open import 1Lab.Type

module 1Lab.Path where

The Interval🔗

In HoTT, the inductively-defined identity type gets a new meaning explanation: continuous paths, in a topological sense. The “key idea” of cubical type theory — and thus, Cubical Agda — is that we can take this as a new definition of the identity type, where we interpret a Path in a type by a function where the domain is the interval type.

open import Prim.Extension public
open import Prim.Interval public
open import Prim.Kan public

The type I is meant to represent the (real, closed) unit interval $[0,1]$, the same unit interval used in the topological definition of path. Because the real unit interval has a least and greatest element — 0 and 1 — the interval type also has two global inhabitants, i0 and i1. This is where the analogy with the reals breaks down: There’s no such thing as i0.5 (much less i1/π). In reality, the interval type internalises an abstract interval object.

Regardless, since all functions definable in type theory are automatically continuous, we can take a path to be any value in the function type I → A. When working with paths, though, it’s useful to mention the endpoints of a path in its type — that is, the values the function takes when applied to i0 and to i1. We can “upgrade” any function f : I → A to a Path, using a definition that looks suspiciously like the identity function:

private
  to-path : ∀ {ℓ} {A : Type ℓ} → (f : I → A) → Path A (f i0) (f i1)
  to-path f i = f i

refl : ∀ {ℓ} {A : Type ℓ} {x : A} → x ≡ x
refl {x = x} = to-path (λ i → x)

The type Path A x y is also written x ≡ y, when A is not important - i.e. when it can be inferred from x and y. Under this interpretation, proof that identification is reflexive (i.e. that $x = x$) is given by a Path which yields the same element everywhere on I: The function that is constantly $x$.

If we have a Path, we can apply it to a value of the interval type to get an element of the underlying type. When a path is applied to one of the endpoints, the result is the same as declared in its type — even when we’re applying a path we don’t know the definition of.²

module _ {ℓ} {A : Type ℓ} {x y : A} {p : x ≡ y} where
  private
    left-endpoint : p i0 ≡ x
    left-endpoint i = x

    right-endpoint : p i1 ≡ y
    right-endpoint i = y

In addition to the two endpoints i0 and i1, the interval has the structure of a De Morgan algebra. All the following equations are respected (definitionally), but they can not be expressed internally as a Path because I is not in Type.³

• $x \land \mathrm{i0} = \mathrm{i0}$, $x \land \mathrm{i1} = x$
• $x \lor \mathrm{i0} = x$, $x \lor \mathrm{i1} = \mathrm{i1}$
• $\neg(x \land y) = \neg x \lor \neg y$
• $\neg\mathrm{i0} = \mathrm{i1}$, $\neg\mathrm{i1} = \mathrm{i0}$, $\neg\neg x = x$
• $\land$ and $\lor$ are both associative, commutative and idempotent, and distribute over eachother.

Note that, in the formalisation, $\neg x$ is written ~ x. As a more familiar description, a De Morgan algebra is a Boolean algebra that does not (necessarily) satisfy the law of excluded middle. This is necessary to maintain type safety.

Raising Dimension🔗

To wit: In cubical type theory, a term in a context with $n$ interval variables expresses a way of mapping an $n$-cube into that type. One very important class of these maps are the $1$-cubes — lines or paths — which represent identifications between terms of that type.

Iterating this construction, a term in a context with 2 interval variables represents a square in the type, which can be read as saying that some paths (specialising one of the variables to $i0$ or $i1$) in that space are identical: A path between paths, which we call a homotopy.

The structural operations on contexts, and the $\land$ and $\lor$ operations on the interval, give a way of extending from $n$-dimensional cubes to $n+k$-dimensional cubes. For instance, if we have a path like the one below, we can extend it to any of a bunch of different squares:

module _ {ℓ} {A : Type ℓ} {a b : A} {p : Path A a b} where

The first thing we can do is introduce another interval variable and ignore it, varying the path over the non-ignored variable. These give us squares where either the top/bottom or left/right faces are the path p, and the other two are refl.

  private
    drop-j : PathP (λ i → p i ≡ p i) refl refl
    drop-j i j = p i

    drop-i : PathP (λ i → a ≡ b) p p
    drop-i i j = p j

These squares can be drawn as below. Take a moment to appreciate how the types of drop-j and drop-i specify the boundary of the diagram — A PathP (λ i → p i ≡ p i) refl refl corresponds to a square whose top/bottom faces are both p, and whose left/right faces are both refl (by convention). Similarly, PathP (λ i → a ≡ b) p p has refl as top/bottom faces (recall that refl is the constant function regarded as a path), and p as both left/right faces.

The other thing we can do is use one of the binary operators on the interval to get squares called connections, where two adjacent faces are p and the other two are refl:

    ∧-conn : PathP (λ i → a ≡ p i) refl p
    ∧-conn i j = p (i ∧ j)

    ∨-conn : PathP (λ i → p i ≡ b) p refl
    ∨-conn i j = p (i ∨ j)

These correspond to the following two squares:

Since iterated paths are used a lot in homotopy type theory, we introduce a shorthand for 2D non-dependent paths. A Square in a type is exactly what it says on the tin: a square.

Square : ∀ {ℓ} {A : Type ℓ} {a00 a01 a10 a11 : A}
       → (p : a00 ≡ a01)
       → (q : a00 ≡ a10)
       → (s : a01 ≡ a11)
       → (r : a10 ≡ a11)
       → Type ℓ
Square p q s r = PathP (λ i → p i ≡ r i) q s

The arguments to Square are as in the following diagram, listed in the order “PQSR”. This order is a bit unusual (it’s one off from being alphabetical, for instance) but it does have a significant benefit: If you imagine that the letters are laid out in a circle, identical paths are adjacent. Reading the square in the left-right direction, it says that $p$ and $r$ are identical — these are adjacent if you “fold up” the sequence p q s r. Similarly, reading top-down, it says that $q$ and $s$ are identical - these are directly adjacent.

