Finitely presented types

Content created by Egbert Rijke, Fredrik Bakke and Jonathan Prieto-Cubides.

Created on 2022-02-17.
Last modified on 2023-09-06.

module univalent-combinatorics.finitely-presented-types where
open import elementary-number-theory.natural-numbers

open import foundation.dependent-pair-types
open import foundation.equivalences
open import foundation.fibers-of-maps
open import foundation.function-types
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.set-presented-types
open import foundation.set-truncations
open import foundation.subtypes
open import foundation.universe-levels

open import univalent-combinatorics.finite-choice
open import univalent-combinatorics.finite-connected-components
open import univalent-combinatorics.finite-types
open import univalent-combinatorics.standard-finite-types


A type is said to be finitely presented if it is presented by a standard finite type.


To have a presentation of cardinality k

has-presentation-of-cardinality-Prop :
  {l1 : Level} (k : ) (A : UU l1)  Prop l1
has-presentation-of-cardinality-Prop k A =
  has-set-presentation-Prop (Fin-Set k) A

has-presentation-of-cardinality : {l1 : Level} (k : ) (A : UU l1)  UU l1
has-presentation-of-cardinality k A =
  type-Prop (has-presentation-of-cardinality-Prop k A)

Finitely presented types

is-finitely-presented : {l1 : Level}  UU l1  UU l1
is-finitely-presented A =
  Σ   k  has-presentation-of-cardinality k A)


A type has finitely many connected components if and only if it has a finite presentation

has-presentation-of-cardinality-has-cardinality-components :
  {l : Level} (k : ) {A : UU l}  has-cardinality-components k A 
  has-presentation-of-cardinality k A
has-presentation-of-cardinality-has-cardinality-components {l} k {A} H =
  apply-universal-property-trunc-Prop H
    ( has-presentation-of-cardinality-Prop k A)
    ( λ e 
        ( P2 e)
        ( has-presentation-of-cardinality-Prop k A)
        ( λ g 
            ( pair
              ( λ x  pr1 (g x))
              ( is-equiv-htpy-equiv e  x  pr2 (g x))))))
  P1 :
    (e : Fin k  type-trunc-Set A) (x : Fin k) 
    type-trunc-Prop (fiber unit-trunc-Set (map-equiv e x))
  P1 e x = is-surjective-unit-trunc-Set A (map-equiv e x)
  P2 :
    (e : Fin k  type-trunc-Set A) 
    type-trunc-Prop ((x : Fin k)  fiber unit-trunc-Set (map-equiv e x))
  P2 e = finite-choice-Fin k (P1 e)

has-cardinality-components-has-presentation-of-cardinality :
  {l : Level} (k : ) {A : UU l}  has-presentation-of-cardinality k A 
  has-cardinality-components k A
has-cardinality-components-has-presentation-of-cardinality {l} k {A} H =
  apply-universal-property-trunc-Prop H
    ( has-cardinality-components-Prop k A)
    ( λ { (pair f E)  unit-trunc-Prop (pair (unit-trunc-Set  f) E)})

To be finitely presented is a property

all-elements-equal-is-finitely-presented :
  {l1 : Level} {A : UU l1}  all-elements-equal (is-finitely-presented A)
all-elements-equal-is-finitely-presented {l1} {A} (pair k K) (pair l L) =
    ( λ n  has-set-presentation-Prop (Fin-Set n) A)
    ( eq-cardinality
      ( has-cardinality-components-has-presentation-of-cardinality k K)
      ( has-cardinality-components-has-presentation-of-cardinality l L))

is-prop-is-finitely-presented :
  {l1 : Level} {A : UU l1}  is-prop (is-finitely-presented A)
is-prop-is-finitely-presented =
  is-prop-all-elements-equal all-elements-equal-is-finitely-presented

is-finitely-presented-Prop : {l : Level} (A : UU l)  Prop l
pr1 (is-finitely-presented-Prop A) = is-finitely-presented A
pr2 (is-finitely-presented-Prop A) = is-prop-is-finitely-presented

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