# Automorphisms ```agda module foundation.automorphisms where ``` <details><summary>Imports</summary> ```agda open import foundation.dependent-pair-types open import foundation.sets open import foundation.universe-levels open import foundation-core.equivalences open import structured-types.pointed-types ``` </details> ## Idea An automorphism on a type `A` is an equivalence `A ≃ A`. We will just reuse the infrastructure of equivalences for automorphisms. ## Definitions ### The type of automorphisms on a type ```agda Aut : {l : Level} → UU l → UU l Aut Y = Y ≃ Y is-set-Aut : {l : Level} {A : UU l} → is-set A → is-set (Aut A) is-set-Aut H = is-set-equiv-is-set H H Aut-Set : {l : Level} → Set l → Set l pr1 (Aut-Set A) = Aut (type-Set A) pr2 (Aut-Set A) = is-set-Aut (is-set-type-Set A) Aut-Pointed-Type : {l : Level} → UU l → Pointed-Type l pr1 (Aut-Pointed-Type A) = Aut A pr2 (Aut-Pointed-Type A) = id-equiv ```