Library HoTT.PathGroupoids

The groupid structure of paths


Require Import Overture.

Local Open Scope path_scope.

Naming conventions

We need good naming conventions that allow us to name theorems without looking them up. The names should indicate the structure of the theorem, but they may sometimes be ambiguous, in which case you just have to know what is going on. We shall adopt the following principles:
  • we are not afraid of long names
  • we are not afraid of short names when they are used frequently
  • we use underscores
  • name of theorems and lemmas are lower-case
  • records and other types may be upper or lower case
Theorems about concatenation of paths are called concat_XXX where XXX tells us what is on the left-hand side of the equation. You have to guess the right-hand side. We use the following symbols in XXX:
  • 1 means the identity path
  • p means 'the path'
  • V means 'the inverse path'
  • A means 'ap'
  • M means the thing we are moving across equality
  • x means 'the point' which is not a path, e.g. in transport p x
  • 2 means relating to 2-dimensional paths
  • 3 means relating to 3-dimensional paths, and so on
Associativity is indicated with an underscore. Here are some examples of how the name gives hints about the left-hand side of the equation.
Laws about inverse of something are of the form inv_XXX, and those about ap are of the form ap_XXX, and so on. For example:
Then we have laws which move things around in an equation. The naming scheme here is moveD_XXX. The direction D indicates where to move to: L means that we move something to the left-hand side, whereas R means we are moving something to the right-hand side. The part XXX describes the shape of the side from which we are moving where the thing that is getting moves is called M. The presence of 1 next to an M generally indicates an *implied* identity path which is inserted automatically after the movement. Examples:
  • moveL_pM means that we transform p = q @ r to p @ r^ = q because we are moving something to the left-hand side, and we are moving the right argument of concat.
  • moveR_Mp means that we transform p @ q = r to q = p^ @ r because we move to the right-hand side, and we are moving the left argument of concat.
  • moveR_1M means that we transform p = q (rather than p = 1 @ q) to p × q^ = 1.
There are also cancellation laws called cancelR and cancelL, which are inverse to the 2-dimensional 'whiskering' operations whiskerR and whiskerL.
We may now proceed with the groupoid structure proper.

The 1-dimensional groupoid structure.

The identity path is a right unit.
Definition concat_p1 {A : Type} {x y : A} (p : x = y) :
  p @ 1 = p
  :=
  match p with idpath ⇒ 1 end.

The identity is a left unit.
Definition concat_1p {A : Type} {x y : A} (p : x = y) :
  1 @ p = p
  :=
  match p with idpath ⇒ 1 end.

Concatenation is associative.
Definition concat_p_pp {A : Type} {x y z t : A} (p : x = y) (q : y = z) (r : z = t) :
  p @ (q @ r) = (p @ q) @ r :=
  match r with idpath
    match q with idpath
      match p with idpath ⇒ 1
      end end end.

Definition concat_pp_p {A : Type} {x y z t : A} (p : x = y) (q : y = z) (r : z = t) :
  (p @ q) @ r = p @ (q @ r) :=
  match r with idpath
    match q with idpath
      match p with idpath ⇒ 1
      end end end.

The left inverse law.
Definition concat_pV {A : Type} {x y : A} (p : x = y) :
  p @ p^ = 1
  :=
  match p with idpath ⇒ 1 end.

The right inverse law.
Definition concat_Vp {A : Type} {x y : A} (p : x = y) :
  p^ @ p = 1
  :=
  match p with idpath ⇒ 1 end.

Several auxiliary theorems about canceling inverses across associativity. These are somewhat redundant, following from earlier theorems.

Definition concat_V_pp {A : Type} {x y z : A} (p : x = y) (q : y = z) :
  p^ @ (p @ q) = q
  :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition concat_p_Vp {A : Type} {x y z : A} (p : x = y) (q : x = z) :
  p @ (p^ @ q) = q
  :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition concat_pp_V {A : Type} {x y z : A} (p : x = y) (q : y = z) :
  (p @ q) @ q^ = p
  :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition concat_pV_p {A : Type} {x y z : A} (p : x = z) (q : y = z) :
  (p @ q^) @ q = p
  :=
  (match q as i return p, (p @ i^) @ i = p with
    idpath
    fun p
      match p with idpath ⇒ 1 end
  end) p.

