If $E$ is additionally the kernel pair of the map $X \to Q$, then $Q$ is called an effective quotient (and $E$ is called an effective congruence, with the map $X \to Q$ being an effective epimorphism).

Sometimes the term is used more loosely to mean an arbitrary coequalizer. It may also refer to a co-subobject of $X$ (that is, a subobject of $X$ in the opposite category$C^\op$), without reference to any congruence on $X$. Note that in a regular category, any regular epimorphism (i.e. a “regular quotient” in the co-subobject sense) is in fact the quotient (= coequalizer) of its kernel pair.

In higher category theory

These notions have generalizations when $C$ is an (∞,1)-category: