Library UniMath.Induction.M.Limits

Limits in the precategory of types

This is a partial reconstruction of the results of "Homotopy limits in type theory" by Jeremy Avigad, Chris Kapulkin, and Peter LeFanu Lumsdaine (arXiv:1304.0680v3).

Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Sets.
Require Import UniMath.MoreFoundations.Univalence.
Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.categories.Type.Core.

Require Import UniMath.CategoryTheory.limits.graphs.limits.
Require Import UniMath.CategoryTheory.limits.graphs.colimits.
Require Import UniMath.CategoryTheory.Chains.Cochains.

Local Open Scope cat.

Section StandardLimits.

  Context {g : graph} (d : diagram g type_precat).

  Definition standard_limit : UU :=
     (x : (v : vertex g), dob d v),
     (u v : vertex g) (e : edge u v), dmor d e (x u) = x v.

The condition that standard_limit is a cone is basically a rephrasing of its definition.
  Lemma type_cone : cone d standard_limit.
    use make_cone; cbn.
    - exact (λ n l, pr1 l n).
    - intros u v f.
      apply funextsec; intro l; unfold funcomp; cbn.
      apply (pr2 l).

End StandardLimits.

Section StandardLimitHomot.

  Context {g : graph} {d : diagram g type_precat} (x y : standard_limit d).

A homotopy of cones
  Definition standard_limit_homot : UU :=
     h : pr1 x ¬ pr1 y,
       (u v : vertex g) (ed : edge u v),
        (maponpaths (dmor d ed) (h u) @ (pr2 y _ _) ed = pr2 x _ _ ed @ (h v)).

Such homotopies can be made into paths
  Lemma type_cone_homot_to_path (h : standard_limit_homot) : x = y.
    apply (total2_paths_f (funextsec _ _ _ (pr1 h))).

transport_lemma in peterlefanulumsdaine/hott-limits/Limits1.v.
    assert (transport_lemma :
               p : pr1 x = pr1 y,
                transportf _ p (pr2 x) = λ u v (ed : edge u v),
                maponpaths (dmor d ed) (!(toforallpaths _ _ _ p u))
                @ pr2 x _ _ ed
                @ toforallpaths _ _ _ p v).
      intros p; induction p; cbn; unfold idfun.
      do 3 (apply funextsec; intro).
      exact (!(pathscomp0rid _)).
    refine (transport_lemma _ @ _).
    apply funextsec; intro u; apply funextsec; intro v; apply funextsec; intro ed.
    rewrite toforallpaths_funextsec.
    replace (pr2 y u v ed) with (idpath _ @ (pr2 y u v ed)) by reflexivity.
    refine (_ @ maponpaths (λ p, p @ _) (pathsinv0l (maponpaths _ (pr1 h u)))).
    refine (_ @ (path_assoc (! maponpaths _ _) (maponpaths _ _) _)).
    rewrite maponpathsinv0.
    apply maponpaths, pathsinv0.
    exact (pr2 h u v ed).
End StandardLimitHomot.

The canonical cone given by an arrow X → Y where Y has a cone

Definition into_cone_to_cone {X Y : UU} {g : graph} {d : diagram g _}
            (coneY : cone d (Y : ob type_precat)) (f : X Y) : cone d X.
  use make_cone.
  - intro ver.
    exact (pr1 coneY ver (f : type_precat X, Y )).
  - intros ver1 ver2 ed; cbn.
    apply funextsec; intro x.
    apply (toforallpaths _ _ _ (pr2 coneY ver1 ver2 ed)).

Section StandardLimitUP.

  Context {g : graph} {d : diagram g type_precat}.

A rephrasing of the universal property: the canonical map that makes a cone out of a map X → L is an equivalence.
  Definition is_limit_cone {L} (C : cone d L) :=
     (X : UU), isweq (@into_cone_to_cone X L g d C).

