module 1Lab.Equiv where


The key principle that distinguishes HoTT from other type theories is the univalence principle: isomorphic types can be identified. In cubical type theory, this turns out to be a theorem; but from a neutral point of view, it should be read as the definition of the identity types of universes — which are otherwise left unspecified by MLTT. However, the notion of isomorphism is not well-behaved when higher dimensional types are involved, so we can’t actually use it to formulate univalence.

The key issue is that any identity system is equipped with a path induction principle, which, if extended to the type of isomorphisms, would let us conclude that being an isomorphism is a proposition: but this is provably not the case. The result of formulating univalence using isomorphisms is sometimes called isovalence — that page has a formalisation of its failure.

We must therefore define a property of functions which refines being an isomorphism but is an isomorphism: functions satisfying this property will be called equivalences. More specifically, we’re looking for a family which

  • is implied by so that any function with a two-sided inverse is an equivalence. Apart from preserving the connection with the intuitive motivation behind univalence, exhibiting a two-sided inverse to a given map is simply easier than proving that it’s a coherent equivalence.

  • implies so that knowing that a map is an equivalence also lets you extract a two-sided inverse. Most notions of equivalence will also tell you additional properties about the connection between and its inverse map.

  • and finally, is an actual proposition.

Put concisely, we’re looking to define a propositional truncation of — one which allows us to conveniently project out the underlying data. Since propositional truncations are unique, this means that all type families satisfying the three bullet points above are themselves equivalent; so any choice made must be meta-mathematical, driven by concerns like ease of use, and efficiency of the formalisation.

Here in the 1Lab, we formalise three acceptable notions of equivalence:

  • biinvertible maps, which store two inverses to the given function;
  • half-adjoint equivalences, which, in addition to the inverse map and the witnesses of invertibility, have an extra coherence datum; and
  • the notion defined in this file, contractible maps, which repackage the notion of equivalence in a homotopically-inspired way.


Before we set about defining and working with equivalences, we’ll warm up by defining, and proving basic things about, isomorphisms. First, we define what it means for functions to be inverses of eachother, on both the left and the right.

is-left-inverse : (B → A) → (A → B) → Type _
is-left-inverse g f = (x : _) → g (f x) ≡ x

is-right-inverse : (B → A) → (A → B) → Type _
is-right-inverse g f = (x : _) → f (g x) ≡ x

A function which is both a left- and right inverse to is called a two-sided inverse (to To say that a function is an isomorphism is to equip it with a two-sided inverse.

\ Warning

Despite the name is-iso, a two-sided inverse is actual data we can equip a function with: it’s not, in general, valued in propositions.

record is-iso (f : A → B) : Type (level-of A ⊔ level-of B) where
  constructor iso
    inv  : B → A
    rinv : is-right-inverse inv f
    linv : is-left-inverse inv f

It’s immediate from the symmetry of the definition that if is a two-sided inverse to then also inverts an isomorphism’s inverse is again an isomorphism.

  inverse : is-iso inv
  inverse .inv  = f
  inverse .rinv = linv
  inverse .linv = rinv

Second, a composite of isomorphisms is an isomorphism: if we can undo and individually, we can undo them sequentially, by first undoing then undoing The observation that the inverse is in the opposite order is sometimes humorously referred to as the socks-and-shoes principle.

  ∘-is-iso : {f : B → C} {g : A → B} → is-iso f → is-iso g → is-iso (f ∘ g)
  ∘-is-iso f-im g-im .inv x = g-im .inv (f-im .inv x)
  ∘-is-iso {f = f} {g = g} f-im g-im .rinv x =
    f (g (g-im .inv (f-im .inv x))) ≡⟨ ap f (g-im .rinv _) ⟩≡
    f (f-im .inv x)                 ≡⟨ f-im .rinv _ ⟩≡
    x                               ∎
  ∘-is-iso {g = g} f-im g-im .linv x =
    ap (g-im .inv) (f-im .linv (g x)) ∙ g-im .linv x

Finally, the identity map is its own two-sided inverse, so it is an isomorphism. Keep in mind that, again, there are multiple ways for a map to be an isomorphism. It would be more correct to say that there is a parametric way to make the identity map into an isomorphism: if we know more about the specific type, there might be other ways.

