Library HoTT.WildCat.Opposite

Require Import Basics.Overture Basics.Tactics.
Require Import WildCat.Core.

Opposite categories

Definition op (A : Type) := A.
Notation "A ^op" := (op A).

This stops typeclass search from trying to unfold op.

#[global] Typeclasses Opaque op.

Instance isgraph_op {A : Type} `{IsGraph A}
  : IsGraph A ^op.
Proof.
  apply Build_IsGraph; cbv.
  intros a b; exact (Hom b a).
Defined.

Instance is01cat_op {A : Type} `{Is01Cat A} : Is01Cat A ^op.
Proof.
  apply Build_Is01Cat; cbv.
  + exact Id.
  + intros a b c f g; exact (cat_comp g f).
Defined.

We don't invert 2-cells as this is op on the first level.

Instance is2graph_op {A : Type} `{Is2Graph A} : Is2Graph A ^op.
Proof.
  cbv.
  intros a b; exact (isgraph_hom b a).
Defined.

Instance is1cat_op {A : Type} `{Is1Cat A} : Is1Cat A ^op.
Proof.
  snapply Build_Is1Cat; cbv.
  - intros a b; exact (is01cat_hom b a).
  - intros a b; exact (is0gpd_hom b a).
  - intros a b c; exact (is0functor_precomp c b a).
  - intros a b c; exact (is0functor_postcomp c b a).
  - intros a b c d f g h; exact (cat_assoc_opp h g f).
  - intros a b c d f g h; exact (cat_assoc h g f).
  - intros a b; exact cat_idr.
  - intros a b; exact cat_idl.
Defined.

Instance is1cat_strong_op A `{Is1Cat_Strong A}
  : Is1Cat_Strong (A ^op).
Proof.
  snapply Build_Is1Cat_Strong; cbv.
  - intros a b; exact (is01cat_hom_strong b a).
  - intros a b; exact (is0gpd_hom_strong b a).
  - intros a b c; exact (is0functor_precomp_strong c b a).
  - intros a b c; exact (is0functor_postcomp_strong c b a).
  - intros a b c d f g h; exact (cat_assoc_opp_strong h g f).
  - intros a b c d f g h; exact (cat_assoc_strong h g f).
  - intros a b; exact cat_idr_strong.
  - intros a b; exact cat_idl_strong.
Defined.

Opposite groupoids

Instance is0gpd_op A `{Is0Gpd A} : Is0Gpd (A ^op).
Proof.
  srapply Build_Is0Gpd; cbv.
  intros a b.
  exact gpd_rev.
Defined.

Instance op0gpd_fun A `{Is0Gpd A} :
  Is0Functor((fun x ⇒ x) : A ^op → A).
Proof.
  srapply Build_Is0Functor; cbv.
  intros a b.
  exact (fun f ⇒ f ^$).
Defined.

Opposite functors

Instance is0functor_op A B (F : A → B)
  `{IsGraph A, IsGraph B, x : !Is0Functor F}
  : Is0Functor (F : A ^op → B ^op).
Proof.
  apply Build_Is0Functor; cbv.
  intros a b; exact (fmap F).
Defined.

Instance is1functor_op A B (F : A → B)
  `{Is1Cat A, Is1Cat B, !Is0Functor F, !Is1Functor F}
  : Is1Functor (F : A ^op → B ^op).
Proof.
  apply Build_Is1Functor; cbv.
  - intros a b f g; exact (fmap2 F).
  - exact (fmap_id F).
  - intros a b c f g; exact (fmap_comp F g f).
Defined.

Since Is01Cat structures are definitionally involutive (see test/WildCat/Opposite.v), we can use is0functor_op to transform in the reverse direction as well. This result makes that much easier to use in practice.

Instance is0functor_op' A B (F : A ^op → B ^op)
  `{IsGraph A, IsGraph B, Fop : !Is0Functor (F : A ^op → B ^op)}
  : Is0Functor (F : A → B)
  := is0functor_op A^op B^op F.

Is1Cat structures are also definitionally involutive.

Instance is1functor_op' A B (F : A ^op → B ^op)
  `{Is1Cat A, Is1Cat B, !Is0Functor (F : A ^op → B ^op ), Fop2 : !Is1Functor (F : A ^op → B ^op)}
  : Is1Functor (F : A → B)
  := is1functor_op A^op B^op F.

Instance hasmorext_op {A : Type} `{H0 : HasMorExt A}
  : HasMorExt A ^op.
Proof.
  unfold op.
  intros a b; exact (isequiv_Htpy_path b a).
Defined.

Instance isinitial_op_isterminal {A : Type} `{Is1Cat A} (x : A)
  {t : IsTerminal x} : IsInitial (A := A ^op) x
  := t.

Instance isterminal_op_isinitial {A : Type} `{Is1Cat A} (x : A)
  {i : IsInitial x} : IsTerminal (A := A ^op) x
  := i.