Editing Equivalence relation (section)

=== Categories and groupoids ===

Let ''G'' be a set and let "~" denote an equivalence relation over ''G''. Then we can form a [[groupoid]] representing this equivalence relation as follows. The objects are the elements of ''G'', and for any two elements ''x'' and ''y'' of ''G'', there exists a unique morphism from ''x'' to ''y'' [[if and only if]] <math>x \sim y.</math>

The advantages of regarding an equivalence relation as a special case of a groupoid include:
*Whereas the notion of "free equivalence relation" does not exist, that of a [[Free object|free groupoid]] on a [[directed graph]] does. Thus it is meaningful to speak of a "presentation of an equivalence relation," i.e., a presentation of the corresponding groupoid;
* Bundles of groups, [[Group action (mathematics)|group action]]s, sets, and equivalence relations can be regarded as special cases of the notion of groupoid, a point of view that suggests a number of analogies;
*In many contexts "quotienting," and hence the appropriate equivalence relations often called [[Congruence relation|congruences]], are important. This leads to the notion of an internal groupoid in a [[Category (mathematics)|category]].<ref>Borceux, F. and Janelidze, G., 2001. ''Galois theories'', Cambridge University Press, {{ISBN|0-521-80309-8}}</ref>