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 concept of coequalizer in a general category is the generalization of the construction where for two functions between sets and
one forms the set of equivalence classes induced by the equivalence relation generated by the relation
for all . This means that the projection function satisfies
and in fact is universal with this property, hence it “co-equalizes” and .
In this form this may be phrased generally in any category,
In some category , the coequalizer of two parallel morphisms and between two objects and is (if it exists), the colimit under the diagram formed by these two morphisms
In a category a diagram
is called a coequalizer diagram if
is universal for this property: i.e. if is a morphism of such that , then there is a unique morphism such that
Relation to pushouts
Coequalizers are closely related to pushouts:
is a coequalizer diagram, def. 2, precisely if
is a pushout diagram.
is a pushout square, precisely if
is a coequalizer diagram.