This is an archived version pinned as of the submission of my master's thesis. An up-to-date version may be found online.

Bands

module foundation.bands where

open import foundation.set-truncations
open import foundation.universe-levels

open import foundation-core.equivalences

Idea

A band from $X$ to $Y$ is an element of the set-truncation of the type of equivalences from $X$ to $Y$.

Definition

band : {l1 l2 : Level} → UU l1 → UU l2 → UU (l1 ⊔ l2)
band A B = type-trunc-Set (A ≃ B)

unit-band : {l1 l2 : Level} {A : UU l1} {B : UU l2} → (A ≃ B) → band A B
unit-band = unit-trunc-Set

refl-band : {l : Level} (A : UU l) → band A A
refl-band A = unit-band id-equiv