Quotient sets

category theory

# Quotient sets

## Definitions

### In set theory

Given a set $S$ and an equivalence relation $\equiv$ on $S$, the quotient set of $S$ by $\equiv$ is the set $S/{\equiv}$ whose elements are the elements of $S$ but where two elements are now considered equal if in $S$ they were merely equivalent.

If changing the definition of equality like this is not allowed in the given foundations of mathematics, then one can still define $S/{\equiv}$ as a subset of the power set of $S$.

• In material set theory the definition is as follows:
(1)$A \in S/{\equiv} \;\Leftrightarrow\; \exists (x: S),\; A = \{y: S \;\mid\; x \equiv y\}$
• In structural set theory the definition is slightly different, because sets are not elements of sets and one cannot in general compare sets for strict equality.

If there are also no (extensional) power sets in the foundations of mathematics, then there are two alternatives:

• One could use setoids (in which case, perhaps one would do better to change your terminology as described there).
• One could just include the existence of quotient objects in Set as an axiom (the axiom of quotient sets.

The latter is the case if one assumes that $\Set$ is a pretopos or a Grothendieck topos as given by Giraud's axioms). There are actually two possible axiom of quotient sets, depending on what definition of relation one uses.

• The weaker version of the axiom of quotient sets is the one which states that the coherent category $Set$ is a pretopos: that every internal equivalence relation on a set $A$, defined as a subset $\equiv$ of the Cartesian product $A \times A$ satisfying reflexivity, transitivity, and symmetry, has a quotient set $A/\equiv$.

• The stronger version of the axiom of quotient sets states that every proposition $x \equiv y$ in the context of the variables $x \in A$ and $y \in A$ which satisfies reflexivity, transitivity, and symmetry has a quotient set $A/\equiv$. In some presentations of set theory, this is an axiom schema of quotient sets.

In any case, the element of $S/{\equiv}$ that comes from the element $x$ of $S$ may be denoted $[x]_{\equiv}$, or simply $[x]$ if ${\equiv}$ is understood, or simply $x$ if there will be no confusion as to which set it is an element of. This $[x]$ is called the equivalence class of $x$ with respect to $\equiv$; the term ‘class’ here is an old word for ‘set’ (in the sense of ‘subset’) and refers to the definition (1) above, where $[x]$ is literally the set $A$.

#### As initial objects in a category

Let $S$ be a set and let $\equiv$ be an equivalence relation on $S$. Let us define a quotient set algebra of $S$ and $\equiv$ to be a set $A$ with a function $\iota S \to A$ such that for all $a \in S$ and $b \in S$, $(a \equiv b)$ implies $(\iota(a) = \iota(b))$.

A quotient set algebra homomorphism of $S$ and $\equiv$ is a function $f: A \to B$ between two quotient set algebras $A$ and $B$ such that for all $a \in S$, $f(\iota_A(a)) = \iota_B(a)$.

The category of quotient set algebras of $S$ and $\equiv$ is the category $QSA(S, \equiv)$ whose objects $Ob(QSA(S, \equiv))$ are quotient set algebras of $S$ and $\equiv$ and whose morphisms $Mor(A, B)$ for $A \in Ob(QSA(S, \equiv))$ and $B \in Ob(QSA(S, \equiv))$ are quotient set algebra homomorphisms of $S$ and $\equiv$. The quotient set of $S$ and $\equiv$, denoted $S/\equiv$, is defined as the initial object in the category of quotient set algebras of $S$ and $\equiv$.

### In dependent type theory

Similar to set theory, there are multiple ways of defining quotient sets in dependent type theory. If the type theory has a univalent type of all propositions $(\mathrm{Prop}, \mathrm{El})$ and thus power sets $\mathcal{P}T \coloneqq T \to \mathrm{Prop}$ of a type $T$, then given a type $T$ with an equivalence relation $(-)\equiv(-):\mathcal{P}(T \times T)$, one could construct the quotient set $T/\equiv$ as a subtype of the power set, as follows: we first define the predicate on the powerset $\mathcal{P}T$:

$P:\mathcal{P}T \vdash \mathrm{isEquivalenceClass}(P) \coloneqq \left[\sum_{x:T} \prod_{y:T} P(x) =_\mathrm{Prop} (x \equiv y) \right]$

since by univalence there is an equivalence between the types $P(x) =_\mathrm{Prop} (x \equiv y)$ and $\mathrm{El}(P(x)) \simeq \mathrm{El}(x \equiv y)$, which for propositions is the same as $\mathrm{El}(P(x)) \iff \mathrm{El}(x \equiv y)$. Then we define the subtype

$T/\equiv \coloneqq \sum_{P:\mathcal{P}T} isEquivalenceClass(P)$

Alternatively, if the dependent type theory does not have a type of all propositions, then the quotient set $T/\equiv$ can still nevertheless be constructed. The analogue of the axiom of quotient sets in dependent type theory is the rules for the quotient set higher inductive type. In particular, the quotient set $T/\equiv$ is a higher inductive type inductively generated by the following:

• a function $\iota: T \to T/\equiv$

• a family of dependent terms

$a \colon T ,\; b \colon T \;\;\; \vdash \;\;\; eq_{T/\equiv}(a, b) \,\colon\, (a \equiv b) \;\to\; (\iota(a) =_{T/\equiv} \iota(b))$
• a family of dependent terms

$a:T/\equiv, b:T/\equiv \vdash \tau(a, b): isProp(a =_{T/\equiv} b)$

In general, if one doesn’t have impredicative quotient sets or quotient sets as a higher inductive type, than one could only construct quotient sets of equivalence relations with a choice of unique representatives, which for equivalence relation $R(x, y)$ indexed by $x:A$ and $y:A$ consists of a type family $C(x)$ indexed by $x:A$, and an element of type

$\prod_{x:A} \exists!y:A.C(x) \times R(x, y)$

Then the type $\sum_{x:A} C(x)$ is the quotient set of $A$ with respect to $R$.

Equivalently, one could use functions instead of type families, and say that a choice of unique representatives is a type $C$ with a function $f:C \to A$ and an element of type

$\prod_{x:A} \exists!y:A.\left(\sum_{z:C} f(z) =_A x\right) \times R(x, y)$

Then

$C \simeq \sum_{x:A} \sum_{z:C} f(z) =_A x$

is the quotient set of $A$ with respect to $R$.

## Generalisations

Quotient sets in Set generalise to quotient objects in other categories. In particular, an exact category is a regular category in which every congruence on every object has an effective quotient object.