    id-iso : is-iso {A = A} id
    id-iso .inv    = id
    id-iso .rinv x = refl
    id-iso .linv x = refl

Equivalences proper🔗

With that out of the way, we can actually define the property of functions that we’ll adopt under the name equivalence. The notion of contractible maps is due to Voevodsky, and it very elegantly packages the data of an inverse in a way that is immediately seen to be coherent. However, it turns out to have a very simple motivating explanation at the level of sets:

Recall that the preimage of a function over a point written is the set The properties of being injective and surjective can be profitably rephrased in terms of its preimages:

  • a function is injective if each of its preimages has at most one element. For if we have and then we can look at the set it has elements given by and For these to be the same, we must have so is injective.

  • a function is surjective if each of its preimages has at least one element, in the sense of propositional truncations. This is the actual definition: for the preimage to be inhabited means that there exists with

Putting these together, a function is a bijection if each of its fibres has exactly one element. Phrased as above, we’ve actually arrived exactly at the definition of equivalence proposed by Voevodsky: we just have to rephrase a few things.

Rather than talking about preimages, in the context of homotopy type theory, we refer to them as (homotopy) fibres over a given point. Not only is this an objectively more aesthetic name, but it reminds us that the path is actual structure we are equipping the point with, rather than being a property of points.

fibre : (A → B) → B → Type _
fibre {A = A} f y = Σ[ x ∈ A ] (f x ≡ y)

Finally, rather than talking of types with exactly one element, we already have a rephrasing which works at the level of homotopy type theory: contractibility. This is exactly the conjunction of the assertions above, just packaged in a way that does not require us to first define propositional truncations. A function is an equivalence if all of its fibres are contractible:

record is-equiv (f : A → B) : Type (level-of A ⊔ level-of B) where
    is-eqv : (y : B) → is-contr (fibre f y)

To warm up, let’s show that the identity map is an equivalence, working with the definition directly. First, we have to show that the fibre over has an element, for which we can take since what we’re looking for boils down to an element which is mapped to by the identity function.

id-equiv : is-equiv {A = A} id
id-equiv .is-eqv y .centre = y , refl

Then, given any pair with and we must show that we have But this is exactly what path induction promises us! Rather than an appeal to transport, we can directly use a connection:

id-equiv .is-eqv y .paths (x , p) i = p (~ i) , λ j → p (~ i ∨ j)

Since Cubical Agda has primitives that refer to equivalences, we have to take some time to teach the system about the type we just defined above. In addition to teaching it how to project the underlying function of an equivalence, we must prove that an equivalence has contractible fibres — but what the system asks of us is that each partial fibre be extensible: the cubical phrasing of contractibility.

{-# BUILTIN EQUIV _≃_ #-}

is-eqv' : ∀ {a b} (A : Type a) (B : Type b)
        → (w : A ≃ B) (a : B)
        → (ψ : I)
        → (p : Partial ψ (fibre (w .fst) a))
        → fibre (w .fst) a [ ψ ↦ p ]
is-eqv' A B (f , is-equiv) a ψ u0 = inS (
  hcomp (∂ ψ) λ where
    i (ψ = i0) → c .centre
    i (ψ = i1) → c .paths (u0 1=1) i
    i (i = i0) → c .centre)
  where c = is-equiv .is-eqv a

{-# BUILTIN EQUIVPROOF is-eqv' #-}

is-equiv is a proposition🔗

Since being contractible is a proposition, and being an equivalence simply quantifies contractibility over each fibre, we can directly conclude that being an equivalence is a proposition.

is-equiv-is-prop : (f : A → B) → is-prop (is-equiv f)
is-equiv-is-prop f x y i .is-eqv p = is-contr-is-prop (x .is-eqv p) (y .is-eqv p) i

Equivalences are invertible🔗

We can now show that equivalences are invertible. This will amount to taking apart the proof that each fibre is contractible. If we have then the centre of contraction of gives us a distinguished and a path

equiv→inverse : {f : A → B} → is-equiv f → B → A
equiv→inverse eqv y = eqv .is-eqv y .centre .fst

  : ∀ {f : A → B} (eqv : is-equiv f) (y : B) → f (equiv→inverse eqv y) ≡ y
equiv→counit eqv y = eqv .is-eqv y .centre .snd

Now we have to show that applying the inverse to gives us back But note that has two inhabitants: the centre of contraction, which is projected by the inverse, but also the simpler Since the fibre is contractible, we have a path

  : ∀ {f : A → B} (eqv : is-equiv f) x → equiv→inverse eqv (f x) ≡ x
equiv→unit {f = f} eqv x i = eqv .is-eqv (f x) .paths (x , refl) i .fst

Contractibility gives us, in addition to a path between the points a particular higher-dimensional square. Note that we have two ways of proving that corresponding to either cancelling the outer or the inner and The square we obtain tells us that these are actually the same:

  : ∀ {f : A → B} (eqv : is-equiv f) (x : A)
  → Square (ap f (equiv→unit eqv x)) (equiv→counit eqv (f x)) refl refl
equiv→square {f = f} eqv x i j = eqv .is-eqv (f x) .paths (x , refl) i .snd j

However, the square above is slightly skewed: we would have preferred for the two non-trivial paths to be in opposite sides, so that it would correspond to a Path between the paths. We can use hcomp to slide one of the faces around to get us the more useful direct proof that

  : ∀ {f : A → B} (eqv : is-equiv f) x
  → ap f (equiv→unit eqv x) ≡ equiv→counit eqv (f x)
equiv→zig {f = f} eqv x i j = hcomp (∂ i ∨ ∂ j) λ where
  k (k = i0) → equiv→square eqv x j i
  k (i = i0) → f (equiv→unit eqv x j)
  k (i = i1) → equiv→counit eqv (f x) (j ∨ ~ k)
  k (j = i0) → equiv→counit eqv (f x) (i ∧ ~ k)
  k (j = i1) → f x
Note that the law established above has a symmetric counterpart, showing that However, we can prove that this follows from the law above, using another cubical argument. Since the extent of the proof is another lifting problem, we won’t expand on the details.
  : ∀ {f : A → B} (eqv : is-equiv f) x
  → ap (equiv→inverse eqv) (equiv→counit eqv x)
  ≡ equiv→unit eqv (equiv→inverse eqv x)
equiv→zag {f = f} eqv b =
  subst (λ b → ap g (ε b) ≡ η (g b)) (ε b) (helper (g b)) where
  g = equiv→inverse eqv
  ε = equiv→counit eqv
  η = equiv→unit eqv

  helper : ∀ a → ap g (ε (f a)) ≡ η (g (f a))
  helper a i j = hcomp (∂ i ∨ ∂ j) λ where
    k (i = i0) → g (ε (f a) (j ∨ ~ k))
    k (i = i1) → η (η a (~ k)) j
    k (j = i0) → g (equiv→zig eqv a (~ i) (~ k))
    k (j = i1) → η a (i ∧ ~ k)
    k (k = i0) → η a (i ∧ j)

Finally, it’ll be convenient for us to package some of the theorems above into a proof that every equivalence is an isomorphism:

is-equiv→is-iso : {f : A → B} → is-equiv f → is-iso f
is-equiv→is-iso eqv .is-iso.inv  = equiv→inverse eqv
is-equiv→is-iso eqv .is-iso.rinv = equiv→counit eqv
is-equiv→is-iso eqv .is-iso.linv = equiv→unit eqv

Improving isomorphisms🔗

Having shown that being an equivalence is a proposition and implies being an isomorphism, we’re left with the trickiest part: showing that it’s implied by being an isomorphism. The argument presented here is natively cubical; we do everything in terms of lifting properties. There is a more “algebraic” version, factoring through the intermediate notion of half-adjoint equivalence, available on that page.

We’ll write for the the function its inverse and the two homotopies. Our construction definitionally preserves and the homotopy However, it can not preserve the proof since that would imply that is-iso is a proposition, which, again, is provably not the case.

module _ {f : A → B} (i : is-iso f) where
  open is-iso i renaming (inv to g ; rinv to s ; linv to t)

We want to show that, for any the fibre of over is contractible. According to our decomposition above, it will suffice to show that the fibre is propositional, and that it is inhabited. Inhabitation is easy — belongs to the fibre Let’s focus on propositionality.

