⚠️ This module is not part of this project and is only included for reference.
It is either part of the 1lab, the cubical library, or a built-in Agda module.
---
description: |
We make explicit the higher groupoid structure of type formers, which
is derived from their Kan operations.
---
<!--
```agda
open import 1Lab.Path hiding (_∙_)
open import 1Lab.Type
```
-->
```agda
module 1Lab.Path.Groupoid where
```
<!--
```agda
_ = Path
_ = hfill
_ = ap-refl
_ = ap-∙
_ = ap-sym
```
-->
# Types are groupoids
The `Path`{.Agda} types equip every `Type`{.Agda} with the structure of
an _$\infty$-groupoid_. The higher structure of a type begins with its
inhabitants (the 0-cells); Then, there are the paths between inhabitants
- these are inhabitants of the type `Path A x y`, which are the 1-cells
in `A`. Then, we can consider the inhabitants of `Path (Path A x y) p
q`, which are _homotopies_ between paths.
This construction iterates forever - and, at every stage, we have that
the $n$-cells in `A` are the 0-cells of some other type (e.g.: The
1-cells of `A` are the 0-cells of `Path A ...`). Furthermore, this
structure is _weak_: The laws, e.g. associativity of composition, only
hold up to a homotopy. These laws satisfy their own laws --- again using
associativity as an example, the associator satisfies the pentagon
identity.
These laws can be proved in two ways: Using path induction, or directly
with a cubical argument. Here, we do both.
## Book-style
This is the approach taken in the HoTT book. We fix a type, and some
variables of that type, and some paths between variables of that type,
so that each definition doesn't start with 12 parameters.
```agda
module WithJ where
private variable
ℓ : Level
A : Type ℓ
w x y z : A
```
First, we (re)define the operations using `J`{.Agda}. These will be closer to the
structure given in the book.
```agda
_∙_ : x ≡ y → y ≡ z → x ≡ z
_∙_ {x = x} {y} {z} = J (λ y _ → y ≡ z → x ≡ z) (λ x → x)
```
First we define path composition. Then, we can prove that the identity
path - `refl`{.Agda} - acts as an identity for path composition.
```agda
∙-idr : (p : x ≡ y) → p ∙ refl ≡ p
∙-idr {x = x} {y = y} p =
J (λ _ p → p ∙ refl ≡ p)
(happly (J-refl (λ y _ → y ≡ y → x ≡ y) (λ x → x)) _)
p
```
This isn't as straightforward as it would be in "Book HoTT" because -
remember - `J`{.Agda} doesn't compute definitionally, only up to the path
`J-refl`{.Agda}. Now the other identity law:
```agda
∙-idl : (p : y ≡ z) → refl ∙ p ≡ p
∙-idl {y = y} {z = z} p = happly (J-refl (λ y _ → y ≡ z → y ≡ z) (λ x → x)) p
```
This case we get for less since it's essentially the computation rule for `J`{.Agda}.
```agda
∙-assoc : (p : w ≡ x) (q : x ≡ y) (r : y ≡ z)
→ p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r
∙-assoc {w = w} {x = x} {y = y} {z = z} p q r =
J (λ x p → (q : x ≡ y) (r : y ≡ z) → p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r) lemma p q r
where
lemma : (q : w ≡ y) (r : y ≡ z)
→ (refl ∙ (q ∙ r)) ≡ ((refl ∙ q) ∙ r)
lemma q r =
(refl ∙ (q ∙ r)) ≡⟨ ∙-idl (q ∙ r) ⟩
q ∙ r ≡⟨ sym (ap (λ e → e ∙ r) (∙-idl q)) ⟩
(refl ∙ q) ∙ r ∎
```
The associativity rule is trickier to prove, since we do inductive where
the motive is a dependent product. What we're doing can be summarised
using words: By induction, it suffices to assume `p` is refl. Then, what
we want to show is `(refl ∙ (q ∙ r)) ≡ ((refl ∙ q) ∙ r)`. But both of
those compute to `q ∙ r`, so we are done. This computation isn't
automatic - it's expressed by the `lemma`{.Agda}.
