quotienting

Quotienting is a construction in mathematics that produces new objects by identifying elements that are considered equivalent and collapsing each equivalence class to a single point. Given a set X and an equivalence relation ~, the quotient set X/~ consists of the equivalence classes [x]. There is a natural projection map pi: X -> X/~ that sends each x to its class [x].

To carry additional structure through to the quotient, the relation must be a congruence or compatible with the structure. In groups, quotienting by a normal subgroup N yields the quotient group G/N whose elements are cosets gN and whose operation is well-defined because N is normal. In rings or algebras, quotienting by an ideal I or a two-sided ideal likewise produces a well-defined ring or algebra with elements a+I, and in vector spaces quotienting by a subspace W yields the quotient space V/W.

In topology, quotienting forms a quotient space X/~ by declaring a subset U of X/~ open if and only if the preimage of U under pi is open in X; the projection pi is a quotient map. This construction often reflects identifications or symmetries in a space and has a universal property: any map from X that is constant on equivalence classes factors uniquely through pi.

Common examples include the integers mod n, written Z/nZ, the quotient of a ring by an ideal, and the quotient of a vector space by a subspace. Quotienting also appears in many areas of mathematics and in category theory as a coequalizer.