Symmetry🔗

The involution ~_ on the interval type gives a way of inverting paths — a proof that identification is symmetric.

sym : ∀ {ℓ₁} {A : Type ℓ₁} {x y : A}
    → x ≡ y → y ≡ x
sym p i = p (~ i)

As a minor improvement over “Book HoTT”, this operation is definitionally involutive:

module _ {ℓ} {A : Type ℓ} {x y : A} {p : x ≡ y} where
  private
    sym-invol : sym (sym p) ≡ p
    sym-invol i = p

Paths🔗

While the basic structure of the path type is inherited from its nature as functions out of an internal De Morgan algebra, the structure of identifications presented by paths is more complicated. For starters, let’s see how paths correspond to identifications in that they witness the logical principle of “indiscernibility of identicals”.

Transport🔗

A basic principle of identity is that identicals are indiscernible: if $x = y$ and $P(x)$ holds, then $P(y)$ also holds, for any choice of predicate $P$. In type theory, this is generalised, as $P$ can be not only a predicate, but any type family.

The way this is incarnated is by an operation called transport, which says that every path between A and B gives rise to a function A → B.

transport : ∀ {ℓ} {A B : Type ℓ} → A ≡ B → A → B
transport p = transp (λ i → p i) i0

The transport operation is the earliest case of when thinking of p : A ≡ B as merely saying “A and B are equal” goes seriously wrong. A path gives a specific identification of A and B, which can be highly non-trivial.

As a concrete example, it can be shown that the type Bool ≡ Bool has exactly two inhabitants (see here), which is something like saying “the set of booleans is equal to itself in two ways”. That phrase is nonsensical, which is why “there are two paths Bool → Bool” is preferred: it’s not nonsense.

In Cubical Agda, transport is a derived notion, with the actual primitive being transp. Unlike transport, which has two arguments (the path, and the point to transport), transp has three:

• The first argument to transp is a line of types, i.e. a function A : I → Type, just as for transport.

• The second argument to transp has type I, but it’s not playing the role of an endpoint of the interval. It’s playing the role of a formula, which specifies where the transport is constant: In transp P i1, P is required to be constant, and the transport is the identity function:

  _ : ∀ {ℓ} {A : Type ℓ} → transp (λ i → A) i1 ≡ id
  _ = refl
  

• The third argument is an inhabitant of A i0, as for transport.

This second argument, which lets us control where transp is constant, brings a lot of power to the table! For example, the proof that transporting along refl is id is as follows:

transport-refl : ∀ {ℓ} {A : Type ℓ} (x : A)
               → transport (λ i → A) x ≡ x
transport-refl {A = A} x i = transp (λ _ → A) i x

Since λ i → A is a constant function, the definition of transport-refl is well-typed, and it has the stated endpoints because transport is defined to be transp P i0, and transp P i1 is the identity function.

In fact, this generalises to something called the filler of transport: transport p x and x are identical, but they’re identical over the given path:

transport-filler : ∀ {ℓ} {A B : Type ℓ}
                 → (p : A ≡ B) (x : A)
                 → PathP (λ i → p i) x (transport p x)
transport-filler p x i = transp (λ j → p (i ∧ j)) (~ i) x

transport-filler-ext : ∀ {ℓ} {A B : Type ℓ} (p : A ≡ B)
                     → PathP (λ i → A → p i) (λ x → x) (transport p)
transport-filler-ext p i x = transport-filler p x i

transport⁻-filler-ext : ∀ {ℓ} {A B : Type ℓ} (p : A ≡ B)
                     → PathP (λ i → p i → A) (λ x → x) (transport (sym p))
transport⁻-filler-ext p i x = transp (λ j → p (i ∧ ~ j)) (~ i) x

transport⁻transport : ∀ {ℓ} {A B : Type ℓ} (p : A ≡ B) (a : A)
                    → transport (sym p) (transport p a) ≡ a
transport⁻transport p a i =
  transport⁻-filler-ext p (~ i) (transport-filler-ext p (~ i) a)

The path is constant when i = i0 because (λ j → p (i0 ∧ j)) is (λ j → p i0) (by the reduction rules for _∧_). It has the stated endpoints, again, because transp P i1 is the identity function.

By altering a path p using a predicate P, we get the promised principle of indiscernibility of identicals:

subst : ∀ {ℓ₁ ℓ₂} {A : Type ℓ₁} (P : A → Type ℓ₂) {x y : A}
      → x ≡ y → P x → P y
subst P p x = transp (λ i → P (p i)) i0 x

Computation🔗

In “Book HoTT”, transport is defined using path induction, and it computes definitionally on refl. We have already seen that this is not definitional in cubical type theory, which might lead you to ask: When does transport compute? The answer is: By cases on the path. The structure of the path P is what guides reduction of transport. Here are some reductions:

For the natural numbers, and other inductive types without parameters, transport is always the identity function. This is justified because there’s nothing to vary in Nat, so we can just ignore the transport:

_ : {x : Nat} → transport (λ i → Nat) x ≡ x
_ = refl

For other type formers, the definition is a bit more involved. Let’s assume that we have two lines, A and B, to see how transport reduces in types built out of A and B:

module _ {A : I → Type} {B : I → Type} where private

For non-dependent products, the reduction rule says that “transport is homomorphic over forming products”:

  _ : {x : A i0} {y : B i0}
    → transport (λ i → A i × B i) (x , y)
    ≡ (transport (λ i → A i) x , transport (λ i → B i) y)
  _ = refl

For non-dependent functions, we have a similar situation, except one of the transports is backwards. This is because, given an f : A i0 → B i0, we have to turn an A i1 into an A i0 to apply f!

  _ : {f : A i0 → B i0}
    → transport (λ i → A i → B i) f
    ≡ λ x → transport (λ i → B i) (f (transport (λ i → A (~ i)) x))
  _ = refl

module _ {A : I → Type} {B : (i : I) → A i → Type} where private

In the dependent cases, we have slightly more work to do. Suppose that we have a line A : I → Type ℓ and a dependent line B : (i : I) → A i → Type ℓ. Let’s characterise transport in the lines (λ i → (x : A i) → B i x). A first attempt would be to repeat the non-dependent construction: Given an f : (x : A i0) → B i0 x and an argument x : A i1, we first get x' : A i0 by transporting along λ i → A (~ i), compute f x' : B i0 x, then transport along (λ i → B i x') to g- Wait.