This expresses that the paths behave like morphisms in a category. For a
groupoid, we also need inverses and cancellation:
```agda
inv : x ≡ y → y ≡ x
inv {x = x} = J (λ y _ → y ≡ x) refl
```
The operation which assigns inverses has to be involutive, which follows
from two computations.
```agda
inv-inv : (p : x ≡ y) → inv (inv p) ≡ p
inv-inv {x = x} =
J (λ y p → inv (inv p) ≡ p)
(ap inv (J-refl (λ y _ → y ≡ x) refl) ∙ J-refl (λ y _ → y ≡ x) refl)
```
And we have to prove that composing with an inverse gives the reflexivity path.
```agda
∙-invr : (p : x ≡ y) → p ∙ inv p ≡ refl
∙-invr {x = x} = J (λ y p → p ∙ inv p ≡ refl)
(∙-idl (inv refl) ∙ J-refl (λ y _ → y ≡ x) refl)
∙-invl : (p : x ≡ y) → inv p ∙ p ≡ refl
∙-invl {x = x} = J (λ y p → inv p ∙ p ≡ refl)
(∙-idr (inv refl) ∙ J-refl (λ y _ → y ≡ x) refl)
```
## Cubically
Now we do the same using `hfill`{.Agda} instead of path induction.
```agda
module _ where
private variable
ℓ : Level
A B : Type ℓ
w x y z : A
a b c d : A
open 1Lab.Path
```
<!--
```agda
∙-filler₂ : ∀ {ℓ} {A : Type ℓ} {x y z : A} (q : x ≡ y) (r : y ≡ z)
→ Square q (q ∙ r) r refl
∙-filler₂ q r k i = hcomp (k ∨ ∂ i) λ where
l (l = i0) → q (i ∨ k)
l (k = i1) → r (l ∧ i)
l (i = i0) → q k
l (i = i1) → r l
```
-->
The left and right identity laws follow directly from the two fillers
for the composition operation.
```agda
∙-idr : (p : x ≡ y) → p ∙ refl ≡ p
∙-idr p = sym (∙-filler p refl)
∙-idl : (p : x ≡ y) → refl ∙ p ≡ p
∙-idl p = sym (∙-filler' refl p)
```
For associativity, we use both:
```agda
∙-assoc : (p : w ≡ x) (q : x ≡ y) (r : y ≡ z)
→ p ∙ (q ∙ r) ≡ (p ∙ q) ∙ r
∙-assoc p q r i = ∙-filler p q i ∙ ∙-filler' q r (~ i)
```
For cancellation, we need to sketch an open cube where the missing
square expresses the equation we're looking for. Thankfully, we only
have to do this once!
```agda
∙-invr : ∀ {x y : A} (p : x ≡ y) → p ∙ sym p ≡ refl
∙-invr {x = x} p i j = hcomp (∂ j ∨ i) λ where
k (k = i0) → p (j ∧ ~ i)
k (i = i1) → x
k (j = i0) → x
k (j = i1) → p (~ k ∧ ~ i)
```
For the other direction, we use the fact that `p` is definitionally
equal to `sym (sym p)`. In that case, we show that `sym p ∙ sym (sym p)
≡ refl` - which computes to the thing we want!
```agda
∙-invl : (p : x ≡ y) → sym p ∙ p ≡ refl
∙-invl p = ∙-invr (sym p)
```
In addition to the groupoid identities for paths in a type, it has been
established that functions behave like functors: These are the lemmas
`ap-refl`{.Agda}, `ap-∙`{.Agda} and `ap-sym`{.Agda} in the
[1Lab.Path] module.
[1Lab.Path]: 1Lab.Path.html#functorial-action
### Convenient helpers
Since a _lot_ of Homotopy Type Theory is dealing with paths, this
section introduces useful helpers for dealing with $n$-ary compositions.