Inverse distributes over concatenation
Definition inv_pp {A : Type} {x y z : A} (p : x = y) (q : y = z) :
  (p @ q)^ = q^ @ p^
  :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition inv_Vp {A : Type} {x y z : A} (p : y = x) (q : y = z) :
  (p^ @ q)^ = q^ @ p
  :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition inv_pV {A : Type} {x y z : A} (p : x = y) (q : z = y) :
  (p @ q^)^ = q @ p^.
Proof.
  destruct p. destruct q. reflexivity.
Defined.

Definition inv_VV {A : Type} {x y z : A} (p : y = x) (q : z = y) :
  (p^ @ q^)^ = q @ p.
Proof.
  destruct p. destruct q. reflexivity.
Defined.

Inverse is an involution.
Definition inv_V {A : Type} {x y : A} (p : x = y) :
  p^^ = p
  :=
  match p with idpath ⇒ 1 end.


Definition moveR_Mp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) :
  p = r^ @ q r @ p = q.
Proof.
  destruct r.
  intro h. exact (concat_1p _ @ h @ concat_1p _).
Defined.

Definition moveR_pM {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) :
  r = q @ p^ r @ p = q.
Proof.
  destruct p.
  intro h. exact (concat_p1 _ @ h @ concat_p1 _).
Defined.

Definition moveR_Vp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) :
  p = r @ q r^ @ p = q.
Proof.
  destruct r.
  intro h. exact (concat_1p _ @ h @ concat_1p _).
Defined.

Definition moveR_pV {A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) :
  r = q @ p r @ p^ = q.
Proof.
  destruct p.
  intro h. exact (concat_p1 _ @ h @ concat_p1 _).
Defined.

Definition moveL_Mp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) :
  r^ @ q = p q = r @ p.
Proof.
  destruct r.
  intro h. exact ((concat_1p _)^ @ h @ (concat_1p _)^).
Defined.

Definition moveL_pM {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : y = x) :
  q @ p^ = r q = r @ p.
Proof.
  destruct p.
  intro h. exact ((concat_p1 _)^ @ h @ (concat_p1 _)^).
Defined.

Definition moveL_Vp {A : Type} {x y z : A} (p : x = z) (q : y = z) (r : x = y) :
  r @ q = p q = r^ @ p.
Proof.
  destruct r.
  intro h. exact ((concat_1p _)^ @ h @ (concat_1p _)^).
Defined.

Definition moveL_pV {A : Type} {x y z : A} (p : z = x) (q : y = z) (r : y = x) :
  q @ p = r q = r @ p^.
Proof.
  destruct p.
  intro h. exact ((concat_p1 _)^ @ h @ (concat_p1 _)^).
Defined.

Definition moveL_1M {A : Type} {x y : A} (p q : x = y) :
  p @ q^ = 1 p = q.
Proof.
  destruct q.
  intro h. exact ((concat_p1 _)^ @ h).
Defined.

Definition moveL_M1 {A : Type} {x y : A} (p q : x = y) :
  q^ @ p = 1 p = q.
Proof.
  destruct q.
  intro h. exact ((concat_1p _)^ @ h).
Defined.

Definition moveL_1V {A : Type} {x y : A} (p : x = y) (q : y = x) :
  p @ q = 1 p = q^.
Proof.
  destruct q.
  intro h. exact ((concat_p1 _)^ @ h).
Defined.

Definition moveL_V1 {A : Type} {x y : A} (p : x = y) (q : y = x) :
  q @ p = 1 p = q^.
Proof.
  destruct q.
  intro h. exact ((concat_1p _)^ @ h).
Defined.

