Types equipped with endomorphisms

module structured-types.types-equipped-with-endomorphisms where

open import foundation.dependent-pair-types
open import foundation.endomorphisms
open import foundation.functions
open import foundation.unit-type
open import foundation.universe-levels

Idea

A type equipped with an endomorphism consists of a type A equipped with a map A → A.

Definitions

Types equipped with endomorphisms

Endo : (l : Level) → UU (lsuc l)
Endo l = Σ (UU l) endo

module _
  {l : Level} (X : Endo l)
  where

  type-Endo : UU l
  type-Endo = pr1 X

  endomorphism-Endo : type-Endo → type-Endo
  endomorphism-Endo = pr2 X

Example

Unit type equipped with endomorphisms

trivial-Endo : {l : Level} → Endo l
pr1 (trivial-Endo {l})= raise-unit l
pr2 trivial-Endo = id