58 lines
1.7 KiB
Text
58 lines
1.7 KiB
Text
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/-
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Copyright (c) 2015 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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Authors: Floris van Doorn
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Terminal category
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-/
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import .indiscrete
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open functor is_trunc unit eq
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namespace category
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definition terminal_precategory [constructor] : precategory unit :=
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indiscrete_precategory unit
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definition Terminal_precategory [constructor] : Precategory :=
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precategory.Mk terminal_precategory
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notation 1 := Terminal_precategory
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definition one_op : 1ᵒᵖ = 1 := idp
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definition terminal_functor [constructor] (C : Precategory) : C ⇒ 1 :=
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functor.mk (λx, star)
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(λx y f, star)
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(λx, idp)
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(λx y z g f, idp)
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definition is_contr_functor_one [instance] (C : Precategory) : is_contr (C ⇒ 1) :=
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is_contr.mk (terminal_functor C)
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begin
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intro F, fapply functor_eq,
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{ intro x, apply @is_hprop.elim unit},
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{ intro x y f, apply @is_hprop.elim unit}
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end
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definition terminal_functor_op (C : Precategory)
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: (terminal_functor C)ᵒᵖ = terminal_functor Cᵒᵖ := idp
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definition terminal_functor_comp {C D : Precategory} (F : C ⇒ D)
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: (terminal_functor D) ∘f F = terminal_functor C := idp
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definition point (C : Precategory) (c : C) : 1 ⇒ C :=
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functor.mk (λx, c)
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(λx y f, id)
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(λx, idp)
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(λx y z g f, !id_id⁻¹)
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-- we need id_id in the declaration of precategory to make this to hold definitionally
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definition point_op (C : Precategory) (c : C) : (point C c)ᵒᵖ = point Cᵒᵖ c := idp
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definition point_comp {C D : Precategory} (F : C ⇒ D) (c : C)
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: F ∘f point C c = point D (F c) := idp
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end category
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