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equalizer (Definition)

Let $ f,g$ be two morphisms in $ \operatorname{Hom}(A,B)$, where $ A$ and $ B$ are objects of a category $ \mathcal{C}$. A morphism $ d\colon X\to A$ is said to equalize $ f$ and $ g$ if $ fd=gd$. In other words, the following diagrams are equal:

$\displaystyle \xymatrix@1{X\ar[r]^d&A\ar[r]^f&B}=\xymatrix@1{X\ar[r]^d&A\ar[r]^g&B}$

An equalizer of $ f$ and $ g$ is a morphism $ d$ from an object $ X \in \mathcal{C}$ to $ A$, such that

  1. $ d$ equalizes $ f$ and $ g$
  2. $ d$ is universal among all morphisms that equalize $ f$ and $ g$. Specifically, if $ e$ is a morphism from an object $ Y\in\mathcal{C}$ to $ A$ such that $ e$ equalizes $ f$ and $ g$, then there exists a unique morphism $ h:Y\to X$ and a commutative diagram:
    $\displaystyle \xymatrix@1{Y \ar[d]_h \ar[dr]^e \\ X \ar[r]_d & A}$
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 $ f,g\in\operatorname{Hom}(A,B)$, a coequalizer is a morphism $ c$ from $ B$ to an object $ Z\in\mathcal{C}$ such that
  1. $ \xymatrix@1{A\ar[r]^f&B\ar[r]^c&Z}=\xymatrix@1{A\ar[r]^g&B\ar[r]^c&Z}$. Such a morphism is said to coequalize $ f$ and $ g$.
  2. $ c$ is universal among all morphisms that coequalizes $ f$ and $ g$. This means that given a morphism $ r$ from $ B$ to an object $ Y\in\mathcal{C}$, there exists a unique morphism $ r\in\operatorname{Hom}(Z,Y)$ so the following diagram commutes:
    $\displaystyle \xymatrix@1{B \ar[dr]_e \ar[r]^c & Z \ar[d]^r \\ & Y}$

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 $ X\to A$ is an equalizer of $ f,g\colon A\to B$, then $ [X\to A]$ is a subobject of $ A$. Furthermore, by the universality of the equalizer, it is the “largest” such subobject. Similarly, If $ B\to Z$ is a coequalizer of $ f,g$, then $ [B\to Z]$ is the “largest" quotient object of $ B$.
  • From the above discussion, we can safely say the equalizer of $ f$ and $ g$ and the coequalizer of $ f$ and $ g$.
  • One can also define an equalizer of an arbitrary set of morphisms with a common domain and a common codomain: if $ \lbrace f_i:A\to B\mid i\in I\rbrace$ is a set of morphisms from $ A$ to $ B$, indexed by a set $ I$, then an equalizer of the $ f_i$'s is a morphism $ d$ from an object $ X$ to $ A$ such that $ d$ equalizes every pair of morphisms $ f_i$ and $ f_j$ and that $ d$ is universal among all morphisms with such a property.

Example - Using equalizer and coequalizer to define kernel and cokernel.

If a category $ \mathcal{C}$ contains a zero object $ O$, then given objects $ A,B$, we can define a zero morphism, or null morphism to be the unique morphism $ o$ of the composition of the two unique morphisms $ A\to O$ and $ O\to B$ in $ \operatorname{Hom}(A,B)$:

$\displaystyle \xymatrix@1{A\ar[r]^o&B}=\xymatrix@1{A\ar[r]&O\ar[r]&B}.$
With the zero morphism, we can define kernel of a morphism $ f$, $ \operatorname{ker}(f)$, to be the equalizer of $ f$ and the zero morphism $ o$. Dually, we can also define the cokernel of a morphism $ g$, $ \operatorname{coker}(g)$, to be the coequalizer of $ g$ and $ o$. Kernels and cokernels are necessarily unique by the universality of equalizers and coequalizers.

An equalizer is also known as a difference kernel. This name is justifiably given as we recognize that a kernel of a morphism $ f$ is, in a way, the “difference" between $ f$ and $ o$, the zero morphism.



"equalizer" is owned by CWoo.
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See Also: properties of regular and extremal monomorphisms

Other names:  difference kernel, difference cokernel, equaliser, coequaliser
Also defines:  coequalizer, zero morphism, null morphism, regular monomorphism, regular epimorphism

Attachments:
proof that an equalizer is a monomorphism (Proof) by rmilson
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Cross-references: difference, composition, zero object, contains, cokernel, kernel, property, indexed by, codomain, domain, quotient object, universality, subobject, definitions, epimorphism, monomorphism, commutative diagram, universal, category, objects, morphisms
There are 11 references to this entry.

This is version 8 of equalizer, born on 2004-10-22, modified 2008-06-17.
Object id is 6404, canonical name is Equalizer.
Accessed 7005 times total.

Classification:
AMS MSC: 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms)
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