nLab subquotient

Definition

A subquotient of an object $X$ in some category is

• a quotient object of a subobject of $X$,

or

• a subobject of a quotient object of $X$.

Properties

• The two definitions are equivalent whenever quotients are stable under pullback and subobjects are stable under pushout, such as in a topos.

• Just as with subobjects and quotient objects, we have that $X$ is a subquotient of itself, and subquotients of subquotients of $X$ are themselves subquotients of $X$ in a natural way.

• Just as subobjects of a set $X$ are in correspondence with predicates on $X$ and quotients of $X$ are in correspondence with equivalence relations on $X$, subquotients of $X$ are in correspondence with partial equivalence relations on $X$.

Examples

• In Grp, all but six of the 26 sporadic finite simple groups are subquotients of the Monster group, the largest such group.