nLab
quotient object

Context

Category theory

Limits and colimits

Contents

Definition

The quotient object QQ of a congruence (an internal equivalence relation) EE on an object XX in a category CC is the coequalizer QQ of the induced pair of maps EXE \to X.

If EE is additionally the kernel pair of the map XQX \to Q, then QQ is called an effective quotient (and EE is called an effective congruence, with the map XQX \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 XX (that is, a subobject of XX in the opposite category C opC^\op), without reference to any congruence on XX. 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 CC is an (∞,1)-category:

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

Examples

Revised on November 4, 2013 02:09:52 by Urs Schreiber (89.204.154.47)