  Lemma isaprop_isLimCone {L} (C : cone d L) : isaprop (is_limit_cone C).
    repeat (apply impred; intro).
    apply isapropiscontr.

A weak equivalence expressing the above universal property.
  Definition limit_up_weq {X L} {C : cone d L} {is : is_limit_cone C} :
    (X L) cone d X := make_weq (into_cone_to_cone C) (is X).

The universal property of a limit.
  • Proposition 4.2.8 (limit_universal) in Avigad, Kapulkin, and Lumsdaine
  • Generalizes Lemma 10 in Ahrens, Capriotti, and Spadotti
  • Generalizes univ-iso in HoTT/M-types
  Lemma limit_universal : is_limit_cone (type_cone d).
    intro X.
    use isweq_iso.
    - intros xcone x.
      unfold standard_limit.
      use tpair.
      + exact (λ ver, pr1 xcone ver x).
      + intros ver1 ver2 ed.
        apply (toforallpaths _ _ _ (pr2 xcone _ _ _)).
    - intros f.
      apply funextfun; intro xcone.
      use total2_paths_f; cbn; [reflexivity|].
      cbn; unfold idfun.
      apply funextsec; intro ver1.
      apply funextsec; intro ver2.
      apply funextsec; intro ed.
      do 2 (rewrite toforallpaths_funextsec).
    - intro conex.
      unfold into_cone_to_cone; cbn.
      use total2_paths_f; cbn.
      + reflexivity.
      + apply funextsec; intro ver1.
        apply funextsec; intro ver2.
        apply funextsec; intro ed.
        unfold funcomp; cbn; unfold idfun.
        rewrite toforallpaths_funextsec; cbn.
        rewrite funextsec_toforallpaths.

The above weak equivalence specialized to the case of standard_limits
In the case of cochains, we can provide a somewhat simpler limit.
Section CochainLimit.
  Context (coch : cochain type_precat).
  Let X := (pr1 coch).
  Let π := (pr2 coch).
  Let π' n := π (S n) n (idpath _).

  Definition cochain_limit :=
     (x : n : nat, X n), n, π' n (x (S n)) = x n.

  Lemma simplify_cochain_step
        {u v : nat}
        (x : n, X n)
        (e : S v = u) :
    dmor coch e (x u) = π' v (x (S v)).
    unfold π', π in *; unfold dmor.
    (induction e; auto).

  Lemma simplify_cochain_step_idpath {u : nat} (x : n, X n) :
    @simplify_cochain_step (S u) u x (idpath _) =
    idpath _.

  Definition cochain_limit_cone :
    cone coch cochain_limit.
    use make_cone; cbn.
    - intros v cochain_limit_element.
      apply (pr1 cochain_limit_element).
    - intros u v e.
      apply funextsec; intro l; unfold funcomp; cbn.
      refine (_ @ pr2 l _).
      unfold dmor, π', π.
      apply simplify_cochain_step.

  Lemma lim_equiv : cochain_limit (standard_limit coch).
    unfold cochain_limit, standard_limit; cbn.
    apply weqfibtototal; intro.
    use weq_iso.
    - intros eq.
      intros u v e.
      specialize (eq v).
      unfold π', π in *; unfold dmor.
      refine (_ @ eq).
      apply simplify_cochain_step.
    - intros eq; intro; apply eq.
    - abstract ( intro; apply funextsec; intro; reflexivity ).
    - abstract ( intro; cbn;
      do 2 (apply funextsec; intro);
      apply funextsec; intro p;
      induction p;
      reflexivity ).

  Lemma cochain_limit_is_limit :
     c : cone coch cochain_limit, is_limit_cone c.
    assert (H :
      ( c : cone coch cochain_limit, is_limit_cone c)
       c : cone coch (standard_limit coch), is_limit_cone c
      induction (weqtopaths lim_equiv).
      apply idweq.
    apply H.
    eapply tpair.
    apply limit_universal.

End CochainLimit.