Library UniMath.CategoryTheory.limits.graphs.zero

Zero Objects

Zero objects are objects of precategory which are both initial objects and terminal object.

Contents

  • Definition of Zero
  • Coincides with the direct definition

Require Import UniMath.Foundations.PartD.
Require Import UniMath.Foundations.Propositions.
Require Import UniMath.Foundations.Sets.

Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Isos.
Require Import UniMath.CategoryTheory.limits.graphs.colimits.
Require Import UniMath.CategoryTheory.limits.graphs.limits.
Require Import UniMath.CategoryTheory.limits.graphs.initial.
Require Import UniMath.CategoryTheory.limits.graphs.terminal.
Require Import UniMath.CategoryTheory.limits.zero.

Local Open Scope cat.

Definition of zero using limits and colimits

Section def_zero.

  Context {C : precategory}.

An object c is zero if it initial and terminal.
  Definition isZero (c : C) : UU := (isInitial C c) × (isTerminal C c).

Construction of isZero for an object c from the conditions that the space of all morphisms from c to any object d is contractible and and the space of all morphisms from any object d to c is contractible.
  Definition make_isZero (c : C) (H : (∏ (d : C), iscontr (c --> d))
                                      × (∏ (d : C), iscontr (d --> c))) :
    isZero c := make_isInitial c (dirprod_pr1 H),,make_isTerminal c (dirprod_pr2 H).

Definition of Zero.
  Definition Zero : UU := ∑ c : C, isZero c.
  Definition make_Zero (c : C) (H : isZero c) : Zero := tpair _ c H.
  Definition ZeroObject (Z : Zero) : C := pr1 Z.

Construction of Initial and Terminal from Zero.
  Definition Zero_to_Initial (Z : Zero) : Initial C :=
    make_Initial (pr1 Z) (dirprod_pr1 (pr2 Z)).
  Definition Zero_to_Terminal (Z : Zero) : Terminal C :=
    make_Terminal (pr1 Z) (dirprod_pr2 (pr2 Z)).

The following lemmas show that the underlying objects of Initial and Terminal, constructed above, are equal to ZeroObject.
  Lemma ZeroObject_equals_InitialObject (Z : Zero) :
    ZeroObject Z = InitialObject (Zero_to_Initial Z).
  Proof.
    apply idpath.
  Defined.

  Lemma ZeroObject_equals_TerminalObject (Z : Zero) :
    ZeroObject Z = TerminalObject (Zero_to_Terminal Z).
  Proof.
    apply idpath.
  Defined.

We construct morphisms from ZeroObject to any other object c and from any other object c to the ZeroObject.
  Definition ZeroArrowFrom (Z : Zero) (c : C) : C⟦ZeroObject Z, c⟧ :=
    InitialArrow (Zero_to_Initial Z) c.
  Definition ZeroArrowTo (Z : Zero) (c : C) : C⟦c, ZeroObject Z⟧ :=
    TerminalArrow (Zero_to_Terminal Z) c.

In particular, we get a zero morphism between any objects.
  Definition ZeroArrow (Z : Zero) (c d : C) : C⟦c, d⟧ :=
    @compose C _ (ZeroObject Z) _ (ZeroArrowTo Z c) (ZeroArrowFrom Z d).

We show that the above morphisms from ZeroObject and to ZeroObject are unique by using uniqueness of the morphism from InitialObject and uniqueness of the morphism to TerminalObject.
  Lemma ZeroArrowFromUnique (Z : Zero) (c : C) (f : C⟦ZeroObject Z, c⟧) :
    f = (ZeroArrowFrom Z c).
  Proof.
    apply (InitialArrowUnique (Zero_to_Initial Z) c f).
  Defined.

  Lemma ZeroArrowToUnique (Z : Zero) (c : C) (f : C⟦c, ZeroObject Z⟧) :
    f = (ZeroArrowTo Z c).
  Proof.
    apply (TerminalArrowUnique (Zero_to_Terminal Z) c f).
  Defined.

Therefore, any two morphisms from the ZeroObject to an object c are equal and any two morphisms from an object c to the ZeroObject are equal.
  Corollary ArrowsFromZero (Z : Zero) (c : C) (f g : C⟦ZeroObject Z, c⟧) :
    f = g.
  Proof.
    eapply pathscomp0.
    apply (ZeroArrowFromUnique Z c f).
    apply pathsinv0.
    apply (ZeroArrowFromUnique Z c g).
  Defined.

  Corollary ArrowsToZero (Z : Zero) (c : C) (f g : C⟦c, ZeroObject Z⟧) :
    f = g.
  Proof.
    eapply pathscomp0.
    apply (ZeroArrowToUnique Z c f).
    apply pathsinv0.
    apply (ZeroArrowToUnique Z c g).
  Defined.

It follows that any morphism which factors through 0 is the ZeroArrow.
  Corollary ZeroArrowUnique (Z : Zero) (c d : C) (f : C⟦c, ZeroObject Z⟧)
            (g : C⟦ZeroObject Z, d⟧) : f · g = ZeroArrow Z c d.
  Proof.
    rewrite (ZeroArrowToUnique Z c f).
    rewrite (ZeroArrowFromUnique Z d g).
    apply idpath.
  Defined.

Compose any morphism with the ZeroArrow and you get the ZeroArrow.
  Lemma precomp_with_ZeroArrow (Z : Zero) (a b c : C) (f : C⟦a, b⟧) :
    f · ZeroArrow Z b c = ZeroArrow Z a c.
  Proof.
    unfold ZeroArrow at 1. rewrite assoc.
    apply ZeroArrowUnique.
  Defined.

