Library UniMath.CategoryTheory.ProductCategory


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Anders Mörtberg
2016
For a specialization to binary products, see precategory_binproduct.
Contents:

Require Import UniMath.Foundations.PartD.
Require Import UniMath.MoreFoundations.Tactics.

Require Import UniMath.CategoryTheory.Core.Categories.
Require Import UniMath.CategoryTheory.Core.Functors.
Require Import UniMath.CategoryTheory.opp_precat.
Local Open Scope cat.

Section dep_product_precategory.

  Context {I : UU} (C : I → precategory).

  Definition product_precategory_ob_mor : precategory_ob_mor.
  Proof.
  use tpair.
  - apply (∏ (i : I), ob (C i)).
  - intros f g.
    apply (∏ i, f i --> g i).
  Defined.

  Definition product_precategory_data : precategory_data.
  Proof.
    ∃ product_precategory_ob_mor.
    split.
    - intros f i; simpl in ×.
      apply (identity (f i)).
    - intros a b c f g i; simpl in ×.
      exact (f i · g i).
  Defined.

  Lemma is_precategory_product_precategory_data :
    is_precategory product_precategory_data.
  Proof.
    repeat split; intros; apply funextsec; intro i.
    - apply id_left.
    - apply id_right.
    - apply assoc.
    - apply assoc'.
  Defined.
needed for the op-related goal below

  Definition product_precategory : precategory
    := tpair _ _ is_precategory_product_precategory_data.

  Definition has_homsets_product_precategory (hsC : ∏ (i:I), has_homsets (C i)) :
    has_homsets product_precategory.
  Proof.
  intros ? ?; simpl.
  apply impred_isaset; intro; apply hsC.
  Qed.

End dep_product_precategory.

Goal ∏ (I:UU) (C:I→precategory), product_precategory (λ i, (C i)^op) = (product_precategory C)^op.
  reflexivity.
Qed.

The product of categories is again a category.
Definition product_category {I : UU} (C : I → category) : category.
  use make_category.
  - exact (product_precategory C).
  - apply has_homsets_product_precategory.
    intro; exact (homset_property (C _)).
Defined.

Section power_precategory.
  Context (I : UU) (C : precategory).

  Definition power_precategory : precategory
    := @product_precategory I (λ _, C).

  Definition has_homsets_power_precategory (hsC : has_homsets C) :
    has_homsets power_precategory.
  Proof.
  apply has_homsets_product_precategory.
  intro i; assumption.
  Qed.

End power_precategory.

Functors


Section functors.

Families of functors (family_functor)


Definition family_functor_data (I : UU) {A B : I → precategory}
  (F : ∏ (i : I), (A i) ⟶ (B i)) :
  functor_data (product_precategory A)
               (product_precategory B).
Proof.
use tpair.
- intros a i; apply (F i (a i)).
- intros a b f i; apply (# (F i) (f i)).
Defined.

Definition family_functor (I : UU) {A B : I → precategory}
  (F : ∏ (i : I), (A i) ⟶ (B i)) :
  (product_precategory A) ⟶ (product_precategory B).
Proof.
apply (tpair _ (family_functor_data I F)).
abstract
  (split; [ intro x; apply funextsec; intro i; simpl; apply functor_id
          | intros x y z f g; apply funextsec; intro i; apply functor_comp]).
Defined.

Projections (pr_functor)


Definition pr_functor_data (I : UU) (C : I → precategory) (i : I) :
  functor_data (product_precategory C) (C i).
Proof.
use tpair.
- intro a; apply (a i).
- intros x y f; simpl; apply (f i).
Defined.

Definition pr_functor (I : UU) (C : I → precategory) (i : I) :
  (product_precategory C) ⟶ (C i).
Proof.
apply (tpair _ (pr_functor_data I C i)).
abstract (split; intros x *; apply idpath).
Defined.

Delta functor (delta_functor)


Definition delta_functor_data (I : UU) (C : precategory) :
  functor_data C (power_precategory I C).
Proof.
use tpair.
- intros x i; apply x.
- intros x y f i; simpl; apply f.
Defined.

Definition delta_functor (I : UU) (C : precategory) :
  C ⟶ (power_precategory I C).
Proof.
apply (tpair _ (delta_functor_data I C)).
abstract (split; intros x *; apply idpath).
Defined.

Tuple functor (tuple_functor)


Definition tuple_functor_data {I : UU} {A : precategory} {B : I → precategory}
  (F : ∏ i, A ⟶ (B i)) : functor_data A (product_precategory B).
Proof.
use tpair.
- intros a i; exact (F i a).
- intros a b f i; exact (# (F i) f).
Defined.

Lemma tuple_functor_axioms {I : UU} {A : precategory} {B : I → precategory}
  (F : ∏ i, A ⟶ (B i)) : is_functor (tuple_functor_data F).
Proof.
split.
- intros a. apply funextsec; intro i. apply functor_id.
- intros ? ? ? ? ?. apply funextsec; intro i. apply functor_comp.
Qed.

Definition tuple_functor {I : UU} {A : precategory} {B : I → precategory}
  (F : ∏ i, A ⟶ (B i)) : A ⟶ (product_precategory B)
    := (tuple_functor_data F,, tuple_functor_axioms F).

Lemma pr_tuple_functor {I : UU} {A : precategory} {B : I → precategory} (hsB : ∏ i, has_homsets (B i))
  (F : ∏ i, A ⟶ (B i)) (i : I) : tuple_functor F ∙ pr_functor I B i = F i.
Proof.
now apply functor_eq.
Qed.

End functors.

Equivalence between functors into components and functors into product

This is a phrasing of the universal property of the product, compare to weqfuntoprodtoprod.
Lemma functor_into_product_weq {I : UU} {A : category} {B : I → category} :
  A ⟶ (product_category B) ≃ (∏ i : I, A ⟶ (B i)).
Proof.
  use weq_iso.
  - intros ? i.
Compose A ⟶ product_precategory I B ⟶ B i
    apply (functor_composite (C' := product_precategory B)).
    + assumption.
    + exact (pr_functor _ _ i).
  - exact tuple_functor.
  - intro y.
    apply functor_eq; [apply homset_property|].
    reflexivity.
  - intro f; cbn.
    apply funextsec; intro i.
    apply functor_eq; [exact (homset_property (B i))|].
    reflexivity.
Defined.