (x y : ob) : hset :=
hset.mk (hom x y) _
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end precategory
structure Precategory : Type :=
(objects : Type)
(category_instance : precategory objects)
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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] := Precategory.objects
definition category_instance [instance] [coercion] [reducible] := Precategory.category_instance
notation g `∘⁅` C `⁆` f := @compose (objects C) (category_instance C) _ _ _ g f
-- TODO: make this left associative
-- TODO: change this notation?
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end precategory
open precategory
protected definition Precategory.eta (C : Precategory) : Precategory.mk C C = C :=
Precategory.rec (λob c, idp) C