nLab equivalence relation

Definition

An equivalence relation on a set $S$ is a binary relation $\equiv$ on $S$ that is:

• reflexive: $x \equiv x$ for all $x: S$;
• symmetric: $x \equiv y$ if $y \equiv x$; and
• transitive: $x \equiv z$ if $x \equiv y \equiv z$.

Thus, an equivalence relation is a symmetric preorder.

(One can also define it as a relation that is both reflexive and euclidean.) The de Morgan dual of an equivalence relation is an apartness relation.

If $R$ is any relation on $S$, there is a smallest equivalence relation containing $R$ (noting that an intersection of an arbitrary family of equivalence relations is again such; see also Moore closure), called the equivalence relation generated by $R$.

In homotopy type theory

In homotopy type theory, since propositions are subsingletons, an equivalence relation on a type $T$ is a type family $x \equiv y$ indexed by elements $x:T$ and $y:T$ with witnesses of reflexivity, transitivity, and symmetry, and propositional truncation:

$$\mathrm{refl}(x):x \equiv x$$

$$\mathrm{trans}(x, y, z):x \equiv y \times y \equiv z \to x \equiv z$$

$$\mathrm{sym}(x, y):x \equiv y \to y \equiv x$$

$$\mathrm{proptrunc}(x, y):\prod_{p:x \equiv y} \prod_{q:x \equiv y} p = q$$

One then further distinguishes between a type with an equivalence relation (a symmetric preordered type) and an h-set with an equivalence relation (a symmetric proset).

Extensional functions

Let $A \coloneqq (A_0, \equiv_A)$ and $B:\coloneqq (B_0, \equiv_B)$ be two symmetric preordered types. An extensional function $f:A \to B$ consists of a function $f_0:A_0 \to B_0$ and for all elements $a:A_0$ and $b:A_0$ a function $f_\equiv(a, b):(a \equiv_A b) \to (f_0(a) \equiv_B f_0(b))$.

The identity function $id_A:A \to A$ is an extensional function which consists of the identity function $id_{A_0}:A_0 \to A_0$ for all elements $a:A_0$ and $b:A_0$, the identity function $id_\equiv(a, b):(a \equiv_A b) \to (a \equiv_A b))$.

Given symmetric preordered types $A \coloneqq (A_0, \equiv_A)$, $B:\coloneqq (B_0, \equiv_B)$, and $C \coloneqq (C_0, \equiv_C)$, the composition $g \circ f:A \to C$ of extensional functions $f:A \to B$ and $g:B \to C$ is defined as

$$(g \circ f)_0 \coloneqq g_0 \circ f_0$$

and for all objects $a:A_0$ and $b:A_0$,

$$(g \circ f)_\equiv(a, b) \coloneqq g_equiv(f_0(a), f_0(b)) \circ f_equiv(a, b)$$

Injections, surjections, and bijections

An extensional function $f:A \to B$ is a injection if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):f_0(a) \equiv b$ is a subsingleton.

An extensional function $f:A \to B$ is a surjection if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):f_0(a) \equiv b$ is a pointed type.

An extensional function $f:A \to B$ is a bijection if it is both an injection and a surjection; i.e. for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):f_0(a) \equiv b$ is a pointed subsingleton, or a singleton.

Bijections, isomorphisms, and equivalences

An extensional function $f:A \to B$ is a bijection if for all elements $b:B_0$ the type of elements $a:A_0$ with a witness $p(a, b):f_0(a) \equiv b$ is a singleton.

An extensional function $f:A \to B$ is an isomorphism if there exist an extensional functiom $g:B \to A$ with witnesses $\mathrm{ret}(f, g):g \circ f = id_{A}$ and $\mathrm{sec}(f, g):f \circ g = id_{B}$.

An extensional function $f:A \to B$ is an equivalence if the function $f_0:A_0 \to B_0$ is an equivalence of types and for all elements $a:A_0$ and $b:A_0$ the function $f_\equiv(a, b):(a \equiv_A b) \to (f_0(a) \equiv_B f_0(b))$ is an equivalence of types.

The only symmetric preordered types for which bijection, isomorphism, and equivalence are the same are the symmetric posets, which are the h-sets.

A categorical view

A set equipped with an equivalence relation can be regarded as a groupoid enriched over the cartesian monoidal category of truth values. Equivalently, it is a groupoid that is 0-truncated. Then the equivalence relation on $S$ is a way of making $S$ into the set of objects of such a groupoid. Equivalently, a set with an equivalence relation is a (0,1)-category whose each morphism is iso, or a symmetric preordered set.

It may well be useful to consider several possible equivalence relations on a given set. When considering a single equivalence relation once and for all, however, it is common to take the quotient set $S/_{\equiv}$ and use that instead. As a groupoid, any set with an equivalence relation is equivalent to a set in this way (although in the absence of the axiom of choice, it is only a “weak” or ana-equivalence).

A set equipped with an equivalence relation is sometimes called a setoid; however the term setoid is primarily used for a pseudo-equivalence relation instead, and the usage on the nLab follows the latter.

This terminology is particularly common in foundations of mathematics where quotient sets don't always exist and the above equivalence to a set cannot be carried out. However, arguably this is a terminological mismatch, and such people should say ‘set’ where they say ‘setoid’ and something else (such as ‘preset’, ‘type’, or ‘completely presented set’) where they say ‘set’. (See Bishop set and page 9 of these lecture notes.)

Variants and generalizations

• A partial equivalence relation is a symmetric and transitive relation.

• A congruence is a notion of equivalence relation internal to a suitable category.

• The generalization of this to (∞,1)-category theory is that of groupoid object in an (∞,1)-category.

Related concepts

• Bishop set

• congruence

• complete loop graph

• stable equivalence relation

• pseudo-equivalence relation

• quotient equivalence relation

• enriched set

• monoidal symmetric proset

• set-like structure in homotopy type theory?

• tolerance relation

References

For the history of the notion of equivalence relation see this MO discussion.