  _ : {f : (x : A i0) → B i0 x}
    → transport (λ i → (x : A i) → B i x) f
    ≡ λ (x : A i1) →
        let
          x' : A i0
          x' = transport (λ i → A (~ i)) x

We can’t “transport along (λ i → B i x')”, that’s not even a well-formed type! Indeed, B i : A i → Type, but x' : A i1. What we need is some way of connecting our original x and x', so that we may get a B i1 x'. This is where transport-filler comes in:

          x≡x' : PathP (λ i → A (~ i)) x x'
          x≡x' = transport-filler (λ i → A (~ i)) x

By using λ i → B i (x≡x' (~ i)) as our path, we a) get something type-correct, and b) get something with the right endpoints. (λ i → B i (x≡x' (~ i))) connects B i0 x and B i1 x', which is what we wanted.

          fx' : B i0 x'
          fx' = f x'
        in transport (λ i → B i (x≡x' (~ i))) fx'
  _ = refl

The case for dependent products (i.e. general Σ types) is analogous, but without any inverse transports.

Path Induction🔗

The path induction principle, also known as “axiom J”, essentially breaks down as the following two statements:

• Identicals are indiscernible (transport)

• Singletons are contractible. The type Singleton A x is the “subtype of A of the elements identical to x”:

Singleton : ∀ {ℓ} {A : Type ℓ} → A → Type _
Singleton x = Σ[ y ∈ _ ] (x ≡ y)

There is a canonical inhabitant of Singleton x, namely (x, refl). To say that singletons are contractible is to say that every other inhabitant has a path to (x, refl):

Singleton-is-contr : ∀ {ℓ} {A : Type ℓ} {x : A} (y : Singleton x)
                   → Path (Singleton x) (x , refl) y
Singleton-is-contr {x = x} (y , path) i = path i , square i where
  square : Square refl refl path path
  square i j = path (i ∧ j)

Thus, the definition of J: transport + Singleton-is-contr.

J : ∀ {ℓ₁ ℓ₂} {A : Type ℓ₁} {x : A}
    (P : (y : A) → x ≡ y → Type ℓ₂)
  → P x (λ _ → x)
  → {y : A} (p : x ≡ y)
  → P y p
J {x = x} P prefl {y} p = transport (λ i → P (path i .fst) (path i .snd)) prefl where
  path : (x , refl) ≡ (y , p)
  path = Singleton-is-contr (y , p)

This eliminator doesn’t definitionally compute to prefl when p is refl, again since transport (λ i → A) isn’t definitionally the identity. However, since it is a transport, we can use the transport-filler to get a path expressing the computation rule.

J-refl : ∀ {ℓ₁ ℓ₂} {A : Type ℓ₁} {x : A}
           (P : (y : A) → x ≡ y → Type ℓ₂)
       → (pxr : P x refl)
       → J P pxr refl ≡ pxr
J-refl {x = x} P prefl i = transport-filler (λ i → P _ (λ j → x)) prefl (~ i)

Composition🔗

In “Book HoTT”, the primitive operation from which the higher-dimensional structure of types is derived is the J eliminator, with J-refl as a definitional computation rule. This has the benefit of being very elegant: This one elimination rule generates an infinite amount of coherent data. However, it’s very hard to make compute in the presence of higher inductive types and univalence, so much so that, in the book, univalence and HITs only compute up to paths.

In Cubical Agda, types are interpreted as objects called cubical Kan complexes⁴, which are a geometric description of spaces as “sets we can probe by cubes”. In Agda, this “probing” is reflected by mapping the interval into a type: A “probe” of $A$ by an $n$-cube is a term of type $A$ in a context with $n$ variables of type I — points, lines, squares, cubes, etc. This structure lets us “explore” the higher dimensional structure of a type, but it does not specify how this structure behaves.

That’s where the “Kan” part of “cubical Kan complex” comes in: Semantically, every open box extends to a cube. The concept of “open box” might make even less sense than the concept of “cube in a type” initially, so it helps to picture them! Suppose we have three paths $p : w \equiv x$, $q : x \equiv y$, and $r : y \equiv z$. We can pictorially arrange them into an open box like in the diagram below, by joining the paths by their common endpoints:

In the diagram above, we have a square assembled of three lines $w \equiv x$, $x \equiv y$, and $y \equiv z$. Note that in the left face of the diagram, the path was inverted; This is because while we have a path $w \equiv x$, we need a path $x \equiv w$, and all parallel faces of a cube must “point” in the same direction. The way the diagram is drawn strongly implies that there is a face missing — the line $w \equiv z$. The interpretation of types as Kan cubical sets guarantees that the open box above extends to a complete square, and thus the line $w \equiv z$ exists.

Partial Elements🔗

The definition of Kan cubical sets as those having fillers for all open boxes is all well and good, but to use this from within type theory we need a way of reflecting the idea of “open box” as syntax. This is done is by using the Partial type former.

The Partial type former takes two arguments: A formula $\varphi$, and a type $A$. The idea is that a term of type $\mathrm{Partial}\ \varphi\ A$ in a context with $n$ I-typed variables is a $n$-cube that is only defined when $\varphi$ “is true”. In Agda, formulas are represented using the De Morgan structure of the interval, and they are “true” when they are equal to 1. The predicate IsOne represents truth of a formula, and there is a canonical inhabitant 1=1 which says i1 is i1.

For instance, if we have a variable i : I of interval type, we can represent disjoint endpoints of a Path by a partial element with formula $\neg i \lor i$. Note that this is not the same thing as i1! Since elements of I are meant to represent real numbers $r \in [0,1]$, it suffices to find one for which $\max(x, 1 - x)$ is not $1$ — like 0.5.

private
  not-a-path : (i : I) → Partial (~ i ∨ i) Bool
  not-a-path i (i = i0) = true
  not-a-path i (i = i1) = false

This represents the following shape: Two disconnected points, with completely unrelated values at each endpoint of the interval.

More concretely, an element of Partial can be understood as a function where the domain is the predicate IsOne, which has an inhabitant 1=1, stating that one is one. Indeed, we can apply a Partial to an argument of type IsOne to get a value of the underlying type.

