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Let be two morphisms in
, where and are objects of a category
. A morphism
is said to equalize and if . In other words, the following diagrams are equal:
An equalizer of and is a morphism from an object
to , such that
equalizes and 
is universal among all morphisms that equalize and . Specifically, if is a morphism from an object
to such that equalizes and , then there exists a unique morphism and a commutative diagram:
Reversing all the arrows in the previous paragraphs, we have the dual notion of an equalizer: that of a coequalizer. To make this statement explicitly, let there be given two morphisms
, a coequalizer is a morphism from to an object
such that
-
. Such a morphism is said to coequalize and .
is universal among all morphisms that coequalizes and . This means that given a morphism from to an object
, there exists a unique morphism
so the following diagram commutes:
Remarks
- An equalizer is a monomorphism (but not the other way around, a monomorphism that is also an equalizer is called a regular monomorphism). A coequalizer is an epimorphism (and conversely, an epimorphism that is also a coequalizer is called a regular epimorphism). This follows directly from the above definitions and definitions of monomorphisms and epimorphisms.
- If
is an equalizer of
, then is a subobject of . Furthermore, by the universality of the equalizer, it is the “largest” such subobject. Similarly, If is a coequalizer of , then is the “largest" quotient object of .
- From the above discussion, we can safely say the equalizer of
and and the coequalizer of and .
- The equalizer of a morphism
and itself is the identity morphism on .
One can also define an equalizer of an arbitrary set of morphisms with a common domain and a common codomain: if
is a set of morphisms from to , indexed by a set , then an equalizer of the 's is a morphism from an object to
such that equalizes every pair of morphisms and and that is universal among all morphisms with such a property.
Remark. An equalizer (coequalizer) is also known as a difference kernel (difference cokernel). This name is justifiably given as we recognize that a kernel of a morphism is, in a way, the “difference" between and , the zero morphism.
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"equalizer" is owned by CWoo.
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(view preamble)
Cross-references: zero morphism, difference, kernel of a morphism, property, indexed by, codomain, domain, identity, quotient object, universality, subobject, definitions, epimorphism, monomorphism, commutative diagram, universal, category, objects, morphisms
There are 10 references to this entry.
This is version 12 of equalizer, born on 2004-10-22, modified 2008-08-31.
Object id is 6404, canonical name is Equalizer.
Accessed 7585 times total.
Classification:
| AMS MSC: | 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms) | | | 18A30 (Category theory; homological algebra :: General theory of categories and functors :: Limits and colimits ) |
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Pending Errata and Addenda
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