Wide subcategories

Content created by Fredrik Bakke and Egbert Rijke.

Created on 2023-11-01.
Last modified on 2024-02-06.

module category-theory.wide-subcategories where
Imports
open import category-theory.categories
open import category-theory.composition-operations-on-binary-families-of-sets
open import category-theory.faithful-functors-precategories
open import category-theory.functors-categories
open import category-theory.isomorphisms-in-categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.maps-categories
open import category-theory.precategories
open import category-theory.subcategories
open import category-theory.wide-subprecategories

open import foundation.contractible-types
open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.function-types
open import foundation.functoriality-dependent-pair-types
open import foundation.fundamental-theorem-of-identity-types
open import foundation.identity-types
open import foundation.iterated-dependent-product-types
open import foundation.propositions
open import foundation.sets
open import foundation.subtypes
open import foundation.unit-type
open import foundation.universe-levels

Idea

A wide subcategory of a category C is a subcategory that contains all the objects of C.

Lemma

Wide subprecategories of categories are categories

module _
  {l1 l2 l3 : Level}
  (C : Precategory l1 l2)
  (P : Wide-Subprecategory l3 C)
  (is-category-C : is-category-Precategory C)
  where

  is-category-is-category-Wide-Subprecategory :
    is-category-Precategory (precategory-Wide-Subprecategory C P)
  is-category-is-category-Wide-Subprecategory x =
    fundamental-theorem-id
      ( is-contr-equiv _
        ( equiv-tot
          ( λ y 
            inv-compute-iso-Subcategory
              ( C , is-category-C)
              ( subprecategory-Wide-Subprecategory C P)))
        ( is-torsorial-iso-Category (C , is-category-C) x))
      ( iso-eq-Precategory (precategory-Wide-Subprecategory C P) x)

Definitions

The predicate of being a wide subcategory

module _
  {l1 l2 l3 l4 : Level}
  (C : Category l1 l2)
  (P : Subcategory l3 l4 C)
  where

  is-wide-prop-Subcategory : Prop (l1  l3)
  is-wide-prop-Subcategory =
    is-wide-prop-Subprecategory (precategory-Category C) P

  is-wide-Subcategory : UU (l1  l3)
  is-wide-Subcategory = type-Prop is-wide-prop-Subcategory

  is-prop-is-wide-Subcategory : is-prop (is-wide-Subcategory)
  is-prop-is-wide-Subcategory = is-prop-type-Prop is-wide-prop-Subcategory

wide sub-hom-families of categories

module _
  {l1 l2 : Level} (l3 : Level)
  (C : Category l1 l2)
  where

  subtype-hom-wide-Category : UU (l1  l2  lsuc l3)
  subtype-hom-wide-Category =
    (x y : obj-Category C)  subtype l3 (hom-Category C x y)

Categorical predicates on wide sub-hom-families

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P₁ : subtype-hom-wide-Category l3 C)
  where

  contains-id-prop-subtype-hom-wide-Category : Prop (l1  l3)
  contains-id-prop-subtype-hom-wide-Category =
    Π-Prop (obj-Category C)  x  P₁ x x (id-hom-Category C))

  contains-id-subtype-hom-wide-Category : UU (l1  l3)
  contains-id-subtype-hom-wide-Category =
    type-Prop contains-id-prop-subtype-hom-wide-Category

  is-prop-contains-id-subtype-hom-wide-Category :
    is-prop contains-id-subtype-hom-wide-Category
  is-prop-contains-id-subtype-hom-wide-Category =
    is-prop-type-Prop contains-id-prop-subtype-hom-wide-Category

  is-closed-under-composition-subtype-hom-wide-Category : UU (l1  l2  l3)
  is-closed-under-composition-subtype-hom-wide-Category =
    (x y z : obj-Category C) 
    (g : hom-Category C y z) 
    (f : hom-Category C x y) 
    is-in-subtype (P₁ y z) g 
    is-in-subtype (P₁ x y) f 
    is-in-subtype (P₁ x z) (comp-hom-Category C g f)

  is-prop-is-closed-under-composition-subtype-hom-wide-Category :
    is-prop is-closed-under-composition-subtype-hom-wide-Category
  is-prop-is-closed-under-composition-subtype-hom-wide-Category =
    is-prop-iterated-Π 7
      ( λ x y z g f _ _ 
        is-prop-is-in-subtype (P₁ x z) (comp-hom-Category C g f))

  is-closed-under-composition-prop-subtype-hom-wide-Category :
    Prop (l1  l2  l3)
  pr1 is-closed-under-composition-prop-subtype-hom-wide-Category =
    is-closed-under-composition-subtype-hom-wide-Category
  pr2 is-closed-under-composition-prop-subtype-hom-wide-Category =
    is-prop-is-closed-under-composition-subtype-hom-wide-Category

