You are here
Homeequalizer
Primary tabs
equalizer
Let $\mathcal{C}$ be a category. A family of morphisms in $\mathcal{C}$ is said be parallel if they belong to $\operatorname{Hom}(A,B)$, for some objects $A,B$ in $\mathcal{C}$.
Let $f,g$ be a pair of parallel morphisms in $\operatorname{Hom}(A,B)$. 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:
$\xymatrix@1{X\ar[r]^{d}&A\ar[r]^{f}&B}=\xymatrix@1{X\ar[r]^{d}&A\ar[r]^{g}&B}$ 
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:
$\xymatrix@1{Y\ar[d]_{h}\ar[dr]^{e}\\ X\ar[r]_{d}&A}$
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:
$\xymatrix@1{B\ar[dr]_{e}\ar[r]^{c}&Z\ar[d]^{r}\\ &Y}$
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 $A$.

A category is said to have equalizers if every pair of parallel morphisms has an equalizer.
One can also define an equalizer of an arbitrary set of morphisms with a common domain and a common codomain: if $\{f_{i}:A\to B\mid i\in I\}$ 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.
Mathematics Subject Classification
18A20 no label found18A30 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections