nLab quotient set

Quotient sets

Quotient sets

Definitions

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

If changing the definition of equality like this is not allowed in your foundations of mathematics, then you can still define S/S/{\equiv} as a subset of the power set of SS as follows:

(1)AS/(x:S),A={y:Sxy} A \in S/{\equiv} \;\Leftrightarrow\; \exists (x: S),\; A = \{y: S \;\mid\; x \equiv y\}

If you don't have (extensional) power sets either, then you'll have to use setoids (in which case, perhaps you'd do better to change your terminology as described there). Alternatively, don't worry about any of this and just include the existence of quotient objects in Set as an axiom (the axiom of quotient sets, which you have if you assume that Set\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 SetSet is a pretopos: that every internal equivalence relation on a set AA, defined as a subset \equiv of the Cartesian product A×AA \times A satisfying reflexivity, transitivity, and symmetry, has a quotient set A/A/\equiv.

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

In any case, the element of S/S/{\equiv} that comes from the element xx of SS may be denoted [x] [x]_{\equiv}, or simply [x][x] if {\equiv} is understood, or simply xx if there will be no confusion as to which set it is an element of. This [x][x] is called the equivalence class of xx 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][x] is literally the set AA.

As initial objects in a category

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

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

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

In type theory

Let (T,)(T,\equiv) be a symmetric proset. The quotient set T/T/\equiv is a higher inductive type inductively generated by the following:

  • a function ι:TT/\iota: T \to T/\equiv

  • a family of dependent terms

    a:T,b:Teq T/(a,b):(ab)(ι(a)= T/ι(b))a:T, b:T \vdash eq_{T/\equiv}(a, b): (a \equiv b) \to (\iota(a) =_{T/\equiv} \iota(b))
  • a family of dependent terms

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

The axiom of quotient sets in type theory says that every symmetric proset is a set.

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.

See also

Last revised on December 21, 2022 at 02:35:05. See the history of this page for a list of all contributions to it.