Suppose that we have and as below; Note that and are fibres of over What we need to show is that we have some and over

  private module _ (y : B) (x0 x1 : A) (p0 : f x0 ≡ y) (p1 : f x1 ≡ y) where

As an intermediate step in proving that we must show that — without this, we can’t even state that and are identified, since they live in different types! To this end, we will build parts of which will be required to assemble the overall proof that

We’ll detail the construction of for the same method is used. We want to construct a line, which we can do by exhibiting that line as the missing face in a square. We have equations (refl), (the action of g on p0), and by the assumption that is a right inverse to Diagrammatically, these fit together into a square:

The missing line in this square is Since the inside (the filler) will be useful to us later, we also give it a name:

    π₀ : g y ≡ x0
    π₀ i = hcomp (∂ i) λ where
      k (i = i0) → g y
      k (i = i1) → t x0 k
      k (k = i0) → g (p0 (~ i))

    θ₀ : Square (ap g (sym p0)) refl (t x0) π₀
    θ₀ i j = hfill (∂ i) j λ where
      k (i = i0) → g y
      k (i = i1) → t x0 k
      k (k = i0) → g (p0 (~ i))

Since the construction of is analogous, I’ll simply present the square. We correspondingly name the missing face and the filler

Joining these paths by their common face, we obtain This square also has a filler, connecting and over the line

This concludes the construction of and thus, the 2D part of the proof. Now, we want to show that over a path induced by This is a square with a specific boundary, which can be built by constructing an appropriate open cube, where the missing face is that square. As an intermediate step, we define to be the filler for the square above.

    θ : Square refl π₀ π₁ π
    θ i j = hfill (∂ i) j λ where
      k (i = i1) → π₁ k
      k (i = i0) → π₀ k
      k (k = i0) → g y

Observe that we can coherently alter to get below, which expresses that and are identified.

    ι : Square (ap (g ∘ f) π) (ap g p0) (ap g p1) refl
    ι i j = hcomp (∂ i ∨ ∂ j) λ where
      k (k = i0) → θ i (~ j)
      k (i = i0) → θ₀ (~ j) (~ k)
      k (i = i1) → θ₁ (~ j) (~ k)
      k (j = i0) → t (π i) (~ k)
      k (j = i1) → g y

This composition can be visualised as the red (front) face in the diagram below. The back face is i.e. (θ i (~ j)) in the code. Similarly, the (bottom) face is g y, the (top) face is t (π i) (~ k), and similarly for (left) and (right).

The fact that only appears as can be understood as the diagram above being upside-down. Indeed, and in the boundary of (the inner, blue face) are inverted when their types are considered. We’re in the home stretch: Using our assumption we can cancel all of the in the diagram above to get what we wanted:

    sq1 : Square (ap f π) p0 p1 refl
    sq1 i j = hcomp (∂ i ∨ ∂ j) λ where
      k (i = i0) → s (p0 j) k
      k (i = i1) → s (p1 j) k
      k (j = i0) → s (f (π i)) k
      k (j = i1) → s y k
      k (k = i0) → f (ι i j)

The composition above can be visualised as the front (red) face in the cubical diagram below. Once more, left is right is up is and down is

Putting all of this together, we get that Since there were no assumptions on any of the variables under consideration, this indeed says that the fibre over is a proposition for any choice of

    is-iso→fibre-is-prop : (x0 , p0) ≡ (x1 , p1)
    is-iso→fibre-is-prop i .fst = π i
    is-iso→fibre-is-prop i .snd = sq1 i

Since the fibre over is inhabited by we get that any isomorphism has contractible fibres:

  is-iso→is-equiv : is-equiv f
  is-iso→is-equiv .is-eqv y .centre .fst = g y
  is-iso→is-equiv .is-eqv y .centre .snd = s y
  is-iso→is-equiv .is-eqv y .paths z =
    is-iso→fibre-is-prop y (g y) (fst z) (s y) (snd z)

If we package this differently, then we can present it as a map between the types of isomorphisms and equivalences

Iso→Equiv : Iso A B → A ≃ B
Iso→Equiv (f , is-iso) = f , is-iso→is-equiv is-iso
inverse-is-equiv : {f : A → B} (eqv : is-equiv f) → is-equiv (equiv→inverse eqv)
inverse-is-equiv {f = f} eqv .is-eqv x .centre = record
  { fst = f x ; snd = equiv→unit eqv x }
inverse-is-equiv {A = A} {B = B} {f = f} eqv .is-eqv x .paths (y , p) = q where
  g = equiv→inverse eqv
  η = equiv→unit eqv
  ε = equiv→counit eqv
  zag = equiv→zag eqv

  q : (f x , η x) ≡ (y , p)
  q i .fst = (ap f (sym p) ∙ ε y) i
  q i .snd j = hcomp (∂ i ∨ ∂ j) λ where
    k (k = i0) → zag y j i
    k (i = i0) → η (p k) (j ∧ k)
    k (i = i1) → p (k ∧ j)
    k (j = i0) → g (∙-filler' (ap f (sym p)) (ε y) k i)
    k (j = i1) → η (p k) (i ∨ k)

module Equiv {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} (f : A ≃ B) where
  to = f .fst
  from = equiv→inverse (f .snd)
  η    = equiv→unit (f .snd)
  ε    = equiv→counit (f .snd)
  zig  = equiv→zig (f .snd)
  zag  = equiv→zag (f .snd)

  injective : ∀ {x y} → to x ≡ to y → x ≡ y
  injective p = ap fst $ is-contr→is-prop (f .snd .is-eqv _) (_ , refl) (_ , sym p)

  injective₂ : ∀ {x y z} → to x ≡ z → to y ≡ z → x ≡ y
  injective₂ p q = ap fst $ is-contr→is-prop (f .snd .is-eqv _)
    (_ , p) (_ , q)

  inverse : B ≃ A
  inverse = from , inverse-is-equiv (f .snd)

  adjunctl : ∀ {x y} → to x ≡ y → x ≡ from y
  adjunctl p = sym (η _) ∙ ap from p

  adjunctr : ∀ {x y} → x ≡ from y → to x ≡ y
  adjunctr p = ap to p ∙ ε _

  open is-iso

  adjunct : ∀ {x y} → (to x ≡ y) ≃ (x ≡ from y)
  adjunct {x} {y} .fst = adjunctl
  adjunct {x} {y} .snd = is-iso→is-equiv λ where
    .inv    → adjunctr
    .rinv p → J (λ _ p → sym (η _) ∙ ap from (ap to (sym p) ∙ ε _) ≡ sym p)
      (sym (∙-swapl (∙-idr _ ∙ sym (zag _) ∙ sym (∙-idl _) ∙ sym (ap-∙ from _ _))))
      (sym p)
    .linv → J (λ _ p → ap to (sym (η _) ∙ ap from p) ∙ ε _ ≡ p)
      (sym (∙-swapr (∙-idl _ ∙ ap sym (sym (zig _)) ∙ sym (∙-idr _) ∙ sym (ap-∙ to _ _))))

module Iso {ℓ ℓ'} {A : Type ℓ} {B : Type ℓ'} ((f , f-iso) : Iso A B) where
  open is-iso f-iso renaming (inverse to inverse-iso)

  injective : ∀ {x y} → f x ≡ f y → x ≡ y
  injective p = sym (linv _) ·· ap inv p ·· linv _

  inverse : Iso B A
  inverse = inv , inverse-iso

Simple constructions🔗

Having established the three desiderata for a notion of equivalence, we will spend the rest of this module constructing readily-available equivalences, and establishing basic facts about them.

Contractible types🔗

If and are contractible types, then any function must be homotopic to the function which sends everything in to the centre of This function is invertible by sending everything in to the centre of Therefore, any function between contractible types is an equivalence.

is-contr→is-equiv : is-contr A → is-contr B → {f : A → B} → is-equiv f
is-contr→is-equiv cA cB = is-iso→is-equiv f-is-iso where
  f-is-iso : is-iso _
  f-is-iso .is-iso.inv  _ = cA .centre
  f-is-iso .is-iso.rinv _ = is-contr→is-prop cB _ _
  f-is-iso .is-iso.linv _ = is-contr→is-prop cA _ _

Pairing this with the “canonical” function, we obtain an equivalence between any contractible types. Since the unit type is contractible, any contractible type is equivalent to the unit.

is-contr→≃ : is-contr A → is-contr B → A ≃ B
is-contr→≃ cA cB = (λ _ → cB .centre) , is-contr→is-equiv cA cB

is-contr→≃⊤ : is-contr A → A ≃ ⊤
is-contr→≃⊤ c = is-contr→≃ c ⊤-is-contr

Strictness of the empty type🔗

We say that an initial object is strict if every map into it is an equivalence. This holds of the empty type, and moreover, the proof is delightfully simple! Since showing that is an equivalence boils down to showing something about given a point , it’s immediate that any map into the empty type is an equivalence.

¬-is-equiv : (f : A → ⊥) → is-equiv f
¬-is-equiv f .is-eqv ()

is-empty→≃⊥ : ¬ A → A ≃ ⊥
is-empty→≃⊥ ¬a = _ , ¬-is-equiv ¬a

Propositional extensionality (redux)🔗

Following the trend set by contractible types, above, another h-level for which constructing equivalences is easy are the propositions. If and are propositions, then any map (resp. must be homotopic to the identity, and consequently any pair of functions and is a pair of inverses. Put another way, any biimplication between propositions is an equivalence.

  biimp-is-equiv : (f : P → Q) → (Q → P) → is-equiv f
  biimp-is-equiv f g .is-eqv y .centre .fst = g y
  biimp-is-equiv f g .is-eqv y .centre .snd = qprop (f (g y)) y
  biimp-is-equiv f g .is-eqv y .paths (p' , path) = Σ-pathp
    (pprop (g y) p')
    (is-prop→squarep (λ _ _ → qprop) _ _ _ _)

  prop-ext : (P → Q) → (Q → P) → P ≃ Q
  prop-ext p→q q→p .fst = p→q
  prop-ext p→q q→p .snd = biimp-is-equiv p→q q→p

Groupoid operations🔗

Since types are higher groupoids, we have certain algebraic laws regarding the behaviour of path operations which can be expressed as saying that they form equivalences. First, the inverse path operation is definitionally involutive, so it’s its own two-sided inverse:

  sym-equiv : ∀ {ℓ} {A : Type ℓ} {x y : A} → (x ≡ y) ≃ (y ≡ x)
  sym-equiv .fst = sym
  sym-equiv .snd = is-iso→is-equiv (iso sym (λ _ → refl) (λ _ → refl))

If we have a path then and Viewing this as a function of it says that the operation precompose with is inverted on both sides by precomposition with

  ∙-pre-equiv : ∀ {ℓ} {A : Type ℓ} {x y z : A} → x ≡ y → (y ≡ z) ≃ (x ≡ z)
  ∙-pre-equiv p .fst q = p ∙ q
  ∙-pre-equiv p .snd = is-iso→is-equiv λ where
    .inv q  → sym p ∙ q
    .rinv q → ∙-assoc p _ _       ·· ap (_∙ q) (∙-invr p) ·· ∙-idl q
    .linv q → ∙-assoc (sym p) _ _ ·· ap (_∙ q) (∙-invl p) ·· ∙-idl q

Similarly, postcomposition with is inverted on both sides by postcomposition with so it too is an equivalence.

  ∙-post-equiv : ∀ {ℓ} {A : Type ℓ} {x y z : A} → y ≡ z → (x ≡ y) ≃ (x ≡ z)
  ∙-post-equiv p .fst q = q ∙ p
  ∙-post-equiv p .snd = is-iso→is-equiv λ where
    .inv q  → q ∙ sym p
    .rinv q → sym (∙-assoc q _ _) ·· ap (q ∙_) (∙-invl p) ·· ∙-idr q
    .linv q → sym (∙-assoc q _ _) ·· ap (q ∙_) (∙-invr p) ·· ∙-idr q

The Lift type🔗

Because Agda’s universes are not cumulative, we can not freely move a type to conclude that or to higher universes. To work around this, we have a Lift type, which, given a small type and some universe gives us a name for in To know that this operation is coherent, we can prove that the lifting map

is an equivalence: the name of in really is equivalent to the type we started with. Because Lift is a very well-behaved record type, the proof that this is an equivalence looks very similar to the proof that the identity function is an equivalence:

Lift-≃ : ∀ {a ℓ} {A : Type a} → Lift ℓ A ≃ A
Lift-≃ .fst (lift a) = a
Lift-≃ .snd .is-eqv a .centre = lift a , refl
Lift-≃ .snd .is-eqv a .paths (x , p) i .fst = lift (p (~ i))
Lift-≃ .snd .is-eqv a .paths (x , p) i .snd j = p (~ i ∨ j)

Finally, while we’re here, we can easily prove that the Lift type respects equivalences, in that if we have small then their liftings and will still be equivalent.

