lean2/hott/algebra/category/strict.hlean

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/-
Copyright (c) 2015 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Module: algebra.category.strict
Authors: Floris van Doorn, Jakob von Raumer
-/
import .precategory .functor
open is_trunc eq
namespace category
structure strict_precategory [class] (ob : Type) extends precategory ob :=
mk' :: (is_hset_ob : is_hset ob)
attribute strict_precategory.is_hset_ob [instance]
definition strict_precategory.mk [reducible] {ob : Type} (C : precategory ob)
(H : is_hset ob) : strict_precategory ob :=
precategory.rec_on C strict_precategory.mk' H
structure Strict_precategory : Type :=
(carrier : Type)
(struct : strict_precategory carrier)
attribute Strict_precategory.struct [instance] [coercion]
definition Strict_precategory.to_Precategory [coercion] [reducible]
(C : Strict_precategory) : Precategory :=
Precategory.mk (Strict_precategory.carrier C) C
open functor
definition precat_strict_precat : precategory Strict_precategory :=
precategory.mk (λ a b, functor a b)
(λ a b c g f, functor.compose g f)
(λ a, functor.id)
(λ a b c d h g f, !functor.assoc)
(λ a b f, !functor.id_left)
(λ a b f, !functor.id_right)
definition Precat_of_strict_precats := precategory.Mk precat_strict_precat
namespace ops
abbreviation SPreCat := Precat_of_strict_precats
--attribute precat_strict_precat [instance]
end ops
end category
/-section
open decidable unit empty
variables {A : Type} [H : decidable_eq A]
include H
definition set_hom (a b : A) := decidable.rec_on (H a b) (λh, unit) (λh, empty)
theorem set_hom_subsingleton [instance] (a b : A) : subsingleton (set_hom a b) := _
definition set_compose {a b c : A} (g : set_hom b c) (f : set_hom a b) : set_hom a c :=
decidable.rec_on
(H b c)
(λ Hbc g, decidable.rec_on
(H a b)
(λ Hab f, rec_on_true (trans Hab Hbc) ⋆)
(λh f, empty.rec _ f) f)
(λh (g : empty), empty.rec _ g) g
omit H
definition discrete_precategory (A : Type) [H : decidable_eq A] : precategory A :=
mk (λa b, set_hom a b)
(λ a b c g f, set_compose g f)
(λ a, decidable.rec_on_true rfl ⋆)
(λ a b c d h g f, @subsingleton.elim (set_hom a d) _ _ _)
(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
(λ a b f, @subsingleton.elim (set_hom a b) _ _ _)
definition Discrete_category (A : Type) [H : decidable_eq A] := Mk (discrete_category A)
end
section
open unit bool
definition category_one := discrete_category unit
definition Category_one := Mk category_one
definition category_two := discrete_category bool
definition Category_two := Mk category_two
end-/