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Definition

The quotient object $Q$ of a congruence (an internal equivalence relation) $E$ on an object $X$ in a category $C$ is the coequalizer $Q$ of the induced pair of maps $E\to X$.

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:

• an equivalence relation is then a groupoid object in an (∞,1)-category

• it has an “effective quotient” if it is deloopable.

For instance an action groupoid is a quotient of a group action in 2-category theory.

Examples

• quotient set

• quotient group

• quotient module

Related concepts

• subquotient

• quotient space, geometric invariant theory

• quotient stack

• In type theory/homotopy type theory the analogous concept is that of quotient types.