  : ∀ {a b ℓ ℓ'} {A : Type a} {B : Type b}
  → A ≃ B → Lift ℓ A ≃ Lift ℓ' B
Lift-ap (f , eqv) .fst (lift x) = lift (f x)
Lift-ap f .snd .is-eqv (lift y) .centre = lift (Equiv.from f y) , ap lift (Equiv.ε f y)
Lift-ap f .snd .is-eqv (lift y) .paths (lift x , q) i = lift (p i .fst) , λ j → lift (p i .snd j)
  where p = f .snd .is-eqv y .paths (x , ap Lift.lower q)


Finally, we can show here that any involution — a function which inverts itself — is an equivalence.

is-involutive→is-equiv : {f : A → A} → (∀ a → f (f a) ≡ a) → is-equiv f
is-involutive→is-equiv inv = is-iso→is-equiv (iso _ inv inv)

Closure properties🔗

We will now show a rather fundamental property of equivalences: they are closed under two-out-of-three. This means that, considering and as a set of three things, if any two are an equivalence, then so is the third:

  ∙-is-equiv    : is-equiv f → is-equiv g → is-equiv (g ∘ f)
  equiv-cancell : is-equiv g → is-equiv (g ∘ f) → is-equiv f
  equiv-cancelr : is-equiv f → is-equiv (g ∘ f) → is-equiv g

We have already shown the first of these, when the individual functions are equivalences, since being an equivalence is logically equivalent to being an isomorphism. Supposing that and are, then we can show that

is an inverse to the other case is symmetric. Since both of the proofs are just calculations, we will not comment on them.

However, we will render them for the curious reader. It’s an instructive exercise to work these out for yourself!
  ∙-is-equiv ef eg = is-iso→is-equiv (∘-is-iso (is-equiv→is-iso eg) (is-equiv→is-iso ef))

  equiv-cancell eg egf = is-iso→is-equiv (iso inv right left) where
    inv : B → A
    inv x = equiv→inverse egf (g x)
      right : is-right-inverse inv f
      right x =
        f (equiv→inverse egf (g x))                        ≡˘⟨ equiv→unit eg _ ⟩≡˘
        equiv→inverse eg (g (f (equiv→inverse egf (g x)))) ≡⟨ ap (equiv→inverse eg) (equiv→counit egf _) ⟩≡
        equiv→inverse eg (g x)                             ≡⟨ equiv→unit eg _ ⟩≡
        x                                                  ∎
      left : is-left-inverse inv f
      left x = equiv→unit egf x

  equiv-cancelr ef egf = is-iso→is-equiv (iso inv right left) where
    inv : C → B
    inv x = f (equiv→inverse egf x)
    right : is-right-inverse inv g
    right x = equiv→counit egf x
    left : is-left-inverse inv g
    left x =
      f (equiv→inverse egf (g x))                        ≡˘⟨ ap (f ∘ equiv→inverse egf ∘ g) (equiv→counit ef _) ⟩≡˘
      f (equiv→inverse egf (g (f (equiv→inverse ef x)))) ≡⟨ ap f (equiv→unit egf _) ⟩≡
      f (equiv→inverse ef x)                             ≡⟨ equiv→counit ef _ ⟩≡
      x                                                  ∎

Specialising these, any left- or right- inverse of an equivalence must be homotopic to the specified one, so that it too is an equivalence.