The predicate on subtypes of hom-sets of being a wide subcategory

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P₁ : subtype-hom-wide-Category l3 C)
  where

  is-wide-subcategory-Prop : Prop (l1  l2  l3)
  is-wide-subcategory-Prop =
    product-Prop
      ( contains-id-prop-subtype-hom-wide-Category C P₁)
      ( is-closed-under-composition-prop-subtype-hom-wide-Category C P₁)

  is-wide-subcategory : UU (l1  l2  l3)
  is-wide-subcategory = type-Prop is-wide-subcategory-Prop

  is-prop-is-wide-subcategory : is-prop (is-wide-subcategory)
  is-prop-is-wide-subcategory = is-prop-type-Prop is-wide-subcategory-Prop

  contains-id-is-wide-subcategory :
    is-wide-subcategory  contains-id-subtype-hom-wide-Category C P₁
  contains-id-is-wide-subcategory = pr1

  is-closed-under-composition-is-wide-subcategory :
    is-wide-subcategory 
    is-closed-under-composition-subtype-hom-wide-Category C P₁
  is-closed-under-composition-is-wide-subcategory = pr2

Wide subcategories

Wide-Subcategory :
  {l1 l2 : Level} (l3 : Level)
  (C : Category l1 l2) 
  UU (l1  l2  lsuc l3)
Wide-Subcategory l3 C =
  Σ (subtype-hom-wide-Category l3 C) (is-wide-subcategory C)

Objects in wide subcategories

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  subtype-obj-Wide-Subcategory : subtype lzero (obj-Category C)
  subtype-obj-Wide-Subcategory _ = unit-Prop

  obj-Wide-Subcategory : UU l1
  obj-Wide-Subcategory = obj-Category C

  inclusion-obj-Wide-Subcategory :
    obj-Wide-Subcategory  obj-Category C
  inclusion-obj-Wide-Subcategory = id

Morphisms in wide subcategories

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  subtype-hom-Wide-Subcategory : subtype-hom-wide-Category l3 C
  subtype-hom-Wide-Subcategory = pr1 P

  hom-Wide-Subcategory : (x y : obj-Wide-Subcategory C P)  UU (l2  l3)
  hom-Wide-Subcategory x y =
    type-subtype (subtype-hom-Wide-Subcategory x y)

  inclusion-hom-Wide-Subcategory :
    (x y : obj-Wide-Subcategory C P) 
    hom-Wide-Subcategory x y 
    hom-Category C
      ( inclusion-obj-Wide-Subcategory C P x)
      ( inclusion-obj-Wide-Subcategory C P y)
  inclusion-hom-Wide-Subcategory x y =
    inclusion-subtype (subtype-hom-Wide-Subcategory x y)

The predicate on morphisms between any objects of being contained in the wide subcategory:

  is-in-hom-Wide-Subcategory :
    (x y : obj-Category C) (f : hom-Category C x y)  UU l3
  is-in-hom-Wide-Subcategory x y =
    is-in-subtype (subtype-hom-Wide-Subcategory x y)

  is-prop-is-in-hom-Wide-Subcategory :
    (x y : obj-Category C) (f : hom-Category C x y) 
    is-prop (is-in-hom-Wide-Subcategory x y f)
  is-prop-is-in-hom-Wide-Subcategory x y =
    is-prop-is-in-subtype (subtype-hom-Wide-Subcategory x y)

  is-in-hom-inclusion-hom-Wide-Subcategory :
    (x y : obj-Wide-Subcategory C P)
    (f : hom-Wide-Subcategory x y) 
    is-in-hom-Wide-Subcategory
      ( inclusion-obj-Wide-Subcategory C P x)
      ( inclusion-obj-Wide-Subcategory C P y)
      ( inclusion-hom-Wide-Subcategory x y f)
  is-in-hom-inclusion-hom-Wide-Subcategory x y =
    is-in-subtype-inclusion-subtype (subtype-hom-Wide-Subcategory x y)

