43 lines
1.5 KiB
Text
43 lines
1.5 KiB
Text
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-- 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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infixr `∘` := compose
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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 id_left2 {A B : ob} {f : mor A B} : id ∘ f = f :=
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rec (λ comp id assoc idr idl, idl A B f) Cat
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end
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end category
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