  _ : not-a-path i0 1=1 ≡ true
  _ = refl

Note that if we did have (~i ∨ i) = i1 (i.e. our De Morgan algebra was a Boolean algebra), the partial element above would give us a contradiction, since any I → Partial i1 T extends to a path:

  _ : (f : I → Partial i1 Bool) → Path Bool (f i0 1=1) (f i1 1=1)
  _ = λ f i → f i 1=1

Extensibility🔗

A partial element in a context with $n$-variables gives us a way of mapping some subobject of the $n$-cube into a type. A natural question to ask, then, is: Given a partial element $e$ of $A$, can we extend that to a honest-to-god element of $A$, which agrees with $e$ where it is defined?

Specifically, when this is the case, we say that $x : A$ extends $e : \mathrm{Partial}\ \varphi\ A$. We could represent this very generically as a lifting problem, i.e. trying to find a map $\square^n$ which agrees with $e$ when restricted to $\varphi$, but I believe a specific example will be more helpful.

Suppose we have a partial element of Bool which is true on the left endpoint of the interval, and undefined elsewhere. This is a partial element with one interval variable, so it would be extended by a path — a 1-dimensional cube. The reflexivity path is a line in Bool, which is true on the left endpoint of the interval (in fact, it is true everywhere), so we say that refl extends the partial element.

In the diagram, we draw the specific partial element being extended in red, and the total path extending it in black. In Agda, extensions are represented by the type former _[_↦_].⁵

We can formalise the red-black extensibility diagram above by defining the partial element left-true and giving refl to inS, the constructor for _[_↦_].

private
  left-true : (i : I) → Partial (~ i) Bool
  left-true i (i = i0) = true

  refl-extends : (i : I) → Bool [ (~ i) ↦ left-true i ]
  refl-extends i = inS (refl {x = true} i)

The constructor inS expresses that any totally-defined cube $u$ can be seen as a partial cube, one that agrees with $u$ for any choice of formula $\varphi$. This might be a bit abstract, so let’s diagram the case where we have some square $a$, and the partial element has formula $i \lor j$. This extension can be drawn as in the diagram below: The red “backwards L” shape is the partial element, which is “extended by” the black lines to make a complete square.

  _ : ∀ {ℓ} {A : Type ℓ} {φ : I} (u : A) → A [ φ ↦ (λ _ → u) ]
  _ = inS

Note that since an extension must agree with the partial element everywhere, there are elements that can not be extended at all. Take notAPath from before — since there is no path that is true at i0 and false at i1, it is not extensible. If it were extensible, we would have true ≡ false — a contradiction.⁶

  not-extensible : ((i : I) → Bool [ (~ i ∨ i) ↦ not-a-path i ]) → true ≡ false
  not-extensible ext i = outS (ext i)

This counterexample demonstrates the eliminator for _[_↦_], outS, which turns an A [ φ ↦ u ] to A, with a computation rule saying that, for x : A [ i1 ↦ u ], outS x computes to u 1=1:

  _ : ∀ {A : Type} {u : Partial i1 A} {x : A [ i1 ↦ u ]}
    → outS x ≡ u 1=1
  _ = refl

The notion of partial elements and extensibility captures the specific interface of the Kan operations, which can be summed up in the following sentence: If a partial path is extensible at i0, then it is extensible at i1. Let’s unpack that a bit:

A partial path is anything of type I → Partial φ A – let’s say we have an f in that type. It takes a value at i0 (that’s f i0), and a value at i1. The Kan condition expresses that, if there exists an A [ φ ↦ f i0 ], then we also have an A [ φ ↦ f i1 ]. In other words: Extensibility is preserved by paths.

Recall the open box we drew by gluing paths together at the start of the section (on the left). It has a top face q, and it has a tube — its left/right faces, which can be considered as a partial (in the left-right direction) path going in the top-down direction.

We can make this the construction of this “open box” formal by giving a Partial element of A, which is defined on $\partial i \lor \neg j$ — where $i$ represents the left/right direction, and $j$ the up/down direction, as is done below. So, this is an element that is defined almost everywhere: all three out of four faces of the square exist, but we’re missing the fourth face and an inside.

module _ {A : Type} {w x y z : A} {p : w ≡ x} {q : x ≡ y} {r : y ≡ z} where private
  double-comp-tube : (i j : I) → Partial (~ i ∨ i ∨ ~ j) A
  double-comp-tube i j (i = i0) = sym p j
  double-comp-tube i j (i = i1) = r j
  double-comp-tube i j (j = i0) = q i

The Kan condition on types says that, whenever we have some formula $\phi$ and a partial element $u$ defined along $\phi \lor \neg j$ (for $j$ disjoint from $\phi$; we call it the “direction of composition”, sometimes), then we can extend it to a totally-defined element, which agrees with $u$ along $\phi$.

The idea is that the $\neg j$, being in some sense “orthogonal to” the dimensions in $\phi$, will “connect” the tube given by $\phi$. This is a slight generalization of the classical Kan condition, which would insist $\phi = \bigvee_i \partial i$, where $i$ ranges over all dimensions in the context.

  extensible-at-i1 : (i : I) → A [ (i ∨ ~ i) ↦ double-comp-tube i i1 ]
  extensible-at-i1 i = inS $ₛ hcomp (∂ i) (double-comp-tube i)

Unwinding what it means for this element to exist, we see that the hcomp operation guarantees the existence of a path $w \to z$. It is the face that is hinted at by completing the open box above to a complete square.

  double-comp : w ≡ z
  double-comp i = outS (extensible-at-i1 i)

Note that hcomp gives us the missing face of the open box, but the semantics guarantees the existence of the box itself, as an $n$-cube. From the De Morgan structure on the interval, we can derive the existence of the cubes themselves (called fillers) from the existence of the missing faces:

hfill : ∀ {ℓ} {A : Type ℓ} (φ : I) → I
      → ((i : I) → Partial (φ ∨ ~ i) A)
      → A
hfill φ i u =
  hcomp (φ ∨ ~ i) λ where
    j (φ = i1) → u (i ∧ j) 1=1
    j (i = i0) → u i0 1=1
    j (j = i0) → u i0 1=1

Note: While every inhabitant of Type has a composition operation, not every type (something that can be on the right of type signature e : T) does. We call the types that do have a composition operation “fibrant”, since these are semantically the cubical sets which are Kan complices. Examples of types which are not fibrant include the interval I, the partial elements Partial, and the extensions _[_↦_]⁷.

