This is an archived version pinned as of the submission of my master's thesis.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

Small universes

module foundation.small-universes where

open import foundation.universe-levels

open import foundation-core.cartesian-product-types
open import foundation-core.small-types

Idea

A universe UU l1 is said to be small with respect to UU l2 if UU l1 is a UU l2-small type and each X : UU l1 is a UU l2-small type

is-small-universe :
  (l l1 : Level) → UU (lsuc l1 ⊔ lsuc l)
is-small-universe l l1 = is-small l (UU l1) × ((X : UU l1) → is-small l X)