Definition moveR_M1 {A : Type} {x y : A} (p q : x = y) :
  1 = p^ @ q p = q.
Proof.
  destruct p.
  intro h. exact (h @ (concat_1p _)).
Defined.

Definition moveR_1M {A : Type} {x y : A} (p q : x = y) :
  1 = q @ p^ p = q.
Proof.
  destruct p.
  intro h. exact (h @ (concat_p1 _)).
Defined.

Definition moveR_1V {A : Type} {x y : A} (p : x = y) (q : y = x) :
  1 = q @ p p^ = q.
Proof.
  destruct p.
  intro h. exact (h @ (concat_p1 _)).
Defined.

Definition moveR_V1 {A : Type} {x y : A} (p : x = y) (q : y = x) :
  1 = p @ q p^ = q.
Proof.
  destruct p.
  intro h. exact (h @ (concat_1p _)).
Defined.

Definition moveR_transport_p {A : Type} (P : A Type) {x y : A}
  (p : x = y) (u : P x) (v : P y)
  : u = p^ # v p # u = v.
Proof.
  destruct p.
  exact idmap.
Defined.

Definition moveR_transport_V {A : Type} (P : A Type) {x y : A}
  (p : y = x) (u : P x) (v : P y)
  : u = p # v p^ # u = v.
Proof.
  destruct p.
  exact idmap.
Defined.

Definition moveL_transport_V {A : Type} (P : A Type) {x y : A}
  (p : x = y) (u : P x) (v : P y)
  : p # u = v u = p^ # v.
Proof.
  destruct p.
  exact idmap.
Defined.

Definition moveL_transport_p {A : Type} (P : A Type) {x y : A}
  (p : y = x) (u : P x) (v : P y)
  : p^ # u = v u = p # v.
Proof.
  destruct p.
  exact idmap.
Defined.

Functoriality of functions

Here we prove that functions behave like functors between groupoids, and that ap itself is functorial.
Functions take identity paths to identity paths.
Definition ap_1 {A B : Type} (x : A) (f : A B) :
  ap f 1 = 1 :> (f x = f x)
  :=
  1.

Definition apD_1 {A B} (x : A) (f : x : A, B x) :
  apD f 1 = 1 :> (f x = f x)
  :=
  1.

Functions commute with concatenation.
Definition ap_pp {A B : Type} (f : A B) {x y z : A} (p : x = y) (q : y = z) :
  ap f (p @ q) = (ap f p) @ (ap f q)
  :=
  match q with
    idpath
    match p with idpath ⇒ 1 end
  end.

Definition ap_p_pp {A B : Type} (f : A B) {w x y z : A}
  (r : f w = f x) (p : x = y) (q : y = z) :
  r @ (ap f (p @ q)) = (r @ ap f p) @ (ap f q).
Proof.
  destruct p, q. simpl. exact (concat_p_pp r 1 1).
Defined.

Definition ap_pp_p {A B : Type} (f : A B) {w x y z : A}
  (p : x = y) (q : y = z) (r : f z = f w) :
  (ap f (p @ q)) @ r = (ap f p) @ (ap f q @ r).
Proof.
  destruct p, q. simpl. exact (concat_pp_p 1 1 r).
Defined.

Functions commute with path inverses.
Definition inverse_ap {A B : Type} (f : A B) {x y : A} (p : x = y) :
  (ap f p)^ = ap f (p^)
  :=
  match p with idpath ⇒ 1 end.

Definition ap_V {A B : Type} (f : A B) {x y : A} (p : x = y) :
  ap f (p^) = (ap f p)^
  :=
  match p with idpath ⇒ 1 end.

ap itself is functorial in the first argument.

Definition ap_idmap {A : Type} {x y : A} (p : x = y) :
  ap idmap p = p
  :=
  match p with idpath ⇒ 1 end.

Definition ap_compose {A B C : Type} (f : A B) (g : B C) {x y : A} (p : x = y) :
  ap (g o f) p = ap g (ap f p)
  :=
  match p with idpath ⇒ 1 end.

Definition ap_compose' {A B C : Type} (f : A B) (g : B C) {x y : A} (p : x = y) :
  ap (fun ag (f a)) p = ap g (ap f p)
  :=
  match p with idpath ⇒ 1 end.

The action of constant maps.
Definition ap_const {A B : Type} {x y : A} (p : x = y) (z : B) :
  ap (fun _z) p = 1
  :=
  match p with idpathidpath end.

Naturality of ap.
Definition concat_Ap {A B : Type} {f g : A B} (p : x, f x = g x) {x y : A} (q : x = y) :
  (ap f q) @ (p y) = (p x) @ (ap g q)
  :=
  match q with
    | idpathconcat_1p _ @ ((concat_p1 _) ^)
  end.

Naturality of ap at identity.
Definition concat_A1p {A : Type} {f : A A} (p : x, f x = x) {x y : A} (q : x = y) :
  (ap f q) @ (p y) = (p x) @ q
  :=
  match q with
    | idpathconcat_1p _ @ ((concat_p1 _) ^)
  end.

Definition concat_pA1 {A : Type} {f : A A} (p : x, x = f x) {x y : A} (q : x = y) :
  (p x) @ (ap f q) = q @ (p y)
  :=
  match q as i in (_ = y) return (p x @ ap f i = i @ p y) with
    | idpathconcat_p1 _ @ (concat_1p _)^
  end.

Naturality with other paths hanging around.
Definition concat_pA_pp {A B : Type} {f g : A B} (p : x, f x = g x)
  {x y : A} (q : x = y)
  {w z : B} (r : w = f x) (s : g y = z)
  :
  (r @ ap f q) @ (p y @ s) = (r @ p x) @ (ap g q @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_pA_p {A B : Type} {f g : A B} (p : x, f x = g x)
  {x y : A} (q : x = y)
  {w : B} (r : w = f x)
  :
  (r @ ap f q) @ p y = (r @ p x) @ ap g q.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_A_pp {A B : Type} {f g : A B} (p : x, f x = g x)
  {x y : A} (q : x = y)
  {z : B} (s : g y = z)
  :
  (ap f q) @ (p y @ s) = (p x) @ (ap g q @ s).
Proof.
  destruct q, s; cbn.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.

Definition concat_pA1_pp {A : Type} {f : A A} (p : x, f x = x)
  {x y : A} (q : x = y)
  {w z : A} (r : w = f x) (s : y = z)
  :
  (r @ ap f q) @ (p y @ s) = (r @ p x) @ (q @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_pp_A1p {A : Type} {g : A A} (p : x, x = g x)
  {x y : A} (q : x = y)
  {w z : A} (r : w = x) (s : g y = z)
  :
  (r @ p x) @ (ap g q @ s) = (r @ q) @ (p y @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_pA1_p {A : Type} {f : A A} (p : x, f x = x)
  {x y : A} (q : x = y)
  {w : A} (r : w = f x)
  :
  (r @ ap f q) @ p y = (r @ p x) @ q.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_A1_pp {A : Type} {f : A A} (p : x, f x = x)
  {x y : A} (q : x = y)
  {z : A} (s : y = z)
  :
  (ap f q) @ (p y @ s) = (p x) @ (q @ s).
Proof.
  destruct q, s; cbn.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.

Definition concat_pp_A1 {A : Type} {g : A A} (p : x, x = g x)
  {x y : A} (q : x = y)
  {w : A} (r : w = x)
  :
  (r @ p x) @ ap g q = (r @ q) @ p y.
Proof.
  destruct q; simpl.
  repeat rewrite concat_p1.
  reflexivity.
Defined.

Definition concat_p_A1p {A : Type} {g : A A} (p : x, x = g x)
  {x y : A} (q : x = y)
  {z : A} (s : g y = z)
  :
  p x @ (ap g q @ s) = q @ (p y @ s).
Proof.
  destruct q, s; simpl.
  repeat rewrite concat_p1, concat_1p.
  reflexivity.
Defined.

Action of apD10 and ap10 on paths.

Application of paths between functions preserves the groupoid structure

Definition apD10_1 {A} {B:AType} (f : x, B x) (x:A)
  : apD10 (idpath f) x = 1
:= 1.

Definition apD10_pp {A} {B:AType} {f f' f'' : x, B x}
  (h:f=f') (h':f'=f'') (x:A)
: apD10 (h @ h') x = apD10 h x @ apD10 h' x.
Proof.
  case h, h'; reflexivity.
Defined.

Definition apD10_V {A} {B:AType} {f g : x, B x} (h:f=g) (x:A)
  : apD10 (h^) x = (apD10 h x)^
:= match h with idpath ⇒ 1 end.

Definition ap10_1 {A B} {f:AB} (x:A) : ap10 (idpath f) x = 1
  := 1.

Definition ap10_pp {A B} {f f' f'':AB} (h:f=f') (h':f'=f'') (x:A)
  : ap10 (h @ h') x = ap10 h x @ ap10 h' x
:= apD10_pp h h' x.

Definition ap10_V {A B} {f g : AB} (h : f = g) (x:A)
  : ap10 (h^) x = (ap10 h x)^
:= apD10_V h x.

apD10 and ap10 also behave nicely on paths produced by ap
Definition apD10_ap_precompose {A B C} (f : A B) {g g' : x:B, C x} (p : g = g') a
: apD10 (ap (fun h : x:B, C xh oD f) p) a = apD10 p (f a).
Proof.
  destruct p; reflexivity.
Defined.

Definition ap10_ap_precompose {A B C} (f : A B) {g g' : B C} (p : g = g') a
: ap10 (ap (fun h : B Ch o f) p) a = ap10 p (f a)
:= apD10_ap_precompose f p a.

Definition apD10_ap_postcompose {A B C} (f : x, B x C) {g g' : x:A, B x} (p : g = g') a
: apD10 (ap (fun h : x:A, B xfun xf x (h x)) p) a = ap (f a) (apD10 p a).
Proof.
  destruct p; reflexivity.
Defined.

Definition ap10_ap_postcompose {A B C} (f : B C) {g g' : A B} (p : g = g') a
: ap10 (ap (fun h : A Bf o h) p) a = ap f (ap10 p a)
:= apD10_ap_postcompose (fun af) p a.

Transport and the groupoid structure of paths


Definition transport_1 {A : Type} (P : A Type) {x : A} (u : P x)
  : 1 # u = u
:= 1.

Definition transport_pp {A : Type} (P : A Type) {x y z : A} (p : x = y) (q : y = z) (u : P x) :
  p @ q # u = q # p # u :=
  match q with idpath
    match p with idpath ⇒ 1 end
  end.

Definition transport_pV {A : Type} (P : A Type) {x y : A} (p : x = y) (z : P y)
  : p # p^ # z = z
  := (transport_pp P p^ p z)^
  @ ap (fun rtransport P r z) (concat_Vp p).

Definition transport_Vp {A : Type} (P : A Type) {x y : A} (p : x = y) (z : P x)
  : p^ # p # z = z
  := (transport_pp P p p^ z)^
  @ ap (fun rtransport P r z) (concat_pV p).

In the future, we may expect to need some higher coherence for transport: for instance, that transport acting on the associator is trivial.
Definition transport_p_pp {A : Type} (P : A Type)
  {x y z w : A} (p : x = y) (q : y = z) (r : z = w)
  (u : P x)
  : ap (fun ee # u) (concat_p_pp p q r)
    @ (transport_pp P (p@q) r u) @ ap (transport P r) (transport_pp P p q u)
  = (transport_pp P p (q@r) u) @ (transport_pp P q r (p#u))
  :> ((p @ (q @ r)) # u = r # q # p # u) .
Proof.
  destruct p, q, r. simpl. exact 1.
Defined.

