Identity types

Content created by Fredrik Bakke, Egbert Rijke, Raymond Baker, Jonathan Prieto-Cubides and Vojtěch Štěpančík.

Created on 2022-02-04.
Last modified on 2024-04-17.

module foundation-core.identity-types where

open import foundation.universe-levels

Idea

The equality^¶ relation is defined in type theory by the identity type^¶. The identity type on a type A is a binary family of types

  Id : A → A → 𝒰

equipped with a reflexivity element^¶

  refl : (x : A) → Id x x.

In other words, the identity type is a reflexive type valued relation on A. Furthermore, the identity type on A satisfies the universal property that it maps uniquely into any other reflexive relation.

In type theory, we introduce the identity type as an inductive family of types, where the induction principle can be understood as expressing that the identity type is the least reflexive relation.

Notation of the identity type

We include two notations for the identity type. First, we introduce the identity type using Martin-Löf's original notation Id. Then we introduce as a secondary option the infix notation _＝_.

Note: The equals sign in the infix notation is not the standard equals sign on your keyboard, but it is the full width equals sign. Note that the full width equals sign is slightly wider, and it is highlighted like all the other defined constructions in Agda. In order to type the full width equals sign in Agda's Emacs Mode, you need to add it to your agda input method as follows:

• Type M-x customize-variable and press enter.
• Type agda-input-user-translations and press enter.
• Click the INS button
• Type the regular equals sign = in the Key sequence field.
• Click the INS button
• Type the full width equals sign ＝ in the translations field.
• Click the Apply and save button.

After completing these steps, you can type \= in order to obtain the full width equals sign ＝.

Table of files directly related to identity types

The following table lists files that are about identity types and operations on identifications in arbitrary types.

Definition

Identity types

We introduce identity types as a data type. This is Agda's mechanism of introducing types equipped with induction principles. The only constructor of the identity type Id x : A → 𝒰 is the reflexivity identification

  refl : Id x x.

module _
  {l : Level} {A : UU l}
  where

  data Id (x : A) : A → UU l where
    instance refl : Id x x

  infix  _＝_
  _＝_ : A → A → UU l
  (a ＝ b) = Id a b

{-# BUILTIN EQUALITY Id #-}

We marked refl as an instance to enable Agda to automatically insert refl in definitions that make use of Agda's instance search mechanism.

Furthermore, we marked the identity type as BUILTIN in order to support faster type checking.

The induction principle of identity types

The induction principle of identity types states that given a base point x : A and a family of types over the identity types based at x, B : (y : A) (p : x ＝ y) → UU l2, then to construct a dependent function f : (y : A) (p : x ＝ y) → B y p it suffices to define it at f x refl.

Note that Agda's pattern matching machinery allows us to define many operations on the identity type directly. However, sometimes it is useful to explicitly have the induction principle of the identity type.

ind-Id :
  {l1 l2 : Level} {A : UU l1}
  (x : A) (B : (y : A) (p : x ＝ y) → UU l2) →
  (B x refl) → (y : A) (p : x ＝ y) → B y p
ind-Id x B b y refl = b

Operations on the identity type

The identity types form a weak groupoidal structure on types. Thus they come equipped with concatenation^¶ (x ＝ y) → (y ＝ z) → (x ＝ z) and an inverse^¶ operation (x ＝ y) → (y ＝ x).

There are many more operations on identity types. Some of them are defined in path algebra and whiskering of identifications. For a complete reference to all the files about general identity types, see the table given above.

Concatenation of identifications

The concatenation operation on identifications^¶ is a family of binary operations

  _∙_ : x ＝ y → y ＝ z → x ＝ z

indexed by x y z : A. However, there are essentially three different ways we can define concatenation of identifications, all with different computational behaviours.

1. We can define concatenation by induction on the equality x ＝ y. This gives us the computation rule refl ∙ q ≐ q.
2. We can define concatenation by induction on the equality y ＝ z. This gives us the computation rule p ∙ refl ≐ p.
3. We can define concatenation by induction on both x ＝ y and y ＝ z. This only gives us the computation rule refl ∙ refl ≐ refl.

While the third option may be preferred by some for its symmetry, for practical reasons, we prefer one of the first two, and by convention we use the first alternative.

The second option is considered in foundation.strictly-right-unital-concatenation-identifications, and in foundation.yoneda-identity-types we construct an identity type which satisfies both computation rules among others.

module _
  {l : Level} {A : UU l}
  where

  infixl  _∙_
  _∙_ : {x y z : A} → x ＝ y → y ＝ z → x ＝ z
  refl ∙ q = q

  concat : {x y : A} → x ＝ y → (z : A) → y ＝ z → x ＝ z
  concat p z q = p ∙ q

  concat' : (x : A) {y z : A} → y ＝ z → x ＝ y → x ＝ z
  concat' x q p = p ∙ q

Inverting identifications

module _
  {l : Level} {A : UU l}
  where

  inv : {x y : A} → x ＝ y → y ＝ x
  inv refl = refl

Concatenating with inverse identifications

module _
  {l : Level} {A : UU l}
  where

  inv-concat : {x y : A} (p : x ＝ y) (z : A) → x ＝ z → y ＝ z
  inv-concat p = concat (inv p)

  inv-concat' : (x : A) {y z : A} → y ＝ z → x ＝ z → x ＝ y
  inv-concat' x q = concat' x (inv q)

Properties

Associativity of concatenation

For any three identifications p : x ＝ y, q : y ＝ z, and r : z ＝ w, we have an identification

  assoc p q r : ((p ∙ q) ∙ r) ＝ (p ∙ (q ∙ r)).

The identification assoc p q r is also called the associator^¶.

Note that the associator assoc p q r is an identification in the type x ＝ w, i.e., it is an identification of identifications. Here we make crucial use of the fact that the identity types are defined for all types. In other words, since identity types are themselves types, we can consider identity types of identity types, and so on.

Associators

module _
  {l : Level} {A : UU l}
  where

  assoc :
    {x y z w : A} (p : x ＝ y) (q : y ＝ z) (r : z ＝ w) →
    ((p ∙ q) ∙ r) ＝ (p ∙ (q ∙ r))
  assoc refl q r = refl

The unit laws for concatenation

For any identification p : x ＝ y there is an identification

  left-unit : (refl ∙ p) ＝ p.

Similarly, there is an identification

  right-unit : (p ∙ refl) ＝ p.

In other words, the reflexivity identification is a unit element for concatenation of identifications.

module _
  {l : Level} {A : UU l}
  where

  double-assoc :
    {x y z w v : A} (p : x ＝ y) (q : y ＝ z) (r : z ＝ w) (s : w ＝ v) →
    (((p ∙ q) ∙ r) ∙ s) ＝ p ∙ (q ∙ (r ∙ s))
  double-assoc refl q r s = assoc q r s

  triple-assoc :
    {x y z w v u : A}
    (p : x ＝ y) (q : y ＝ z) (r : z ＝ w) (s : w ＝ v) (t : v ＝ u) →
    ((((p ∙ q) ∙ r) ∙ s) ∙ t) ＝ p ∙ (q ∙ (r ∙ (s ∙ t)))
  triple-assoc refl q r s t = double-assoc q r s t

Unit laws

  left-unit : {x y : A} {p : x ＝ y} → refl ∙ p ＝ p
  left-unit = refl

  right-unit : {x y : A} {p : x ＝ y} → p ∙ refl ＝ p
  right-unit {p = refl} = refl

The inverse laws for concatenation

module _
  {l : Level} {A : UU l}
  where

  left-inv : {x y : A} (p : x ＝ y) → inv p ∙ p ＝ refl
  left-inv refl = refl

  right-inv : {x y : A} (p : x ＝ y) → p ∙ (inv p) ＝ refl
  right-inv refl = refl

Inverting identifications is an involution

module _
  {l : Level} {A : UU l}
  where

  inv-inv : {x y : A} (p : x ＝ y) → inv (inv p) ＝ p
  inv-inv refl = refl

Inverting identifications distributes over concatenation

module _
  {l : Level} {A : UU l}
  where

  distributive-inv-concat :
    {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) →
    inv (p ∙ q) ＝ inv q ∙ inv p
  distributive-inv-concat refl refl = refl

Concatenating with an inverse is inverse to concatenating

We show that the operation q ↦ inv p ∙ q is inverse to the operation q ↦ p ∙ q by constructing identifications

  inv p ∙ (p ∙ q) ＝ q
  p ∙ (inv p ∙ q) ＝ q.

