Gaunt categories

Content created by Fredrik Bakke and Egbert Rijke.

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

module category-theory.gaunt-categories where
Imports
open import category-theory.categories
open import category-theory.composition-operations-on-binary-families-of-sets
open import category-theory.isomorphism-induction-categories
open import category-theory.isomorphisms-in-categories
open import category-theory.isomorphisms-in-precategories
open import category-theory.nonunital-precategories
open import category-theory.precategories
open import category-theory.preunivalent-categories
open import category-theory.rigid-objects-categories
open import category-theory.strict-categories

open import foundation.cartesian-product-types
open import foundation.dependent-pair-types
open import foundation.identity-types
open import foundation.propositions
open import foundation.sets
open import foundation.universe-levels

Idea

A gaunt category is a category such that one of the following equivalent conditions hold:

  1. The isomorphism-sets are propositions.
  2. The objects form a set.
  3. Every object is rigid, meaning its automorphism group is trivial.

Gaunt categories forms the common intersection of (univalent) categories and strict categories. We have the following diagram relating the different notions of "category":

        Gaunt categories
              /   \
            /       \
          v           v
  Categories         Strict categories
          \           /
            \       /
              v   v
      Preunivalent categories
                |
                |
                v
          Precategories

Definitions

The predicate on precategories that there is at most one isomorphism between any two objects

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

  is-prop-iso-prop-Precategory : Prop (l1  l2)
  is-prop-iso-prop-Precategory =
    Π-Prop
      ( obj-Precategory C)
      ( λ x 
        Π-Prop
          ( obj-Precategory C)
          ( λ y  is-prop-Prop (iso-Precategory C x y)))

  is-prop-iso-Precategory : UU (l1  l2)
  is-prop-iso-Precategory = type-Prop is-prop-iso-prop-Precategory

  is-property-is-prop-iso-Precategory : is-prop is-prop-iso-Precategory
  is-property-is-prop-iso-Precategory =
    is-prop-type-Prop is-prop-iso-prop-Precategory

The predicate on precategories of being gaunt

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

  is-gaunt-prop-Precategory : Prop (l1  l2)
  is-gaunt-prop-Precategory =
    product-Prop
      ( is-category-prop-Precategory C)
      ( is-prop-iso-prop-Precategory C)

  is-gaunt-Precategory : UU (l1  l2)
  is-gaunt-Precategory = type-Prop is-gaunt-prop-Precategory

The predicate on categories of being gaunt

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

  is-gaunt-prop-Category : Prop (l1  l2)
  is-gaunt-prop-Category = is-prop-iso-prop-Precategory (precategory-Category C)

  is-gaunt-Category : UU (l1  l2)
  is-gaunt-Category = type-Prop is-gaunt-prop-Category

The type of gaunt categories

Gaunt-Category : (l1 l2 : Level)  UU (lsuc l1  lsuc l2)
Gaunt-Category l1 l2 = Σ (Category l1 l2) (is-gaunt-Category)

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

  category-Gaunt-Category : Category l1 l2
  category-Gaunt-Category = pr1 C

  obj-Gaunt-Category : UU l1
  obj-Gaunt-Category = obj-Category category-Gaunt-Category

  hom-set-Gaunt-Category :
    obj-Gaunt-Category  obj-Gaunt-Category  Set l2
  hom-set-Gaunt-Category =
    hom-set-Category category-Gaunt-Category

  hom-Gaunt-Category :
    obj-Gaunt-Category  obj-Gaunt-Category  UU l2
  hom-Gaunt-Category = hom-Category category-Gaunt-Category

  is-set-hom-Gaunt-Category :
    (x y : obj-Gaunt-Category)  is-set (hom-Gaunt-Category x y)
  is-set-hom-Gaunt-Category =
    is-set-hom-Category category-Gaunt-Category

  comp-hom-Gaunt-Category :
    {x y z : obj-Gaunt-Category} 
    hom-Gaunt-Category y z 
    hom-Gaunt-Category x y 
    hom-Gaunt-Category x z
  comp-hom-Gaunt-Category =
    comp-hom-Category category-Gaunt-Category