  : {f : A → B} {g : B → A}
  → is-left-inverse g f → is-equiv f → is-equiv g
left-inverse→equiv linv ef = equiv-cancelr ef
  (subst is-equiv (sym (funext linv)) id-equiv)

  : {f : A → B} {g : B → A}
  → is-right-inverse g f → is-equiv f → is-equiv g
right-inverse→equiv rinv ef = equiv-cancell ef
  (subst is-equiv (sym (funext rinv)) id-equiv)

Equivalence reasoning🔗

To simplify working with chains of equivalences, we can re-package their closure operations as mixfix operators, so that we have a shorthand notation for working with composites of pairs of equivalences and long chains of equivalences, where making the intermediate steps explicit is more important.

id≃ : ∀ {ℓ} {A : Type ℓ} → A ≃ A
id≃ = id , id-equiv

_∙e_ : A ≃ B → B ≃ C → A ≃ C
_∙e_ (f , ef) (g , eg) = g ∘ f , ∙-is-equiv ef eg

_e⁻¹ : A ≃ B → B ≃ A
((f , ef) e⁻¹) = equiv→inverse ef , inverse-is-equiv ef

≃⟨⟩-syntax : ∀ {ℓ ℓ₁ ℓ₂} (A : Type ℓ) {B : Type ℓ₁} {C : Type ℓ₂}
           → B ≃ C → A ≃ B → A ≃ C
≃⟨⟩-syntax A g f = f ∙e g

_≃˘⟨_⟩_ : ∀ {ℓ ℓ₁ ℓ₂} (A : Type ℓ) {B : Type ℓ₁} {C : Type ℓ₂}
        → B ≃ A → B ≃ C → A ≃ C
A ≃˘⟨ f ⟩≃˘ g = f e⁻¹ ∙e g

_≃⟨⟩_ : ∀ {ℓ ℓ₁} (A : Type ℓ) {B : Type ℓ₁} → A ≃ B → A ≃ B
x ≃⟨⟩ x≡y = x≡y

_≃∎ : ∀ {ℓ} (A : Type ℓ) → A ≃ A
x ≃∎ = id≃
infixr 30 _∙e_
infix 31 _e⁻¹

infixr 2 ≃⟨⟩-syntax _≃⟨⟩_ _≃˘⟨_⟩_
infix  3 _≃∎
infix 21 _≃_

syntax ≃⟨⟩-syntax x q p = x ≃⟨ p ⟩ q

  : ∀ {ℓ ℓ'} {A : Type ℓ} {a b : A}
  → lift {ℓ = ℓ'} a ≡ lift {ℓ = ℓ'} b → a ≡ b
lift-inj p = ap Lift.lower p

-- Fibres of composites are given by pairs of fibres.
  : ∀ {ℓ ℓ' ℓ''} {A : Type ℓ} {B : Type ℓ'} {C : Type ℓ''}
  → {f : B → C} {g : A → B}
  → ∀ c → fibre (f ∘ g) c ≃ (Σ[ (b , _) ∈ fibre f c ] (fibre g b))
fibre-∘-≃ {f = f} {g = g} c = Iso→Equiv (fwd , iso bwd invl invr)
      fwd : fibre (f ∘ g) c → Σ[ (b , _) ∈ fibre f c ] (fibre g b)
      fwd (a , p) = ((g a) , p) , (a , refl)

      bwd : Σ[ (b , _) ∈ fibre f c ] (fibre g b) → fibre (f ∘ g) c
      bwd ((b , p) , (a , q)) = a , ap f q ∙ p

      invl : ∀ x → fwd (bwd x) ≡ x
      invl ((b , p) , (a , q)) i .fst .fst = q i
      invl ((b , p) , (a , q)) i .fst .snd j =
        hcomp (∂ i ∨ ∂ j) λ where
          k (i = i0) → ∙-filler (ap f q) p k j
          k (i = i1) → p (j ∧ k)
          k (j = i0) → f (q i)
          k (j = i1) → p k
          k (k = i0) → f (q (i ∨ j))
      invl ((b , p) , a , q) i .snd .fst = a
      invl ((b , p) , a , q) i .snd .snd j = q (i ∧ j)

      invr : ∀ x → bwd (fwd x) ≡ x
      invr (a , p) i .fst = a
      invr (a , p) i .snd = ∙-idl p i

is-empty→≃ : ¬ A → ¬ B → A ≃ B
is-empty→≃ ¬a ¬b = is-empty→≃⊥ ¬a ∙e is-empty→≃⊥ ¬b e⁻¹