Definition transport_pVp {A} (P : A Type) {x y:A} (p:x=y) (z:P x)
  : transport_pV P p (transport P p z)
  = ap (transport P p) (transport_Vp P p z).
Proof.
  destruct p; reflexivity.
Defined.

Dependent transport in a doubly dependent type.

Definition transportD {A : Type} (B : A Type) (C : a:A, B a Type)
  {x1 x2 : A} (p : x1 = x2) (y : B x1) (z : C x1 y)
  : C x2 (p # y)
  :=
  match p with idpathz end.

Transporting along higher-dimensional paths

Definition transport2 {A : Type} (P : A Type) {x y : A} {p q : x = y}
  (r : p = q) (z : P x)
  : p # z = q # z
  := ap (fun p'p' # z) r.

An alternative definition.
Definition transport2_is_ap10 {A : Type} (Q : A Type) {x y : A} {p q : x = y}
  (r : p = q) (z : Q x)
  : transport2 Q r z = ap10 (ap (transport Q) r) z
  := match r with idpath ⇒ 1 end.

Definition transport2_p2p {A : Type} (P : A Type) {x y : A} {p1 p2 p3 : x = y}
  (r1 : p1 = p2) (r2 : p2 = p3) (z : P x)
  : transport2 P (r1 @ r2) z = transport2 P r1 z @ transport2 P r2 z.
Proof.
  destruct r1, r2; reflexivity.
Defined.

Definition transport2_V {A : Type} (Q : A Type) {x y : A} {p q : x = y}
  (r : p = q) (z : Q x)
  : transport2 Q (r^) z = (transport2 Q r z)^
  := match r with idpath ⇒ 1 end.

Definition concat_AT {A : Type} (P : A Type) {x y : A} {p q : x = y}
  {z w : P x} (r : p = q) (s : z = w)
  : ap (transport P p) s @ transport2 P r w
    = transport2 P r z @ ap (transport P q) s
  := match r with idpath ⇒ (concat_p1 _ @ (concat_1p _)^) end.

Lemma ap_transport {A} {P Q : A Type} {x y : A} (p : x = y) (f : x, P x Q x) (z : P x) :
  f y (p # z) = (p # (f x z)).
Proof.
  by induction p.
Defined.

Transporting in particular fibrations.

One frequently needs lemmas showing that transport in a certain dependent type is equal to some more explicitly defined operation, defined according to the structure of that dependent type. For most dependent types, we prove these lemmas in the appropriate file in the types/ subdirectory. Here we consider only the most basic cases.
Transporting in a constant fibration.
Definition transport_const {A B : Type} {x1 x2 : A} (p : x1 = x2) (y : B)
  : transport (fun xB) p y = y.
Proof.
  destruct p. exact 1.
Defined.

Definition transport2_const {A B : Type} {x1 x2 : A} {p q : x1 = x2}
  (r : p = q) (y : B)
  : transport_const p y = transport2 (fun _B) r y @ transport_const q y
  := match r with idpath(concat_1p _)^ end.

Transporting in a pulled back fibration.
Lemma transport_compose {A B} {x y : A} (P : B Type) (f : A B)
  (p : x = y) (z : P (f x))
  : transport (fun xP (f x)) p z = transport P (ap f p) z.
Proof.
  destruct p; reflexivity.
Defined.

Lemma transport_precompose {A B C} (f : A B) (g g' : B C) (p : g = g')
: transport (fun h : B Cg o f = h o f) p 1 =
  ap (fun hh o f) p.
Proof.
  destruct p; reflexivity.
Defined.

A special case of transport_compose which seems to come up a lot.
Definition transport_idmap_ap A (P : A Type) x y (p : x = y) (u : P x)
: transport P p u = transport idmap (ap P p) u
  := match p with idpathidpath end.

The behavior of ap and apD.

In a constant fibration, apD reduces to ap, modulo transport_const.
Lemma apD_const {A B} {x y : A} (f : A B) (p: x = y) :
  apD f p = transport_const p (f x) @ ap f p.
Proof.
  destruct p; reflexivity.
Defined.