Agda also provides a heterogeneous version of composition (sometimes referred to as “CCHM composition”), called comp. It too has a corresponding filling operation, called fill. The idea behind CCHM composition is — by analogy with hcomp expressing that “paths preserve extensibility” — that PathPs preserve extensibility. Thus we have:

fill : ∀ {ℓ : I → Level} (A : ∀ i → Type (ℓ i)) (φ : I) (i : I)
     → (u : ∀ i → Partial (φ ∨ ~ i) (A i))
     → A i
fill A φ i u = comp (λ j → A (i ∧ j)) (φ ∨ ~ i) λ where
  j (φ = i1) → u (i ∧ j) 1=1
  j (i = i0) → u i0 1=1
  j (j = i0) → u i0 1=1

Given the inputs to a composition — a family of partial paths u and a base u0 — hfill connects the input of the composition (u0) and the output. The cubical shape of iterated identifications causes a slight oddity: The only unbiased definition of path composition we can give is double composition, which corresponds to the missing face for the square at the start of this section.

_··_··_ : ∀ {ℓ} {A : Type ℓ} {w x y z : A}
        → w ≡ x → x ≡ y → y ≡ z
        → w ≡ z
(p ·· q ·· r) i = hcomp (∂ i) λ where
  j (i = i0) → p (~ j)
  j (i = i1) → r j
  j (j = i0) → q i

Since it will be useful later, we also give an explicit name for the filler of the double composition square.

··-filler : ∀ {ℓ} {A : Type ℓ} {w x y z : A}
          → (p : w ≡ x) (q : x ≡ y) (r : y ≡ z)
          → Square (sym p) q (p ·· q ·· r) r
··-filler p q r i j =
  hfill (∂ j) i λ where
    k (j = i0) → p (~ k)
    k (j = i1) → r k
    k (k = i0) → q j

We can define the ordinary, single composition by taking p = refl, as is done below. The square associated with the binary composition operation is obtained as the same open box at the start of the section, the same double-comp-tube, but by setting any of the faces to be reflexivity. For definiteness, we chose the left face:

_∙_ : ∀ {ℓ} {A : Type ℓ} {x y z : A}
    → x ≡ y → y ≡ z → x ≡ z
p ∙ q = refl ·· p ·· q

The ordinary, “single composite” of $p$ and $q$ is the dashed face in the diagram above. Since we bound ··-filler above, and defined _∙_ in terms of _··_··_, we can reuse the latter’s filler to get one for the former:

∙-filler : ∀ {ℓ} {A : Type ℓ} {x y z : A}
         → (p : x ≡ y) (q : y ≡ z)
         → Square refl p (p ∙ q) q
∙-filler {x = x} {y} {z} p q = ··-filler refl p q

The single composition has a filler “in the other direction”, which connects $q$ and $p \bullet q$. This is, essentially, because the choice of setting the left face to refl was completely arbitrary in the definition of _∙_: we could just as well have gone with setting the right face to refl.

∙-filler' : ∀ {ℓ} {A : Type ℓ} {x y z : A}
          → (p : x ≡ y) (q : y ≡ z)
          → Square (sym p) q (p ∙ q) refl
∙-filler' {x = x} {y} {z} p q j i =
  hcomp (∂ i ∨ ~ j) λ where
    k (i = i0) → p (~ j)
    k (i = i1) → q k
    k (j = i0) → q (i ∧ k)
    k (k = i0) → p (i ∨ ~ j)

Uniqueness🔗

A common characteristic of geometric interpretations of higher categories — like the one we have here — when compared to algebraic definitions is that there is no prescription in general for how to find composites of morphisms. Instead, we have that each triple of morphism has a contractible space of composites. We call the proof of this fact ··-unique:

··-unique : ∀ {ℓ} {A : Type ℓ} {w x y z : A}
          → (p : w ≡ x) (q : x ≡ y) (r : y ≡ z)
          → (α β : Σ[ s ∈ (w ≡ z) ] Square (sym p) q s r)
          → α ≡ β

Note that the type of α and β asks for a path w ≡ z which specifically completes the open box for double composition. We would not in general expect that w ≡ z is contractible for an arbitrary a! Note that the proof of this involves filling a cube in a context that already has an interval variable in scope - a hypercube!

··-unique {w = w} {x} {y} {z} p q r (α , α-fill) (β , β-fill) =
  λ i → (λ j → square i j) , (λ j k → cube i j k)
  where
    cube : (i j : I) → p (~ j) ≡ r j
    cube i j k = hfill (∂ i ∨ ∂ k) j λ where
      l (i = i0) → α-fill l k
      l (i = i1) → β-fill l k
      l (k = i0) → p (~ l)
      l (k = i1) → r l
      l (l = i0) → q k

    square : α ≡ β
    square i j = cube i i1 j

The term cube above has the following cube as a boundary. Since it is a filler, there is a missing face at the bottom which has no name, so we denote it by hcomp... in the diagram.

This diagram is quite busy because it is a 3D commutative diagram, but it could be busier: all of the unimportant edges were not annotated. By the way, the lavender face (including the lavender $p\ \neg j$) is the $k = \mathrm{i0}$ face, and the red face is the $k = \mathrm{i1}$ face.

However, even though the diagram is very busy, most of the detail it contains can be ignored. Reading it in the left-right direction, it expresses an identification between α-filler j k and β-filler j k, lying over a homotopy α = β. That homotopy is what you get when you read the bottom square of the diagram in the left-right direction. Explicitly, here is that bottom square:

Note that, exceptionally, this diagram is drawn with the left/right edges going up rather than down. This is to match the direction of the 3D diagram above. The colours are also matching.

