lean2/hott/algebra/category/strict.hlean

/-
Copyright (c) 2015 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jakob von Raumer
-/

import .functor.basic

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) _

  open functor

  -- TODO: move to constructions.cat?
  definition precategory_strict_precategory [constructor] : precategory Strict_precategory :=
  precategory.mk (λ A B, A ⇒ B)
                 (λ A B C G F, G ∘f F)
                 (λ A, 1)
                 (λ A B C D, functor.assoc)
                 (λ A B, functor.id_left)
                 (λ A B, functor.id_right)

  definition Precategory_strict_precategory [constructor] := precategory.Mk precategory_strict_precategory

  namespace ops
    abbreviation Cat := Precategory_strict_precategory
  end ops

end category