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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.

structure equiv_functor (f : Type u₀Type u₁) :
Type (max (u₀+1) u₁)

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.

theorem equiv_functor.map_refl {f : Type u₀Type u₁} [c : equiv_functor f] (α : Type u₀) :
theorem equiv_functor.map_trans {f : Type u₀Type u₁} [c : equiv_functor f] {α β γ : Type u₀} (k : α β) (h : β γ) :
def equiv_functor.map_equiv (f : Type u₀Type u₁) [equiv_functor f] {α β : Type u₀} (e : α β) :
f α f β

An equiv_functor in fact takes every equiv to an equiv.

theorem equiv_functor.map_equiv_apply (f : Type u₀Type u₁) [equiv_functor f] {α β : Type u₀} (e : α β) (x : f α) :
theorem equiv_functor.map_equiv_symm_apply (f : Type u₀Type u₁) [equiv_functor f] {α β : Type u₀} (e : α β) (y : f β) :
theorem equiv_functor.map_equiv_refl (f : Type u₀Type u₁) [equiv_functor f] (α : Type u₀) :
theorem equiv_functor.map_equiv_symm (f : Type u₀Type u₁) [equiv_functor f] {α β : Type u₀} (e : α β) :
theorem equiv_functor.map_equiv_trans (f : Type u₀Type u₁) [equiv_functor f] {α β γ : Type u₀} (ab : α β) (bc : β γ) :

The composition of map_equivs is carried over the equiv_functor. For plain functors, this lemma is named map_map when applied or map_comp_map when not applied.

def equiv_functor.of_is_lawful_functor (f : Type u₀Type u₁) [functor f] [is_lawful_functor f] :
theorem equiv_functor.map_equiv.injective (f : Type u₀Type u₁) [applicative f] [is_lawful_applicative f] {α β : Type u₀} (h : ∀ (γ : Type u₀), function.injective pure) :