  associative-comp-hom-Gaunt-Category :
    {x y z w : obj-Gaunt-Category}
    (h : hom-Gaunt-Category z w)
    (g : hom-Gaunt-Category y z)
    (f : hom-Gaunt-Category x y) 
    comp-hom-Gaunt-Category (comp-hom-Gaunt-Category h g) f 
    comp-hom-Gaunt-Category h (comp-hom-Gaunt-Category g f)
  associative-comp-hom-Gaunt-Category =
    associative-comp-hom-Category category-Gaunt-Category

  inv-associative-comp-hom-Gaunt-Category :
    {x y z w : obj-Gaunt-Category}
    (h : hom-Gaunt-Category z w)
    (g : hom-Gaunt-Category y z)
    (f : hom-Gaunt-Category x y) 
    comp-hom-Gaunt-Category h (comp-hom-Gaunt-Category g f) 
    comp-hom-Gaunt-Category (comp-hom-Gaunt-Category h g) f
  inv-associative-comp-hom-Gaunt-Category =
    inv-associative-comp-hom-Category category-Gaunt-Category

  associative-composition-operation-Gaunt-Category :
    associative-composition-operation-binary-family-Set
      hom-set-Gaunt-Category
  associative-composition-operation-Gaunt-Category =
    associative-composition-operation-Category
      ( category-Gaunt-Category)

  id-hom-Gaunt-Category :
    {x : obj-Gaunt-Category}  hom-Gaunt-Category x x
  id-hom-Gaunt-Category =
    id-hom-Category category-Gaunt-Category

  left-unit-law-comp-hom-Gaunt-Category :
    {x y : obj-Gaunt-Category} (f : hom-Gaunt-Category x y) 
    comp-hom-Gaunt-Category id-hom-Gaunt-Category f  f
  left-unit-law-comp-hom-Gaunt-Category =
    left-unit-law-comp-hom-Category category-Gaunt-Category

  right-unit-law-comp-hom-Gaunt-Category :
    {x y : obj-Gaunt-Category} (f : hom-Gaunt-Category x y) 
    comp-hom-Gaunt-Category f id-hom-Gaunt-Category  f
  right-unit-law-comp-hom-Gaunt-Category =
    right-unit-law-comp-hom-Category category-Gaunt-Category

  is-unital-composition-operation-Gaunt-Category :
    is-unital-composition-operation-binary-family-Set
      hom-set-Gaunt-Category
      comp-hom-Gaunt-Category
  is-unital-composition-operation-Gaunt-Category =
    is-unital-composition-operation-Category
      ( category-Gaunt-Category)

  is-gaunt-Gaunt-Category :
    is-gaunt-Category category-Gaunt-Category
  is-gaunt-Gaunt-Category = pr2 C

The underlying nonunital precategory of a gaunt category

nonunital-precategory-Gaunt-Category :
  {l1 l2 : Level}  Gaunt-Category l1 l2  Nonunital-Precategory l1 l2
nonunital-precategory-Gaunt-Category C =
  nonunital-precategory-Category (category-Gaunt-Category C)

The underlying precategory of a gaunt category

precategory-Gaunt-Category :
  {l1 l2 : Level}  Gaunt-Category l1 l2  Precategory l1 l2
precategory-Gaunt-Category C = precategory-Category (category-Gaunt-Category C)

The underlying preunivalent category of a gaunt category

preunivalent-category-Gaunt-Category :
  {l1 l2 : Level}  Gaunt-Category l1 l2  Preunivalent-Category l1 l2
preunivalent-category-Gaunt-Category C =
  preunivalent-category-Category (category-Gaunt-Category C)

The total hom-type of a gaunt category

total-hom-Gaunt-Category :
  {l1 l2 : Level} (C : Gaunt-Category l1 l2)  UU (l1  l2)
total-hom-Gaunt-Category C =
  total-hom-Category (category-Gaunt-Category C)

obj-total-hom-Gaunt-Category :
  {l1 l2 : Level} (C : Gaunt-Category l1 l2) 
  total-hom-Gaunt-Category C 
  obj-Gaunt-Category C × obj-Gaunt-Category C
obj-total-hom-Gaunt-Category C =
  obj-total-hom-Category (category-Gaunt-Category C)