Readers who are already familiar with the notion of h-level will have noticed that the proof ··-unique expresses that the type of double composites p ·· q ·· r is a proposition, not that it is contractible. However, since it is inhabited (by _··_··_ and its filler), it is contractible:

··-contract : ∀ {ℓ} {A : Type ℓ} {w x y z : A}
            → (p : w ≡ x) (q : x ≡ y) (r : y ≡ z)
            → (β : Σ[ s ∈ (w ≡ z) ] Square (sym p) q s r)
            → (p ·· q ·· r , ··-filler p q r) ≡ β
··-contract p q r β = ··-unique p q r _ β

Functorial Action🔗

This composition structure on paths makes every type into an $\infty$-groupoid, which is discussed in a different module.

It is then reasonable to expect that every function behave like a functor, in that it has an action on objects (the actual computational content of the function) and an action on morphisms — how that function acts on paths. Reading paths as identity, this is a proof that functions take identical inputs to identical outputs.

ap : ∀ {a b} {A : Type a} {B : A → Type b} (f : (x : A) → B x) {x y : A}
   → (p : x ≡ y) → PathP (λ i → B (p i)) (f x) (f y)
ap f p i = f (p i)
{-# NOINLINE ap #-}

The following function expresses the same thing as ap, but for binary functions. The type is huge! That’s because it applies to the most general type of 2-argument dependent function possible: (x : A) (y : B x) → C x y. Even then, the proof is beautifully short:

ap₂ : ∀ {a b c} {A : Type a} {B : A → Type b} {C : (x : A) → B x → Type c}
      (f : (x : A) (y : B x) → C x y)
      {x y : A} {α : B x} {β : B y}
    → (p : x ≡ y)
    → (q : PathP (λ i → B (p i)) α β)
    → PathP (λ i → C (p i) (q i))
            (f x α)
            (f y β)
ap₂ f p q i = f (p i) (q i)

This operation satisfies many identities definitionally that are only propositional when ap is defined in terms of J. For instance:

module _ where
  private variable
    ℓ : Level
    A B C : Type ℓ
    f : A → B
    g : B → C

  ap-comp : {x y : A} {p : x ≡ y}
          → ap (λ x → g (f x)) p ≡ ap g (ap f p)
  ap-comp = refl

  ap-id : {x y : A} {p : x ≡ y}
        → ap (λ x → x) p ≡ p
  ap-id = refl

  ap-sym : {x y : A} {p : x ≡ y}
         → sym (ap f p) ≡ ap f (sym p)
  ap-sym = refl

  ap-refl : {x : A} → ap f (λ i → x) ≡ (λ i → f x)
  ap-refl = refl

The last lemma, that ap respects composition of paths, can be proven by uniqueness: both ap f (p ∙ q) and ap f p ∙ ap f q are valid “lids” for the open box with sides refl, ap f p and ap f q, so they must be equal:

  ap-comp-path : (f : A → B) {x y z : A} (p : x ≡ y) (q : y ≡ z)
               → ap f (p ∙ q) ≡ ap f p ∙ ap f q
  ap-comp-path f p q i = ··-unique refl (ap f p) (ap f q)
    (ap f (p ∙ q)    , λ k j → f (∙-filler p q k j))
    (ap f p ∙ ap f q , ∙-filler _ _)
    i .fst

Syntax Sugar🔗

When constructing long chains of identifications, it’s rather helpful to be able to visualise what is being identified with more “priority” than how it is being identified. For this, a handful of combinators with weird names are defined:

≡⟨⟩-syntax : ∀ {ℓ} {A : Type ℓ} (x : A) {y z} → y ≡ z → x ≡ y → x ≡ z
≡⟨⟩-syntax x q p = p ∙ q

≡⟨⟩≡⟨⟩-syntax
  : ∀ {ℓ} {A : Type ℓ} (w x : A) {y z}
  → (p : w ≡ x)
  → (q : x ≡ y)
  → (r : y ≡ z)
  → w ≡ z
≡⟨⟩≡⟨⟩-syntax w x p q r = p ·· q ·· r

infixr  ≡⟨⟩-syntax
syntax ≡⟨⟩-syntax x q p = x ≡⟨ p ⟩ q

_≡˘⟨_⟩_ : ∀ {ℓ} {A : Type ℓ} (x : A) {y z : A} → y ≡ x → y ≡ z → x ≡ z
x ≡˘⟨ p ⟩≡˘ q = (sym p) ∙ q

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

_∎ : ∀ {ℓ} {A : Type ℓ} (x : A) → x ≡ x
x ∎ = refl

infixr  _∙_
infixr  _≡⟨⟩_ _≡˘⟨_⟩_
infix   _∎

along : ∀ {ℓ} {A : I → Type ℓ} {x : A i0} {y : A i1} → (i : I) → PathP A x y → A i
along i p = p i

These functions are used to make equational reasoning chains. For instance, the following proof that addition of naturals is associative is done in equational reasoning style:

private
  +-associative : (x y z : Nat) → (x + y) + z ≡ x + (y + z)
  +-associative zero y z = refl
  +-associative (suc x) y z =
    suc ((x + y) + z) ≡⟨ ap suc (+-associative x y z) ⟩≡
    suc (x + (y + z)) ∎

If your browser runs JavaScript, these equational reasoning chains, by default, render with the justifications (the argument written between ⟨ ⟩) hidden; There is a checkbox to display them, either on the sidebar or on the top bar depending on how narrow your screen is. For your convenience, it’s here too:

Try pressing it!

Dependent Paths🔗

Surprisingly often, we want to compare inhabitants $a : A$ and $b : B$ where the types $A$ and $B$ are not definitionally equal, but only identified in some specified way. We call these “paths over ppaths”, or PathP for short. In the same way that a Path can be understood as a function I → A with specified endpoints, a PathP (path over path) can be understood as a dependent function (i : I) → A i.

