In a finitely complete category , a congruence on an object is an internal equivalence relation on .
This means that it consists of a subobject equipped with the following morphisms: * internal reflexivity: which is a section both of and of ; * internal symmetry: which interchanges and , namely and ; * internal transitivity: ; where with the notation for the projections in the cartesian square
the following holds: and .
A congruence which is the kernel pair of some morphism is called effective.
An effective congruence is always the kernel pair of its quotient if that quotient exists.
The quotient of an effective congruence is an effective quotient.
The eponymous example is congruence modulo (for a fixed natural number ), which can be considered a congruence on in the category of rigs, or on in the category of rings.
A quotient group by a normal subgroup is the quotient of the relation , where is projection on the first factor and is multiplication in (these are source and target maps in the action groupoid ).
A special case of this is that of a quotient module.
Revised on September 11, 2012 10:10:02
by Urs Schreiber