Equalities induce morphisms

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

  hom-eq-Gaunt-Category :
    (x y : obj-Gaunt-Category C)  x  y  hom-Gaunt-Category C x y
  hom-eq-Gaunt-Category =
    hom-eq-Category (category-Gaunt-Category C)

  hom-inv-eq-Gaunt-Category :
    (x y : obj-Gaunt-Category C)  x  y  hom-Gaunt-Category C y x
  hom-inv-eq-Gaunt-Category =
    hom-inv-eq-Category (category-Gaunt-Category C)

Properties

Preunivalent categories whose isomorphism-sets are propositions are strict categories

is-strict-category-is-prop-iso-Preunivalent-Category :
  {l1 l2 : Level} (C : Preunivalent-Category l1 l2) 
  is-prop-iso-Precategory (precategory-Preunivalent-Category C) 
  is-strict-category-Preunivalent-Category C
is-strict-category-is-prop-iso-Preunivalent-Category C is-prop-iso-C x y =
  is-prop-emb (emb-iso-eq-Preunivalent-Category C) (is-prop-iso-C x y)

Gaunt categories are strict

is-strict-category-is-gaunt-Category :
  {l1 l2 : Level} (C : Category l1 l2) 
  is-gaunt-Category C  is-strict-category-Category C
is-strict-category-is-gaunt-Category C =
  is-strict-category-is-prop-iso-Preunivalent-Category
    ( preunivalent-category-Category C)

A strict category is gaunt if iso-eq is surjective

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

  is-category-is-surjective-iso-eq-Strict-Category :
    is-surjective-iso-eq-Precategory (precategory-Strict-Category C) 
    is-category-Precategory (precategory-Strict-Category C)
  is-category-is-surjective-iso-eq-Strict-Category =
    is-category-is-surjective-iso-eq-Preunivalent-Category
      ( preunivalent-category-Strict-Category C)

  is-prop-iso-is-category-Strict-Category :
    is-category-Precategory (precategory-Strict-Category C) 
    is-prop-iso-Precategory (precategory-Strict-Category C)
  is-prop-iso-is-category-Strict-Category is-category-C x y =
    is-prop-is-equiv' (is-category-C x y) (is-set-obj-Strict-Category C x y)

  is-prop-iso-is-surjective-iso-eq-Strict-Category :
    is-surjective-iso-eq-Precategory (precategory-Strict-Category C) 
    is-prop-iso-Precategory (precategory-Strict-Category C)
  is-prop-iso-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C =
    is-prop-iso-is-category-Strict-Category
      ( is-category-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C)

  is-gaunt-is-surjective-iso-eq-Strict-Category :
    is-surjective-iso-eq-Precategory (precategory-Strict-Category C) 
    is-gaunt-Precategory (precategory-Strict-Category C)
  pr1 (is-gaunt-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C) =
    is-category-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C
  pr2 (is-gaunt-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C) =
    is-prop-iso-is-surjective-iso-eq-Strict-Category is-surj-iso-eq-C

A category is gaunt if and only if every object is rigid

Proof: Using the fact that a type is a proposition if and only if having an inhabitant implies it is contractible, we can apply isomorphism induction to get our result.

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

  is-gaunt-is-rigid-Category :
    ((x : obj-Category C)  is-rigid-obj-Category C x)  is-gaunt-Category C
  is-gaunt-is-rigid-Category is-rigid-obj-C x y =
    is-prop-is-proof-irrelevant (ind-iso-Category C _ (is-rigid-obj-C x))

  is-rigid-is-gaunt-Category :
    is-gaunt-Category C  (x : obj-Category C)  is-rigid-obj-Category C x
  is-rigid-is-gaunt-Category is-gaunt-C x =
    is-proof-irrelevant-is-prop (is-gaunt-C x x) (id-iso-Category C)

See also

Recent changes