In the Book, paths over paths are implemented in terms of the transport operation: A path x ≡ y over p is a path transport p x ≡ y, thus defining dependent identifications using non-dependent ones. Fortunately, a cubical argument shows us that these notions coincide:

PathP≡Path : ∀ {ℓ} (P : I → Type ℓ) (p : P i0) (q : P i1)
           → PathP P p q ≡ Path (P i1) (transport (λ i → P i) p) q
PathP≡Path P p q i = PathP (λ j → P (i ∨ j)) (transport-filler (λ j → P j) p i) q

PathP≡Path⁻ : ∀ {ℓ} (P : I → Type ℓ) (p : P i0) (q : P i1)
            → PathP P p q ≡ Path (P i0) p (transport (λ i → P (~ i)) q)
PathP≡Path⁻ P p q i = PathP (λ j → P (~ i ∧ j)) p
                            (transport-filler (λ j → P (~ j)) q i)

We can see this by substituting either i0 or i1 for the variable i.

• When i = i0, we have PathP (λ j → P j) p q, by the endpoint rule for transport-filler.

• When i = i1, we have PathP (λ j → P i1) (transport P p) q, again by the endpoint rule for transport-filler.

The existence of paths over paths gives another “counterexample” to thinking of paths as equality. For instance, it’s hard to imagine a world in which true and false can be equal in any interesting sense of the word equal — but over the identification $\mathrm{Bool} \equiv \mathrm{Bool}$ that switches the points around, true and false can be identified!

Squeezing and spreading🔗

Using the De Morgan algebra structure on the interval, together with the “$\phi$” argument to transp, we can implement operations that move a point between the concrete endpoints (i0, i1) and arbitrary points on the interval, represented as variables. First, we introduce the following names for transporting forwards and backwards along a path.

coe0→1 : ∀ {ℓ} (A : I → Type ℓ) → A i0 → A i1
coe0→1 A a = transp (λ i → A i) i0 a

coe1→0 : ∀ {ℓ} (A : I → Type ℓ) → A i1 → A i0
coe1→0 A a = transp (λ i → A (~ i)) i0 a

These two operations will “spread” a value which is concentrated at one of the endpoints to cover the entire path. They are named after their type: they move a value from i0/i1 to an arbitrary $i$.

coe0→i : ∀ {ℓ : I → Level} (A : ∀ i → Type (ℓ i)) (i : I) → A i0 → A i
coe0→i A i a = transp (λ j → A (i ∧ j)) (~ i) a

coe1→i : ∀ {ℓ : I → Level} (A : ∀ i → Type (ℓ i)) (i : I) → A i1 → A i
coe1→i A i a = transp (λ j → A (i ∨ ~ j)) i a

In the converse direction, we have “squeeze” operations, which take a value from $A(i)$ to $A(i0)$ or $A(i1)$.

coei→0 : ∀ {ℓ : I → Level} (A : ∀ i → Type (ℓ i)) (i : I) → A i → A i0
coei→0 A i a = transp (λ j → A (i ∧ ~ j)) (~ i) a

coei→1 : ∀ {ℓ : I → Level} (A : ∀ i → Type (ℓ i)) (i : I) → A i → A i1
coei→1 A i a = transp (λ l → A (i ∨ l)) i a

Using squeezes and spreads, we can define maps that convert between PathPs and “book-style” dependent paths. These conversions could also be defined in terms of PathP≡Path, but the following definitions are more efficient.

module _ {ℓ} {A : I → Type ℓ} {x : A i0} {y : A i1} where
  to-pathp : coe0→1 A x ≡ y → PathP A x y
  to-pathp p i = hcomp (∂ i) λ where
    j (j = i0) → coe0→i A i x
    j (i = i0) → x
    j (i = i1) → p j

  from-pathp : PathP A x y → coe0→1 A x ≡ y
  from-pathp p i = transp (λ j → A (i ∨ j)) i (p i)

Note that by composing the functions to-pathp and to-pathp with the reversal on the interval, we obtain a correspondence PathP and paths with a backwards transport on the right-hand side.

module _ {ℓ} {A : I → Type ℓ} {x : A i0} {y : A i1} where
  to-pathp⁻ : x ≡ coe1→0 A y → PathP A x y
  to-pathp⁻ p = symP $ to-pathp {A = λ j → A (~ j)} (λ i → p (~ i))

  from-pathp⁻ : PathP A x y → x ≡ coe1→0 A y
  from-pathp⁻ p = sym $ from-pathp (λ i → p (~ i))

It’s actually fairly complicated to show that the functions to-pathp and from-pathp are inverses. The statements of the theorems are simple:

to-from-pathp
  : ∀ {ℓ} {A : I → Type ℓ} {x y} (p : PathP A x y) → to-pathp (from-pathp p) ≡ p

from-to-pathp
  : ∀ {ℓ} {A : I → Type ℓ} {x y} (p : coe0→1 A x ≡ y)
  → from-pathp {A = A} (to-pathp p) ≡ p

to-from-pathp {A = A} {x} {y} p i j = hcomp-unique (∂ j)
  (λ { k (k = i0) → coe0→i A j x
     ; k (j = i0) → x
     ; k (j = i1) → coei→1 A k (p k)
     })
  (λ k → inS (transp (λ l → A (j ∧ (k ∨ l))) (~ j ∨ k) (p (j ∧ k))))
  i

from-to-pathp {A = A} {x} {y} p i j =
  hcomp (∂ i ∨ ∂ j) λ where
    k (k = i0) →
      coei→1 A (j ∨ ~ i) $
        transp (λ l → A (j ∨ (~ i ∧ l))) (i ∨ j) $
          coe0→i A j x

    k (j = i0) → slide (k ∨ ~ i)
    k (j = i1) → p k

    k (i = i0) → coei→1 A j $ hfill (∂ j) k λ where
      k (k = i0) → coe0→i A j x
      k (j = i0) → x
      k (j = i1) → p k

    k (i = i1) → hcomp (∂ k ∨ ∂ j) λ where
      l (l = i0) → slide (k ∨ j)
      l (k = i0) → slide j
      l (k = i1) → p (j ∧ l)
      l (j = i0) → slide k
      l (j = i1) → p (k ∧ l)
  where
    slide : coe0→1 A x ≡ coe0→1 A x
    slide i = coei→1 A i (coe0→i A i x)

Path Spaces🔗

A large part of the study of HoTT is the characterisation of path spaces. Given a type A, what does Path A x y look like? Hedberg’s theorem says that for types with decidable equality, it’s boring. For the circle, we can prove its loop space is the integers — we have Path S¹ base base ≡ Int.

