Library UniMath.CategoryTheory.Subcategory.Limits

(Co)limits in full subprecategories

C has (co)limits of shape F,

C' : ob C → UU is a proposition on the objects of C, and

C' is closed under the formation of (co)limits of shape F,

then the full subcategory on C'-objects has (co)limits of shape F.

Such proofs are mostly just a lot of insertions of

precategory_object_from_sub_precategory_object

precategory_morphism_from_sub_precategory_morphism

and their "inverses"

precategory_morphisms_in_subcat

precategory_object_in_subcat.

Author: Langston Barrett (@siddharthist) (March 2018)

Limits
- Binary products
Colimits

The subcategory inclusion reflects limits

Corollary reflects_all_limits_sub_precategory_inclusion
          {C : precategory} (C' : hsubtype (ob C)) :
  reflects_all_limits (sub_precategory_inclusion C (full_sub_precategory C')).
Proof.
  apply fully_faithful_reflects_all_limits.
  apply fully_faithful_sub_precategory_inclusion.
Defined.

Section Limits.

Limits

Terminal objects

As long as the predicate holds for the terminal object, it's terminal in the full subcategory.

Lemma terminal_in_full_subcategory {C : precategory} (C' : hsubtype (ob C))
        (TC : Terminal C) (TC' : C' (TerminalObject TC)) :
  Terminal (full_sub_precategory C').
Proof.
  use tpair.
  - use precategory_object_in_subcat.
    + exact (TerminalObject TC).
    + assumption.
  - cbn.
    intros X.
    use make_iscontr.
    + use morphism_in_full_subcat.
      apply TerminalArrow.
    + intro; apply eq_in_sub_precategory; cbn.
      apply TerminalArrowUnique.
Defined.

Binary products

Lemma bin_products_in_full_subcategory {C : category} (C' : hsubtype (ob C))
      (BPC : BinProducts C)
      (all : ∏ c1 c2 : ob C , C' c1 → C' c2 → C' (BinProductObject _ (BPC c1 c2))) :
  BinProducts (full_sub_precategory C').
Proof.
  intros c1' c2'.
  pose (c1'_in_C := (precategory_object_from_sub_precategory_object _ _ c1')).
  pose (c2'_in_C := (precategory_object_from_sub_precategory_object _ _ c2')).
  use tpair; [use tpair|]; [|use make_dirprod|].
  - use precategory_object_in_subcat.
    + apply (BinProductObject _ (BPC c1'_in_C c2'_in_C)).
    + apply all.
      × exact (pr2 c1').
      × exact (pr2 c2').
  - use morphism_in_full_subcat; apply BinProductPr1.
  - use morphism_in_full_subcat; apply BinProductPr2.
  - cbn.
    unfold isBinProduct.
    intros bp' f g.
    use tpair.
    + use tpair.
      × use precategory_morphisms_in_subcat;
          [apply BinProductArrow|exact tt];
          apply (precategory_morphism_from_sub_precategory_morphism _ (full_sub_precategory C'));
          assumption.
      × cbn.
        use make_dirprod; apply eq_in_sub_precategory.
        { apply BinProductPr1Commutes. }
        { apply BinProductPr2Commutes. }
    + intros otherarrow.

This is where we use the condition that C has homsets.

      apply subtypePath.
      { intro. apply isapropdirprod;
          apply is_set_sub_precategory_morphisms, homset_property.
      }
      {
        apply eq_in_sub_precategory.
        cbn.
        apply BinProductArrowUnique.
        - exact (maponpaths pr1 (dirprod_pr1 (pr2 otherarrow))).
        - exact (maponpaths pr1 (dirprod_pr2 (pr2 otherarrow))).
      }
Defined.

General limits

Lift a diagram from a full subcategory into the parent category

Definition lift_diagram_full_subcategory {C : precategory} {C' : hsubtype (ob C)} {g : graph}
      (d : diagram g (full_sub_precategory C')) :
  diagram g C.
Proof.
  use tpair.
  - intros v.
    apply (precategory_object_from_sub_precategory_object _ (full_sub_precategory C')).
    exact (dob d v).
  - intros u v e.
    apply (precategory_morphism_from_sub_precategory_morphism _ (full_sub_precategory C')).
    exact (dmor d e).
Defined.

