Limits and colimits
limits and colimits
limit and colimit
limits and colimits by example
commutativity of limits and colimits
connected limit, wide pullback
preserved limit, reflected limit, created limit
product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
end and coend
The quotient object of a congruence (an internal equivalence relation) on an object in a category is the coequalizer of the induced pair of maps .
If is additionally the kernel pair of the map , then is called an effective quotient (and is called an effective congruence, with the map 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 (that is, a subobject of in the opposite category ), without reference to any congruence on . 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 is an (∞,1)-category:
For instance an action groupoid is a quotient of a group action in 2-category theory.
Revised on July 13, 2014 03:42:37
by Urs Schreiber