Structure

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

Created on 2022-02-16.
Last modified on 2025-08-14.

module foundation.structure where
Imports 
open import foundation.dependent-pair-types
open import foundation.univalence
open import foundation.universe-levels

open import foundation-core.equivalences
open import foundation-core.fibers-of-maps
open import foundation-core.identity-types
open import foundation-core.transport-along-identifications

Idea

Given a type family 𝒫 on the universe, a 𝒫-structured type consists of a type A equipped with an element of type 𝒫 A.

Definitions

Structure at a universe

structure : {l1 l2 : Level} (𝒫 : UU l1 → UU l2) → UU (lsuc l1 ⊔ l2)
structure {l1} 𝒫 = Σ (UU l1) 𝒫

structure-family :
  {l1 l2 l3 : Level} (𝒫 : UU l1 → UU l2) {A : UU l3} →
  (A → UU l1) → UU (l2 ⊔ l3)
structure-family 𝒫 {A} B = (x : A) → 𝒫 (B x)

structured-family :
  {l1 l2 l3 : Level} (𝒫 : UU l1 → UU l2) → UU l3 → UU (lsuc l1 ⊔ l2 ⊔ l3)
structured-family 𝒫 A = A → structure 𝒫

structure-map :
  {l1 l2 l3 : Level} (𝒫 : UU (l1 ⊔ l2) → UU l3) {A : UU l1} {B : UU l2}
  (f : A → B) → UU (l2 ⊔ l3)
structure-map 𝒫 {A} {B} f = structure-family 𝒫 (fiber f)

structured-map :
  {l1 l2 l3 : Level}
  (𝒫 : UU (l1 ⊔ l2) → UU l3)
  (A : UU l1) (B : UU l2) → UU (l1 ⊔ l2 ⊔ l3)
structured-map 𝒫 A B = Σ (A → B) (structure-map 𝒫)

hom-structure :
  {l1 l2 l3 : Level} (𝒫 : UU (l1 ⊔ l2) → UU l3) →
  UU l1 → UU l2 → UU (l1 ⊔ l2 ⊔ l3)
hom-structure 𝒫 A B = Σ (A → B) (structure-map 𝒫)

structure-equality :
  {l1 l2 : Level} (𝒫 : UU l1 → UU l2) → UU l1 → UU (l1 ⊔ l2)
structure-equality 𝒫 A = (x y : A) → 𝒫 (x ＝ y)

Properties

Having structure is closed under equivalences

This is a consequence of the univalence axiom

has-structure-equiv :
  {l1 l2 : Level} (𝒫 : UU l1 → UU l2) {X Y : UU l1} → X ≃ Y → 𝒫 X → 𝒫 Y
has-structure-equiv 𝒫 e = tr 𝒫 (eq-equiv e)

has-structure-equiv' :
  {l1 l2 : Level} (𝒫 : UU l1 → UU l2) {X Y : UU l1} → X ≃ Y → 𝒫 Y → 𝒫 X
has-structure-equiv' 𝒫 e = tr 𝒫 (inv (eq-equiv e))

See also

Recent changes