Wide subcategories are wide subcategories:

  is-wide-subcategory-Wide-Subcategory :
    is-wide-subcategory C subtype-hom-Wide-Subcategory
  is-wide-subcategory-Wide-Subcategory = pr2 P

  contains-id-Wide-Subcategory :
    contains-id-subtype-hom-wide-Category C
      ( subtype-hom-Wide-Subcategory)
  contains-id-Wide-Subcategory =
    contains-id-is-wide-subcategory C
      ( subtype-hom-Wide-Subcategory)
      ( is-wide-subcategory-Wide-Subcategory)

  is-closed-under-composition-Wide-Subcategory :
    is-closed-under-composition-subtype-hom-wide-Category C
      ( subtype-hom-Wide-Subcategory)
  is-closed-under-composition-Wide-Subcategory =
    is-closed-under-composition-is-wide-subcategory C
      ( subtype-hom-Wide-Subcategory)
      ( is-wide-subcategory-Wide-Subcategory)

Wide subcategories are subcategories:

  subtype-hom-subcategory-Wide-Subcategory :
    subtype-hom-Category l3 C (subtype-obj-Wide-Subcategory C P)
  subtype-hom-subcategory-Wide-Subcategory =
    subtype-hom-subprecategory-Wide-Subprecategory (precategory-Category C) P

  is-subcategory-Wide-Subcategory :
    is-subcategory C
      ( subtype-obj-Wide-Subcategory C P)
      ( subtype-hom-subcategory-Wide-Subcategory)
  is-subcategory-Wide-Subcategory =
    is-subprecategory-Wide-Subprecategory (precategory-Category C) P

  subcategory-Wide-Subcategory : Subcategory lzero l3 C
  subcategory-Wide-Subcategory =
    subprecategory-Wide-Subprecategory (precategory-Category C) P

  is-wide-Wide-Subcategory :
    is-wide-Subcategory C (subcategory-Wide-Subcategory)
  is-wide-Wide-Subcategory =
    is-wide-Wide-Subprecategory (precategory-Category C) P

The underlying category of a wide subcategory

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  hom-set-Wide-Subcategory :
    (x y : obj-Wide-Subcategory C P)  Set (l2  l3)
  hom-set-Wide-Subcategory =
    hom-set-Wide-Subprecategory (precategory-Category C) P

  is-set-hom-Wide-Subcategory :
    (x y : obj-Wide-Subcategory C P) 
    is-set (hom-Wide-Subcategory C P x y)
  is-set-hom-Wide-Subcategory =
    is-set-hom-Wide-Subprecategory (precategory-Category C) P

  id-hom-Wide-Subcategory :
    {x : obj-Wide-Subcategory C P}  hom-Wide-Subcategory C P x x
  id-hom-Wide-Subcategory =
    id-hom-Wide-Subprecategory (precategory-Category C) P

  comp-hom-Wide-Subcategory :
    {x y z : obj-Wide-Subcategory C P} 
    hom-Wide-Subcategory C P y z 
    hom-Wide-Subcategory C P x y 
    hom-Wide-Subcategory C P x z
  comp-hom-Wide-Subcategory =
    comp-hom-Wide-Subprecategory (precategory-Category C) P

  associative-comp-hom-Wide-Subcategory :
    {x y z w : obj-Wide-Subcategory C P}
    (h : hom-Wide-Subcategory C P z w)
    (g : hom-Wide-Subcategory C P y z)
    (f : hom-Wide-Subcategory C P x y) 
    comp-hom-Wide-Subcategory (comp-hom-Wide-Subcategory h g) f 
    comp-hom-Wide-Subcategory h (comp-hom-Wide-Subcategory g f)
  associative-comp-hom-Wide-Subcategory =
    associative-comp-hom-Wide-Subprecategory (precategory-Category C) P