The 2-dimensional groupoid structure

Horizontal composition of 2-dimensional paths.
Definition concat2 {A} {x y z : A} {p p' : x = y} {q q' : y = z} (h : p = p') (h' : q = q')
  : p @ q = p' @ q'
:= match h, h' with idpath, idpath ⇒ 1 end.

Notation "p @@ q" := (concat2 p q)%path (at level 20) : path_scope.

2-dimensional path inversion
Definition inverse2 {A : Type} {x y : A} {p q : x = y} (h : p = q)
  : p^ = q^
:= match h with idpath ⇒ 1 end.

Whiskering


Definition whiskerL {A : Type} {x y z : A} (p : x = y)
  {q r : y = z} (h : q = r) : p @ q = p @ r
:= 1 @@ h.

Definition whiskerR {A : Type} {x y z : A} {p q : x = y}
  (h : p = q) (r : y = z) : p @ r = q @ r
:= h @@ 1.

Unwhiskering, a.k.a. cancelling.


Lemma cancelL {A} {x y z : A} (p : x = y) (q r : y = z)
  : (p @ q = p @ r) (q = r).
Proof.
  destruct p, r.
  intro a.
  exact ((concat_1p q)^ @ a).
Defined.

Lemma cancelR {A} {x y z : A} (p q : x = y) (r : y = z)
  : (p @ r = q @ r) (p = q).
Proof.
  destruct r, p.
  intro a.
  exact (a @ concat_p1 q).
Defined.

Whiskering and identity paths.

Definition whiskerR_p1 {A : Type} {x y : A} {p q : x = y} (h : p = q) :
  (concat_p1 p) ^ @ whiskerR h 1 @ concat_p1 q = h
  :=
  match h with idpath
    match p with idpath
      1
    end end.

Definition whiskerR_1p {A : Type} {x y z : A} (p : x = y) (q : y = z) :
  whiskerR 1 q = 1 :> (p @ q = p @ q)
  :=
  match q with idpath ⇒ 1 end.

Definition whiskerL_p1 {A : Type} {x y z : A} (p : x = y) (q : y = z) :
  whiskerL p 1 = 1 :> (p @ q = p @ q)
  :=
  match q with idpath ⇒ 1 end.

Definition whiskerL_1p {A : Type} {x y : A} {p q : x = y} (h : p = q) :
  (concat_1p p) ^ @ whiskerL 1 h @ concat_1p q = h
  :=
  match h with idpath
    match p with idpath
      1
    end end.

Definition concat2_p1 {A : Type} {x y : A} {p q : x = y} (h : p = q) :
  h @@ 1 = whiskerR h 1 :> (p @ 1 = q @ 1)
  :=
  match h with idpath ⇒ 1 end.

Definition concat2_1p {A : Type} {x y : A} {p q : x = y} (h : p = q) :
  1 @@ h = whiskerL 1 h :> (1 @ p = 1 @ q)
  :=
  match h with idpath ⇒ 1 end.

The interchange law for concatenation.
Definition concat_concat2 {A : Type} {x y z : A} {p p' p'' : x = y} {q q' q'' : y = z}
  (a : p = p') (b : p' = p'') (c : q = q') (d : q' = q'') :
  (a @@ c) @ (b @@ d) = (a @ b) @@ (c @ d).
Proof.
  case d.
  case c.
  case b.
  case a.
  reflexivity.
Defined.

The interchange law for whiskering. Special case of concat_concat2.
Definition concat_whisker {A} {x y z : A} (p p' : x = y) (q q' : y = z) (a : p = p') (b : q = q') :
  (whiskerR a q) @ (whiskerL p' b) = (whiskerL p b) @ (whiskerR a q')
  :=
  match b with
    idpath
    match a with idpath
      (concat_1p _)^
    end
  end.

