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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$.
  • The equalizer of a morphism $ f:A\to B$ and itself is the identity morphism $ 1_A$ on $ A$.

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.

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 $ 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, regular monomorphism, regular epimorphism

Attachments:
proof that an equalizer is a monomorphism (Proof) by rmilson
kernel of a morphism (Definition) by CWoo
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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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