Similarly, we show that the operation p ↦ p ∙ inv q is inverse to the operation p ↦ p ∙ q by constructing identifications

  (p ∙ q) ∙ inv q ＝ p
  (p ∙ inv q) ∙ q ＝ p.

In foundation.identity-types we will use these families of identifications to conclude that concat p z and concat' x q are equivalences with inverses concat (inv p) z and concat' x (inv q), respectively.

module _
  {l : Level} {A : UU l}
  where

  is-retraction-inv-concat :
    {x y z : A} (p : x ＝ y) (q : y ＝ z) → (inv p ∙ (p ∙ q)) ＝ q
  is-retraction-inv-concat refl q = refl

  is-section-inv-concat :
    {x y z : A} (p : x ＝ y) (r : x ＝ z) → (p ∙ (inv p ∙ r)) ＝ r
  is-section-inv-concat refl r = refl

  is-retraction-inv-concat' :
    {x y z : A} (q : y ＝ z) (p : x ＝ y) → (p ∙ q) ∙ inv q ＝ p
  is-retraction-inv-concat' refl refl = refl

  is-section-inv-concat' :
    {x y z : A} (q : y ＝ z) (r : x ＝ z) → (r ∙ inv q) ∙ q ＝ r
  is-section-inv-concat' refl refl = refl

Transposing inverses

Consider a triangle of identifications

      p
  x ----> y
   \     /
  r \   / q
     ∨ ∨
      z

in a type A. Then we have maps

  p ∙ q ＝ r → q ＝ inv p ∙ r
  p ∙ q ＝ r → p ＝ r ∙ inv q.

In foundation.identity-types we will show that these maps are equivalences.

module _
  {l : Level} {A : UU l}
  where

  left-transpose-eq-concat :
    {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) (r : x ＝ z) →
    p ∙ q ＝ r → q ＝ inv p ∙ r
  left-transpose-eq-concat refl q r s = s

  inv-left-transpose-eq-concat :
    {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) (r : x ＝ z) →
    q ＝ inv p ∙ r → p ∙ q ＝ r
  inv-left-transpose-eq-concat refl q r s = s

  right-transpose-eq-concat :
    {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) (r : x ＝ z) →
    p ∙ q ＝ r → p ＝ r ∙ inv q
  right-transpose-eq-concat p refl r s = (inv right-unit ∙ s) ∙ inv right-unit

  inv-right-transpose-eq-concat :
    {x y : A} (p : x ＝ y) {z : A} (q : y ＝ z) (r : x ＝ z) →
    p ＝ r ∙ inv q → p ∙ q ＝ r
  inv-right-transpose-eq-concat p refl r s = right-unit ∙ (s ∙ right-unit)

  double-transpose-eq-concat :
    {x y u v : A} (r : x ＝ u) (p : x ＝ y) (s : u ＝ v) (q : y ＝ v) →
    p ∙ q ＝ r ∙ s → inv r ∙ p ＝ s ∙ inv q
  double-transpose-eq-concat refl p s refl α =
    (inv right-unit ∙ α) ∙ inv right-unit

  double-transpose-eq-concat' :
    {x y u v : A} (r : x ＝ u) (p : x ＝ y) (s : u ＝ v) (q : y ＝ v) →
    p ∙ q ＝ r ∙ s → q ∙ inv s ＝ inv p ∙ r
  double-transpose-eq-concat' r refl refl q α = right-unit ∙ (α ∙ right-unit)

Splicing and unsplicing concatenations of identifications

Consider two identifications p : a ＝ b and q : b ＝ c, and consider two further identifications r : b ＝ x and s : x ＝ b equipped with an identification inv r ＝ s, as indicated in the diagram

           x
          ∧ |
        r | | s
          | ∨
  a -----> b -----> c.

