2015-09-24 02:44:36 +00:00
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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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Discrete category
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-/
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2015-10-20 01:42:41 +00:00
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import ..groupoid types.bool ..functor.functor
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2015-09-24 02:44:36 +00:00
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open eq is_trunc iso bool functor
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namespace category
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definition precategory_of_1_type [constructor] (A : Type) [H : is_trunc 1 A] : precategory A :=
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2015-10-12 23:49:10 +00:00
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@precategory.mk _ _ (@is_trunc_eq _ _ H)
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2015-09-24 02:44:36 +00:00
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(λ (a b c : A) (p : b = c) (q : a = b), q ⬝ p)
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(λ (a : A), refl a)
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(λ (a b c d : A) (p : c = d) (q : b = c) (r : a = b), con.assoc r q p)
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(λ (a b : A) (p : a = b), con_idp p)
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(λ (a b : A) (p : a = b), idp_con p)
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definition groupoid_of_1_type [constructor] (A : Type) [H : is_trunc 1 A] : groupoid A :=
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groupoid.mk !precategory_of_1_type
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2015-10-22 22:41:55 +00:00
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(λ (a b : A) (p : a = b), is_iso.mk _ !con.right_inv !con.left_inv)
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2015-09-24 02:44:36 +00:00
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definition Precategory_of_1_type [constructor] (A : Type) [H : is_trunc 1 A] : Precategory :=
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precategory.Mk (precategory_of_1_type A)
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definition Groupoid_of_1_type [constructor] (A : Type) [H : is_trunc 1 A] : Groupoid :=
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groupoid.Mk _ (groupoid_of_1_type A)
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definition discrete_precategory [constructor] (A : Type) [H : is_hset A] : precategory A :=
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precategory_of_1_type A
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definition discrete_groupoid [constructor] (A : Type) [H : is_hset A] : groupoid A :=
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groupoid_of_1_type A
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definition Discrete_precategory [constructor] (A : Type) [H : is_hset A] : Precategory :=
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precategory.Mk (discrete_precategory A)
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definition Discrete_groupoid [constructor] (A : Type) [H : is_hset A] : Groupoid :=
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groupoid.Mk _ (discrete_groupoid A)
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definition c2 [constructor] : Precategory := Discrete_precategory bool
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definition c1 [constructor] : Precategory := Discrete_precategory unit
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definition c2_functor [constructor] (C : Precategory) (x y : C) : c2 ⇒ C :=
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functor.mk (bool.rec x y)
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(bool.rec (bool.rec (λf, id) (by contradiction))
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(bool.rec (by contradiction) (λf, id)))
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abstract (bool.rec idp idp) end
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abstract begin intro b₁ b₂ b₃ g f, induction b₁: induction b₂: induction b₃:
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esimp at *: try contradiction: exact !id_id⁻¹ end end
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end category
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