  Lemma postcomp_with_ZeroArrow (Z : Zero) (a b c : C) (f : C⟦b, c⟧) :
    ZeroArrow Z a b · f = ZeroArrow Z a c.
  Proof.
    unfold ZeroArrow at 1. rewrite <- assoc.
    apply ZeroArrowUnique.
  Defined.

An endomorphism of the ZeroObject is the identity morphism.
  Corollary ZeroEndo_is_identity (Z : Zero)
            (f : C⟦ZeroObject Z, ZeroObject Z⟧) :
    f = identity (ZeroObject Z).
  Proof.
    apply ArrowsFromZero.
  Defined.

The morphism from ZeroObject to ZeroObject is an isomorphisms.
  Lemma isiso_from_Zero_to_Zero (Z Z' : Zero) :
    is_iso (ZeroArrowFrom Z (ZeroObject Z')).
  Proof.
    apply (is_iso_qinv _ (ZeroArrowFrom Z' (ZeroObject Z))).
    split; apply ArrowsFromZero.
  Defined.

Using the above lemma we can construct an isomorphisms between any two ZeroObjects.
  Definition iso_Zeros (Z Z' : Zero) : iso (ZeroObject Z) (ZeroObject Z') :=
    tpair _ (ZeroArrowFrom Z (ZeroObject Z')) (isiso_from_Zero_to_Zero Z Z').

  Definition hasZero := ishinh Zero.

Construct Zero from Initial and Terminal for which the underlying objects are isomorphic.
  Definition Initial_and_Terminal_to_Zero
             (I : Initial C) (T : Terminal C)
             (e: iso (InitialObject I) (TerminalObject T)) : Zero.
  Proof.
    use (make_Zero (InitialObject I)).
    split.
    - use (make_isInitial (InitialObject I)); intro b.
      apply make_iscontr with (x := (InitialArrow I b)), InitialArrowUnique.
    - use (make_isTerminal (InitialObject I)); intro a.
      apply (iscontrretract (postcomp_with (inv_from_iso e))
                            (postcomp_with (morphism_from_iso e))).
      intros y. unfold postcomp_with.
      rewrite <- assoc. rewrite (iso_inv_after_iso e).
      apply (remove_id_right _ _ _ y y _ (idpath _) (idpath _)).
      apply (make_iscontr (TerminalArrow T a)), TerminalArrowUnique.
  Defined.

The following lemma verifies that the ZeroObject of the Zero, constructed from Initial and Terminal with InitialObject isomorphic to TerminalObject, is isomorphic to the InitialObject and isomorphic to the TerminalObject.
  Lemma Initial_and_Terminal_ob_equals_Zero_ob (I : Initial C)
        (T :Terminal C) (e : iso (InitialObject I) (TerminalObject T)) :
    (iso (InitialObject I) (ZeroObject (Initial_and_Terminal_to_Zero I T e)))
      × (iso (TerminalObject T)
             (ZeroObject (Initial_and_Terminal_to_Zero I T e))).
  Proof.
    exact(identity_iso (InitialObject I),,iso_inv_from_iso e).
  Defined.

End def_zero.

Zero coincides with the direct definition

Section zero_coincides.

  Context {C : precategory}.

isZero


  Lemma equiv_isZero1 (c : C) :
    limits.zero.isZero c → isZero c.
  Proof.
    intros X.
    use make_isZero.
    split.
    - intros d. apply ((pr1 X) d).
    - intros d. apply ((pr2 X) d).
  Qed.

  Lemma equiv_isZero2 (c : C) :
    limits.zero.isZero c <- isZero c.
  Proof.
    intros X.
    set (XZ := make_Zero c X).

    split.
    - intros b.
      use tpair.
      apply (InitialArrow (Zero_to_Initial XZ) b).
      intros t.
      use (InitialArrowUnique (Zero_to_Initial XZ) b).
    - intros a.
      use tpair.
      apply (TerminalArrow (Zero_to_Terminal XZ) a).
      intros t.
      use (TerminalArrowUnique (Zero_to_Terminal XZ) a).
  Qed.

Zero


  Definition equiv_Zero1 :
    limits.zero.Zero C → @Zero C.
  Proof.
    intros Z.
    exact (make_Zero Z (equiv_isZero1 _ (pr2 Z))).
  Defined.

  Definition equiv_Zero2 :
    limits.zero.Zero C <- @Zero C.
  Proof.
    intros Z.
    exact (limits.zero.make_Zero
             (ZeroObject Z)
             (equiv_isZero2
                _ ((isInitial_Initial (Zero_to_Initial Z))
                     ,,(isTerminal_Terminal (Zero_to_Terminal Z))))).
  Defined.

Arrows


  Lemma equiv_ZeroArrowTo (x : C) (Z : Zero) :
    @limits.zero.ZeroArrowTo C (equiv_Zero2 Z) x = ZeroArrowTo Z x.
  Proof.
    apply ZeroArrowToUnique.
  Qed.

  Lemma equiv_ZeroArrowFrom (x : C) (Z : Zero) :
    @limits.zero.ZeroArrowFrom C (equiv_Zero2 Z) x = ZeroArrowFrom Z x.
  Proof.
    apply ZeroArrowFromUnique.
  Qed.

  Lemma equiv_ZeroArrow (x y : C) (Z : Zero) :
    @limits.zero.ZeroArrow C (equiv_Zero2 Z) x y = ZeroArrow Z x y.
  Proof.
    unfold limits.zero.ZeroArrow. unfold ZeroArrow.
    rewrite equiv_ZeroArrowTo.
    rewrite equiv_ZeroArrowFrom.
    apply idpath.
  Qed.

End zero_coincides.

Following Initial and Terminal, we clear implicit arguments.
Arguments Zero : clear implicits.
Arguments isZero : clear implicits.