# mathlibdocumentation

control.equiv_functor

# Functions functorial with respect to equivalences

An equiv_functor is a function from Type → Type equipped with the additional data of coherently mapping equivalences to equivalences.

In categorical language, it is an endofunctor of the "core" of the category Type.

@[class]
structure equiv_functor  :
(Type u₀Type u₁)Type (max (u₀+1) u₁)
• map : Π {α β : Type ?}, α βf αf β
• map_refl' : (∀ (α : Type ?), . "obviously"
• map_trans' : (∀ {α β γ : Type ?} (k : α β) (h : β γ), . "obviously"

An equiv_functor is only functorial with respect to equivalences.

To construct an equiv_functor, it suffices to supply just the function f α → f β from an equivalence α ≃ β, and then prove the functor laws. It's then a consequence that this function is part of an equivalence, provided by equiv_functor.map_equiv.

Instances
def equiv_functor.map_equiv (f : Type u₀Type u₁) {α β : Type u₀} :
α βf α f β

An equiv_functor in fact takes every equiv to an equiv.

Equations
@[simp]
theorem equiv_functor.map_equiv_apply (f : Type u₀Type u₁) {α β : Type u₀} (e : α β) (x : f α) :
x =

@[simp]
theorem equiv_functor.map_equiv_symm_apply (f : Type u₀Type u₁) {α β : Type u₀} (e : α β) (y : f β) :
e).symm) y =

@[instance]
def equiv_functor.of_is_lawful_functor (f : Type u₀Type u₁) [functor f]  :

Equations