Most of these characterisations need machinery that is not in this module to be properly stated. Even then, we can begin to outline a few simple cases:

Σ Types🔗

For Σ types, a path between (a , b) ≡ (x , y) consists of a path p : a ≡ x, and a path between b and y laying over p.

Σ-pathp : ∀ {a b} {A : Type a} {B : A → Type b}
        → {x y : Σ A B}
        → (p : x .fst ≡ y .fst)
        → PathP (λ i → B (p i)) (x .snd) (y .snd)
        → x ≡ y
Σ-pathp p q i = p i , q i

We can also use the book characterisation of dependent paths, which is simpler in the case where the Σ represents a subset — i.e., B is a family of propositions.

Σ-path : ∀ {a b} {A : Type a} {B : A → Type b}
       → {x y : Σ A B}
       → (p : x .fst ≡ y .fst)
       → subst B p (x .snd) ≡ (y .snd)
       → x ≡ y
Σ-path {A = A} {B} {x} {y} p q = Σ-pathp p (to-pathp q)

Π types🔗

For dependent functions, the paths are homotopies, in the topological sense: Path ((x : A) → B x) f g is the same thing as a function I → (x : A) → B x — which we could turn into a product if we really wanted to.

happly : ∀ {a b} {A : Type a} {B : A → Type b}
         {f g : (x : A) → B x}
       → f ≡ g → (x : A) → f x ≡ g x
happly p x i = p i x

With this, we have made definitional yet another principle which is propositional in the HoTT book: function extensionality. Functions are identical precisely if they assign the same outputs to every input.

funext : ∀ {a b} {A : Type a} {B : A → Type b}
         {f g : (x : A) → B x}
       → ((x : A) → f x ≡ g x) → f ≡ g
funext p i x = p x i

Furthermore, we know (since types are groupoids, and functions are functors) that, by analogy with 1-category theory, paths in a function type should behave like natural transformations (because they are arrows in a functor category). This is indeed the case:

homotopy-natural : ∀ {a b} {A : Type a} {B : Type b}
                 → {f g : A → B}
                 → (H : (x : A) → f x ≡ g x)
                 → {x y : A} (p : x ≡ y)
                 → H x ∙ ap g p ≡ ap f p ∙ H y
homotopy-natural {f = f} {g = g} H {x} {y} p = ∙-unique _ λ i j →
  hcomp (~ i ∨ ∂ j) λ where
    k (k = i0) → H x (j ∧ i)
    k (i = i0) → f (p (j ∧ k))
    k (j = i0) → f x
    k (j = i1) → H (p k) i

Paths🔗

The groupoid structure of paths is also interesting. While the characterisation of Path (Path A x y) p q is fundamentally tied to the characterisation of A, there are general theorems that can be proven about transport in path spaces. For example, transporting both endpoints of a path is equivalent to a ternary composition:

transport-path : ∀ {ℓ} {A : Type ℓ} {x y x' y' : A}
                → (p : x ≡ y)
                → (left : x ≡ x') → (right : y ≡ y')
                → transport (λ i → left i ≡ right i) p ≡ sym left ∙ p ∙ right
transport-path {A = A} {x} {y} {x'} {y'} p left right =
  lemma ∙ double-composite _ _ _

The argument is slightly indirect. First, we have a proof (omitted for space) that composing three paths using binary composition (twice) is the same path as composing them in one go, using the (ternary) double composition operation. This is used in a second step, as a slight endpoint correction.

The first step, the lemma below, characterises transport in path spaces in terms of the double composite: This is almost definitional, but since Cubical Agda implements only composition for PathP, we need to adjust the path by a bunch of transports:

  where
    lemma : _ ≡ (sym left ·· p ·· right)
    lemma i j = hcomp (~ i ∨ ∂ j) λ where
      k (k = i0) → transp (λ j → A) i (p j)
      k (i = i0) → hfill (∂ j) k λ where
        k (k = i0) → transp (λ i → A) i0 (p j)
        k (j = i0) → transp (λ j → A) k (left k)
        k (j = i1) → transp (λ j → A) k (right k)
      k (j = i0) → transp (λ j → A) (k ∨ i) (left k)
      k (j = i1) → transp (λ j → A) (k ∨ i) (right k)

Special cases can be proven about substitution. For example, if we hold the right endpoint constant, we get something homotopic to composing with the inverse of the adjustment path:

subst-path-left : ∀ {ℓ} {A : Type ℓ} {x y x' : A}
                → (p : x ≡ y)
                → (left : x ≡ x')
                → subst (λ e → e ≡ y) left p ≡ sym left ∙ p
subst-path-left {y = y} p left =
  subst (λ e → e ≡ y) left p     ≡⟨⟩
  transport (λ i → left i ≡ y) p ≡⟨ transport-path p left refl ⟩≡
  sym left ∙ p ∙ refl            ≡⟨ ap (sym left ∙_) (sym (∙-filler _ _)) ⟩≡
  sym left ∙ p                   ∎

If we hold the left endpoint constant instead, we get a respelling of composition:

subst-path-right : ∀ {ℓ} {A : Type ℓ} {x y y' : A}
                 → (p : x ≡ y)
                 → (right : y ≡ y')
                 → subst (λ e → x ≡ e) right p ≡ p ∙ right
subst-path-right {x = x} p right =
  subst (λ e → x ≡ e) right p     ≡⟨⟩
  transport (λ i → x ≡ right i) p ≡⟨ transport-path p refl right ⟩≡
  sym refl ∙ p ∙ right            ≡⟨⟩
  refl ∙ p ∙ right                ≡⟨ sym (∙-filler' _ _) ⟩≡
  p ∙ right                       ∎

Finally, we can apply the same path to both endpoints:

subst-path-both : ∀ {ℓ} {A : Type ℓ} {x x' : A}
                → (p : x ≡ x)
                → (adj : x ≡ x')
                → subst (λ x → x ≡ x) adj p ≡ sym adj ∙ p ∙ adj
subst-path-both p adj = transport-path p adj adj