48 lines
1.3 KiB
Text
48 lines
1.3 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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Initial category
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-/
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import .indiscrete
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open functor is_trunc eq
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namespace category
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definition initial_precategory [constructor] : precategory empty :=
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indiscrete_precategory empty
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definition Initial_precategory [constructor] : Precategory :=
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precategory.Mk initial_precategory
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notation 0 := Initial_precategory
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definition zero_op : 0ᵒᵖ = 0 := idp
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definition initial_functor [constructor] (C : Precategory) : 0 ⇒ C :=
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functor.mk (λx, empty.elim x)
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(λx y f, empty.elim x)
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(λx, empty.elim x)
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(λx y z g f, empty.elim x)
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definition is_contr_zero_functor [instance] (C : Precategory) : is_contr (0 ⇒ C) :=
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is_contr.mk (initial_functor C)
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begin
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intro F, fapply functor_eq,
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{ intro x, exact empty.elim x},
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{ intro x y f, exact empty.elim x}
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end
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definition initial_functor_op (C : Precategory)
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: (initial_functor C)ᵒᵖ = initial_functor Cᵒᵖ :=
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by apply @is_hprop.elim (0 ⇒ Cᵒᵖ)
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definition initial_functor_comp {C D : Precategory} (F : C ⇒ D)
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: F ∘f initial_functor C = initial_functor D :=
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by apply @is_hprop.elim (0 ⇒ D)
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end category
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