Structure corresponding to the coherence equations of a bicategory.
The "pentagonator": the 3-cell witnessing the associativity pentagon.
Definition pentagon {A : Type} {v w x y z : A} (p : v = w) (q : w = x) (r : x = y) (s : y = z)
  : whiskerL p (concat_p_pp q r s)
      @ concat_p_pp p (q@r) s
      @ whiskerR (concat_p_pp p q r) s
  = concat_p_pp p q (r@s) @ concat_p_pp (p@q) r s.
Proof.
  case p, q, r, s. reflexivity.
Defined.

The 3-cell witnessing the left unit triangle.
Definition triangulator {A : Type} {x y z : A} (p : x = y) (q : y = z)
  : concat_p_pp p 1 q @ whiskerR (concat_p1 p) q
  = whiskerL p (concat_1p q).
Proof.
  case p, q. reflexivity.
Defined.

The Eckmann-Hilton argument
Definition eckmann_hilton {A : Type} {x:A} (p q : 1 = 1 :> (x = x)) : p @ q = q @ p :=
  (whiskerR_p1 p @@ whiskerL_1p q)^
  @ (concat_p1 _ @@ concat_p1 _)
  @ (concat_1p _ @@ concat_1p _)
  @ (concat_whisker _ _ _ _ p q)
  @ (concat_1p _ @@ concat_1p _)^
  @ (concat_p1 _ @@ concat_p1 _)^
  @ (whiskerL_1p q @@ whiskerR_p1 p).

The action of functions on 2-dimensional paths

Definition ap02 {A B : Type} (f:AB) {x y:A} {p q:x=y} (r:p=q) : ap f p = ap f q
  := match r with idpath ⇒ 1 end.

Definition ap02_pp {A B} (f:AB) {x y:A} {p p' p'':x=y} (r:p=p') (r':p'=p'')
  : ap02 f (r @ r') = ap02 f r @ ap02 f r'.
Proof.
  case r, r'; reflexivity.
Defined.

Definition ap02_p2p {A B} (f:AB) {x y z:A} {p p':x=y} {q q':y=z} (r:p=p') (s:q=q')
  : ap02 f (r @@ s) = ap_pp f p q
                      @ (ap02 f r @@ ap02 f s)
                      @ (ap_pp f p' q')^.
Proof.
  case r, s, p, q. reflexivity.
Defined.

Definition apD02 {A : Type} {B : A Type} {x y : A} {p q : x = y}
  (f : x, B x) (r : p = q)
  : apD f p = transport2 B r (f x) @ apD f q
  := match r with idpath(concat_1p _)^ end.

Definition apD02_pp {A} (B : A Type) (f : x:A, B x) {x y : A}
  {p1 p2 p3 : x = y} (r1 : p1 = p2) (r2 : p2 = p3)
  : apD02 f (r1 @ r2)
  = apD02 f r1
  @ whiskerL (transport2 B r1 (f x)) (apD02 f r2)
  @ concat_p_pp _ _ _
  @ (whiskerR (transport2_p2p B r1 r2 (f x))^ (apD f p3)).
Proof.
  destruct r1, r2. destruct p1. reflexivity.
Defined.

Tactics, hints, and aliases

concat, with arguments flipped. Useful mainly in the idiom apply (concatR (expression)). Given as a notation not a definition so that the resultant terms are literally instances of concat, with no unfolding required.
Notation concatR := (fun p qconcat q p).

Hint Resolve
  concat_1p concat_p1 concat_p_pp
  inv_pp inv_V
 : path_hints.

Hint Rewrite
@concat_p1
@concat_1p
@concat_p_pp
@concat_pV
@concat_Vp
@concat_V_pp
@concat_p_Vp
@concat_pp_V
@concat_pV_p

@inv_V
@moveR_Mp
@moveR_pM
@moveL_Mp
@moveL_pM
@moveL_1M
@moveL_M1
@moveR_M1
@moveR_1M
@ap_1

@inverse_ap
@ap_idmap

@ap_const

@apD10_1
:paths.

Ltac hott_simpl :=
  autorewrite with paths in × |- × ; auto with path_hints.