Then we have identifications

    splice-concat : p ∙ q ＝ (p ∙ r) ∙ (s ∙ q)
  unsplice-concat : (p ∙ r) ∙ (s ∙ q) ＝ p ∙ q.

module _
  {l : Level} {A : UU l}
  where

  splice-concat :
    {a b c x : A}
    (p : a ＝ b) {r : b ＝ x} {s : x ＝ b} (α : inv r ＝ s) (q : b ＝ c) →
    p ∙ q ＝ (p ∙ r) ∙ (s ∙ q)
  splice-concat refl {r} refl q = inv (is-section-inv-concat r q)

  splice-concat' :
    {a b c x : A}
    (p : a ＝ b) {r : b ＝ x} {s : x ＝ b} (α : r ＝ inv s) (q : b ＝ c) →
    p ∙ q ＝ (p ∙ r) ∙ (s ∙ q)
  splice-concat' refl {.(inv s)} {s} refl q = inv (is-retraction-inv-concat s q)

  unsplice-concat :
    {a b c x : A}
    (p : a ＝ b) {r : b ＝ x} {s : x ＝ b} (α : inv r ＝ s) (q : b ＝ c) →
    (p ∙ r) ∙ (s ∙ q) ＝ p ∙ q
  unsplice-concat p α q = inv (splice-concat p α q)

  unsplice-concat' :
    {a b c x : A}
    (p : a ＝ b) {r : b ＝ x} {s : x ＝ b} (α : r ＝ inv s) (q : b ＝ c) →
    (p ∙ r) ∙ (s ∙ q) ＝ p ∙ q
  unsplice-concat' p α q = inv (splice-concat' p α q)

Concatenation is injective

module _
  {l1 : Level} {A : UU l1}
  where

  is-injective-concat :
    {x y z : A} (p : x ＝ y) {q r : y ＝ z} → p ∙ q ＝ p ∙ r → q ＝ r
  is-injective-concat refl s = s

  is-injective-concat' :
    {x y z : A} (r : y ＝ z) {p q : x ＝ y} → p ∙ r ＝ q ∙ r → p ＝ q
  is-injective-concat' refl s = (inv right-unit) ∙ (s ∙ right-unit)

Equational reasoning

Identifications can be constructed by equational reasoning in the following way:

equational-reasoning
  x ＝ y by eq-1
    ＝ z by eq-2
    ＝ v by eq-3

The resulting identification of this computaion is eq-1 ∙ (eq-2 ∙ eq-3), i.e., the identification is associated fully to the right. For examples of the use of equational reasoning, see addition-integers.

infixl  equational-reasoning_
infixl  step-equational-reasoning

equational-reasoning_ :
  {l : Level} {X : UU l} (x : X) → x ＝ x
equational-reasoning x = refl

step-equational-reasoning :
  {l : Level} {X : UU l} {x y : X} →
  (x ＝ y) → (u : X) → (y ＝ u) → (x ＝ u)
step-equational-reasoning p z q = p ∙ q

syntax step-equational-reasoning p z q = p ＝ z by q

Note. Equational reasoning is a convenient way to construct identifications. However, in some situations it may not be the fastest or cleanest mechanism to construct an identification. Some constructions of identifications naturally involve computations that are more deeply nested in the terms. Furthermore, proofs by equational reasoning tend to require a lot of reassociation.

Some tools that allow us to perform faster computations are the transpositions defined above, the transpositions and splicing operations defined in commuting squares of identifications and commuting triangles of identifications, and the higher concatenation operations defined in path algebra. Each of these operations has good computational behavior, so there is infrastructure for reasoning about identifications that are constructed using them.

We also note that there is similar infrastructure for homotopy reasoning.

References

Our setup of equational reasoning is derived from the following sources:

1. Martín Escardó. https://github.com/martinescardo/TypeTopology/blob/master/source/Id.lagda

2. Martín Escardó. https://github.com/martinescardo/TypeTopology/blob/master/source/UF-Equiv.lagda

3. The Agda standard library. https://github.com/agda/agda-stdlib/blob/master/src/Relation/Binary/PropositionalEquality/Core.agda

Recent changes

• 2024-04-17. Fredrik Bakke. Splitting idempotents (#1105).
• 2024-03-13. Egbert Rijke. Refactoring pointed types (#1056).
• 2024-03-12. Vojtěch Štěpančík. Fail the website build on malformed concept macro invocations (#1066).
• 2024-02-08. Fredrik Bakke. Computational identity types (#1015).
• 2024-02-06. Egbert Rijke and Fredrik Bakke. Refactor files about identity types and homotopies (#1014).