Lifts of types

Content created by Egbert Rijke and Fredrik Bakke.

Created on 2024-01-23.
Last modified on 2024-01-23.

module foundation.lifts-types where
open import foundation.dependent-pair-types
open import foundation.universe-levels


Consider a type X. A lift of X is an object in the slice over X, i.e., it consists of a type Y and a map f : Y → X.

In the above definition of lifts of types our aim is to capture the most general concept of what it means to be an lift of a type. Similarly, in any category we would say that an lift of an object X consists of an object Y equipped with a morphism f : Y → X.


lift-type : {l1 : Level} (l2 : Level) (X : UU l1)  UU (l1  lsuc l2)
lift-type l2 X = Σ (UU l2)  Y  Y  X)

module _
  {l1 l2 : Level} {X : UU l1} (Y : lift-type l2 X)

  type-lift-type : UU l2
  type-lift-type = pr1 Y

  projection-lift-type : type-lift-type  X
  projection-lift-type = pr2 Y

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