Equivalence between cones in the parent category and those in the subcategory

Definition cone_in_full_subcategory {C : precategory} {g : graph} (C' : hsubtype (ob C))
      {d : diagram g (full_sub_precategory C')}
      (c : ob C)
      (tip : C' c) :
  cone (lift_diagram_full_subcategory d) c ≃ cone d (c ,, tip).
Proof.
  unfold cone.
  use weqbandf.
  - apply weqonsecfibers; intros x.
    cbn beta.

The following line works because of the computational behavior of lift_diagram_full_subcategory, namely: (∏ v : vertex g, C' (dob (lift_diagram_full_subcategory d) v)) → C' c → ∏ v : vertex g, pr1 (dob d v) = dob (lift_diagram_full_subcategory d) v

    apply (@weq_hom_in_subcat_from_hom_in_precat C C' (c,, tip) (dob d x)).
  - intro legs.
    cbn beta.
    apply weqonsecfibers; intros u;
      apply weqonsecfibers; intros v;
      apply weqonsecfibers; intros e.
    cbn.
    apply invweq.
    refine (subtypeInjectivity _ _ _ _).
    intro; apply propproperty.
Defined.

A full subcategory has a limit of a given shape if the proposition holds for the tip of the lifted limit diagram in the parent category.

Lemma lim_cone_in_full_subcategory {C : precategory} (C' : hsubtype (ob C))
      {g : graph} {d : diagram g (full_sub_precategory C')}
      (LC : Lims_of_shape g C) :
  C' (lim (LC (lift_diagram_full_subcategory d))) → LimCone d.
Proof.
  intro.
  pose (Z := LC (lift_diagram_full_subcategory d)).
  unfold LimCone.
  use tpair.
  - use tpair.
    + use precategory_object_in_subcat.
      × eapply lim; eassumption.
      × assumption.
    + apply cone_in_full_subcategory, limCone.
  - apply (reflects_all_limits_sub_precategory_inclusion C' g d).
    apply (pr2 Z).
Qed.

Colimits

Initial objects

As long as the predicate holds for the initial object, it's initial in the full subcategory.

Lemma initial_in_full_subcategory {C : precategory} (C' : hsubtype (ob C))
      (IC : Initial C) (IC' : C' (InitialObject IC)) :
  Initial (full_sub_precategory C').
Proof.
  use tpair.
  - use precategory_object_in_subcat.
    + exact (InitialObject IC).
    + assumption.
  - cbn.
    intros X.
    use make_iscontr.
    + use morphism_in_full_subcat.
      apply InitialArrow.
    + intro; apply eq_in_sub_precategory; cbn.
      apply InitialArrowUnique.
Defined.

Binary coproducts

Lemma bin_coproducts_in_full_subcategory {C : category} (C' : hsubtype (ob C))
      (BPC : BinCoproducts C)
      (all : ∏ c1 c2 : ob C , C' c1 → C' c2 → C' (BinCoproductObject _ (BPC c1 c2))) :
  BinCoproducts (full_sub_precategory C').
Proof.
  intros c1' c2'.
  pose (c1'_in_C := (precategory_object_from_sub_precategory_object _ _ c1')).
  pose (c2'_in_C := (precategory_object_from_sub_precategory_object _ _ c2')).
  use tpair; [use tpair|]; [|use make_dirprod|].
  - use precategory_object_in_subcat.
    + apply (BinCoproductObject _ (BPC c1'_in_C c2'_in_C)).
    + apply all.
      × exact (pr2 c1').
      × exact (pr2 c2').
  - use morphism_in_full_subcat; apply BinCoproductIn1.
  - use morphism_in_full_subcat; apply BinCoproductIn2.
  - cbn.
    unfold isBinCoproduct.
    intros bp' f g.
    use tpair.
    + use tpair.
      × use precategory_morphisms_in_subcat;
          [apply BinCoproductArrow|exact tt];
          apply (precategory_morphism_from_sub_precategory_morphism _ (full_sub_precategory C'));
          assumption.
      × cbn.
        use make_dirprod; apply eq_in_sub_precategory.
        { apply BinCoproductIn1Commutes. }
        { apply BinCoproductIn2Commutes. }
    + intros otherarrow.