  inv-associative-comp-hom-Wide-Subcategory :
    {x y z w : obj-Wide-Subcategory C P}
    (h : hom-Wide-Subcategory C P z w)
    (g : hom-Wide-Subcategory C P y z)
    (f : hom-Wide-Subcategory C P x y) 
    comp-hom-Wide-Subcategory h (comp-hom-Wide-Subcategory g f) 
    comp-hom-Wide-Subcategory (comp-hom-Wide-Subcategory h g) f
  inv-associative-comp-hom-Wide-Subcategory =
    inv-associative-comp-hom-Wide-Subprecategory (precategory-Category C) P

  left-unit-law-comp-hom-Wide-Subcategory :
    {x y : obj-Wide-Subcategory C P}
    (f : hom-Wide-Subcategory C P x y) 
    comp-hom-Wide-Subcategory (id-hom-Wide-Subcategory) f  f
  left-unit-law-comp-hom-Wide-Subcategory =
    left-unit-law-comp-hom-Wide-Subprecategory (precategory-Category C) P

  right-unit-law-comp-hom-Wide-Subcategory :
    {x y : obj-Wide-Subcategory C P}
    (f : hom-Wide-Subcategory C P x y) 
    comp-hom-Wide-Subcategory f (id-hom-Wide-Subcategory)  f
  right-unit-law-comp-hom-Wide-Subcategory =
    right-unit-law-comp-hom-Wide-Subprecategory (precategory-Category C) P

  associative-composition-operation-Wide-Subcategory :
    associative-composition-operation-binary-family-Set
      ( hom-set-Wide-Subcategory)
  associative-composition-operation-Wide-Subcategory =
    associative-composition-operation-Wide-Subprecategory
      ( precategory-Category C) P

  is-unital-composition-operation-Wide-Subcategory :
    is-unital-composition-operation-binary-family-Set
      ( hom-set-Wide-Subcategory)
      ( comp-hom-Wide-Subcategory)
  is-unital-composition-operation-Wide-Subcategory =
    is-unital-composition-operation-Wide-Subprecategory
      ( precategory-Category C) P

  precategory-Wide-Subcategory : Precategory l1 (l2  l3)
  precategory-Wide-Subcategory =
    precategory-Wide-Subprecategory (precategory-Category C) P

The underlying category of a wide subcategory

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  category-Wide-Subcategory : Category l1 (l2  l3)
  pr1 category-Wide-Subcategory = precategory-Wide-Subcategory C P
  pr2 category-Wide-Subcategory =
    is-category-is-category-Wide-Subprecategory
      ( precategory-Category C) P (is-category-Category C)

The inclusion functor of a wide subcategory

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  inclusion-map-Wide-Subcategory :
    map-Category (category-Wide-Subcategory C P) C
  inclusion-map-Wide-Subcategory =
    inclusion-map-Wide-Subprecategory (precategory-Category C) P

  is-functor-inclusion-Wide-Subcategory :
    is-functor-map-Category
      ( category-Wide-Subcategory C P)
      ( C)
      ( inclusion-map-Wide-Subcategory)
  is-functor-inclusion-Wide-Subcategory =
    is-functor-inclusion-Wide-Subprecategory (precategory-Category C) P

  inclusion-Wide-Subcategory :
    functor-Category (category-Wide-Subcategory C P) C
  inclusion-Wide-Subcategory =
    inclusion-Wide-Subprecategory (precategory-Category C) P

Properties

The inclusion functor is faithful and an equivalence on objects

module _
  {l1 l2 l3 : Level}
  (C : Category l1 l2)
  (P : Wide-Subcategory l3 C)
  where

  is-faithful-inclusion-Wide-Subcategory :
    is-faithful-functor-Precategory
      ( precategory-Wide-Subcategory C P)
      ( precategory-Category C)
      ( inclusion-Wide-Subcategory C P)
  is-faithful-inclusion-Wide-Subcategory =
    is-faithful-inclusion-Wide-Subprecategory (precategory-Category C) P

  is-equiv-obj-inclusion-Wide-Subcategory :
    is-equiv
      ( obj-functor-Category
        ( category-Wide-Subcategory C P)
        ( C)
        ( inclusion-Wide-Subcategory C P))
  is-equiv-obj-inclusion-Wide-Subcategory =
    is-equiv-obj-inclusion-Wide-Subprecategory (precategory-Category C) P

The inclusion functor is pseudomonic

This remains to be formalized. This is a consequence of repeleteness.

Recent changes