For instance, we know that $p^{-1} ∙ p ∙ q$ is $q$, but this involves
more than a handful of intermediate steps:
```agda
∙-cancell
: ∀ {ℓ} {A : Type ℓ} {x y z : A} (p : x ≡ y) (q : y ≡ z)
→ (sym p ∙ p ∙ q) ≡ q
∙-cancell {y = y} p q i j = hcomp (∂ i ∨ ∂ j) λ where
k (k = i0) → p (i ∨ ~ j)
k (i = i0) → ∙-filler (sym p) (p ∙ q) k j
k (i = i1) → q (j ∧ k)
k (j = i0) → y
k (j = i1) → ∙-filler₂ p q i k
∙-cancelr
: ∀ {ℓ} {A : Type ℓ} {x y z : A} (p : x ≡ y) (q : z ≡ y)
→ ((p ∙ sym q) ∙ q) ≡ p
∙-cancelr {x = x} {y = y} q p = sym $ ∙-unique _ λ i j →
∙-filler q (sym p) (~ i) j
commutes→square
: {p : w ≡ x} {q : w ≡ y} {s : x ≡ z} {r : y ≡ z}
→ p ∙ s ≡ q ∙ r
→ Square p q s r
commutes→square {p = p} {q} {s} {r} fill i j =
hcomp (∂ i ∨ ∂ j) λ where
k (k = i0) → fill j i
k (i = i0) → q (k ∧ j)
k (i = i1) → s (~ k ∨ j)
k (j = i0) → ∙-filler p s (~ k) i
k (j = i1) → ∙-filler₂ q r k i
square→commutes
: {p : w ≡ x} {q : w ≡ y} {s : x ≡ z} {r : y ≡ z}
→ Square p q s r → p ∙ s ≡ q ∙ r
square→commutes {p = p} {q} {s} {r} fill i j = hcomp (∂ i ∨ ∂ j) λ where
k (k = i0) → fill j i
k (i = i0) → ∙-filler p s k j
k (i = i1) → ∙-filler₂ q r (~ k) j
k (j = i0) → q (~ k ∧ i)
k (j = i1) → s (k ∨ i)
∙-cancel'-l : {x y z : A} (p : x ≡ y) (q r : y ≡ z)
→ p ∙ q ≡ p ∙ r → q ≡ r
∙-cancel'-l p q r sq = sym (∙-cancell p q) ·· ap (sym p ∙_) sq ·· ∙-cancell p r
∙-cancel'-r : {x y z : A} (p : y ≡ z) (q r : x ≡ y)
→ q ∙ p ≡ r ∙ p → q ≡ r
∙-cancel'-r p q r sq =
sym (∙-cancelr q (sym p))
·· ap (_∙ sym p) sq
·· ∙-cancelr r (sym p)
```
While [[connections]] give us degenerate squares where two adjacent faces are
some path and the other two are `refl`{.Agda}, it is sometimes also useful to
have a degenerate square with two pairs of equal adjacent faces.
We can build this using `hcomp`{.Agda} and more connections:
```agda
double-connection
: (p : a ≡ b) (q : b ≡ c)
→ Square p p q q
double-connection {b = b} p q i j = hcomp (∂ i ∨ ∂ j) λ where
k (k = i0) → b
k (i = i0) → p (j ∨ ~ k)
k (i = i1) → q (j ∧ k)
k (j = i0) → p (i ∨ ~ k)
k (j = i1) → q (i ∧ k)
```
This corresponds to the following diagram, which expresses the trivial equation
$p \bullet q \equiv p \bullet q$:
~~~{.quiver}
\[\begin{tikzcd}
a & b \\
b & c
\arrow["p", from=1-1, to=1-2]
\arrow["p"', from=1-1, to=2-1]
\arrow["q"', from=2-1, to=2-2]
\arrow["q", from=1-2, to=2-2]
\end{tikzcd}\]
~~~
# Groupoid structure of types (cont.)
A useful fact is that if $H$ is a homotopy $f \sim \id$, then it
"commutes with $f$", in the following sense:
<!--
```agda
open 1Lab.Path
```
-->
```agda
homotopy-invert
: ∀ {a} {A : Type a} {f : A → A}
→ (H : (x : A) → f x ≡ x) {x : A}
→ H (f x) ≡ ap f (H x)
```
The proof proceeds entirely using $H$ itself, together with the De
Morgan algebra structure on the interval.
```agda
homotopy-invert {f = f} H {x} i j = hcomp (∂ i ∨ ∂ j) λ where
k (k = i0) → H x (j ∧ i)
k (j = i0) → H (f x) (~ k)
k (j = i1) → H x (~ k ∧ i)
k (i = i0) → H (f x) (~ k ∨ j)
k (i = i1) → H (H x j) (~ k)
```