Library UniMath.CategoryTheory.DisplayedCats.Limits

Require Import UniMath.Foundations.Sets.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.limits.graphs.colimits.
Require Import UniMath.CategoryTheory.limits.graphs.limits.
Local Open Scope cat.

Require Import UniMath.CategoryTheory.DisplayedCats.Auxiliary.
Require Import UniMath.CategoryTheory.DisplayedCats.Core.
Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.
Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations.

Local Open Scope type_scope.
Local Open Scope mor_disp_scope.

Section Auxiliary.

Definition forms_cone
    {C : precategory} {g : graph} (d : diagram g C)
    (c : C) (f : ∏ (v : vertex g), C⟦c, dob d v⟧)
  : UU
:= ∏ (u v : vertex g) (e : edge u v),
     f u · dmor d e = f v.

Coercion coneOut : cone >-> Funclass.

End Auxiliary.

Section Creates_Limits.

Definition creates_limit
  {C : category}
  (D : disp_cat C)
  {J : graph} (F : diagram J (total_category D))
  {x : C} (L : cone (mapdiagram (pr1_category D) F) x)
  (isL : isLimCone _ x L) : UU
:=
  ∑ (CC : iscontr
      ( ∑ (d : D x)
          (δ : ∏ j : vertex J, d -->[L j] (pr2 (dob F j))),
          forms_cone F (x,,d) (λ j, (L j ,, δ j))))
  , isLimCone _ _ (make_cone _ (pr2 (pr2 (iscontrpr1 CC)))).

Definition creates_limits {C : category} (D : disp_cat C) : UU
:=
  ∏ (J : graph) (F : diagram J (total_category D))
    {x : C} (L : cone (mapdiagram (pr1_category D) F) x)
    (isL : isLimCone _ x L),
  creates_limit _ _ _ isL.

End Creates_Limits.

Section creates_preserves.

Context {C : category}
        (D : disp_cat C)
        (H : creates_limits D)
        (J : graph)
        (X : Lims_of_shape J C).

Notation π := (pr1_category D).

Definition total_limits : Lims_of_shape J (total_category D).
Proof.
  intro d.
  set (πd := mapdiagram π d).
  set (LL := X πd).
  set (L := pr1 LL).
  set (c := pr1 L).
  set (isL := pr2 LL). cbn in isL.
  set (XR := H _ d _ _ isL).
  unfold creates_limit in XR.
  cbn.
  use (make_LimCone _ _ _ (pr2 XR)).
Defined.

Lemma pr1_preserves_limit (d : diagram J (total_category D))
  (x : total_category D) (CC : cone d x) : preserves_limit π _ x CC.
Proof.
  intro H1.
  set (XR := X (mapdiagram π d)).
  use is_iso_isLim.
  - apply homset_property.
  - apply X.
  - match goal with |[ |- is_iso ?foo ] ⇒ set (T:= foo) end.
    destruct X as [[a L] isL]. cbn in isL.
    clear XR.
    set (tL := H _ _ _ _ isL).
    unfold creates_limit in tL.
    set (RR := pr1 tL).
    set (RT1 := pr2 tL).

    set (RX := isLim_is_iso _ (make_LimCone _ _ _ RT1) _ _ H1).
    set (XR := @functor_on_is_iso_is_iso _ _ π _ _ _ RX).
    match goal with |[ H : is_iso ?f |- _ ] ⇒ set (T':= f) end.

    assert (X0 : T' = T).
    {

      clear XR.
      clear RX.
      unfold T.
      unfold T'.
      apply (limArrowUnique (make_LimCone _ _ _ isL)) .
      intro j.
      set (RRt := make_LimCone _ _ _ RT1).
      set (RRtt := limArrowCommutes RRt x CC j).
      set (RH := maponpaths (#π)%Cat RRtt).
      cbn in RH.
      apply RH.
    }
    rewrite <- X0.
    apply XR.
Defined.

End creates_preserves.