Contents
Context
Limits and colimits
limits and colimits
1-Categorical
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limit and colimit
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limits and colimits by example
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commutativity of limits and colimits
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small limit
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filtered colimit
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sifted colimit
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connected limit, wide pullback
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preserved limit, reflected limit, created limit
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product, fiber product, base change, coproduct, pullback, pushout, cobase change, equalizer, coequalizer, join, meet, terminal object, initial object, direct product, direct sum
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finite limit
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Kan extension
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weighted limit
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end and coend
2-Categorical
(∞,1)-Categorical
Model-categorical
Contents
Idea
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 quotient function satisfies
(a map satisfying this equation is said to “co-equalize” and ) and moreover is universal with this property.
In this form this may be phrased generally in any category.
Definition
Definition
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
Equivalently:
Definition
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
Properties
Relation to pushouts
Coequalizers are closely related to pushouts:
Proposition
A diagram
is a coequalizer diagram, def. , precisely if
is a pushout diagram.
Conversely:
Proposition
A diagram
is a pushout square, precisely if
is a coequalizer diagram.
Examples