|
|
|
Viewing Version
5
of
'equalizer'
|
[ view 'equalizer'
|
back to history
]
| Title of object: |
equalizer |
| Canonical Name: |
Equalizer |
| Type: |
Definition |
| Created on: |
2004-10-22 17:50:23 |
| Modified on: |
2006-03-20 14:15:38 |
| Classification: |
msc:18A20 |
| Defines: |
coequalizer, zero morphism, null morphism, regular monomorphism |
| Synonyms: |
equalizer=difference kernel
equalizer=difference cokernel |
Revision comment (for changes between this and next version):
| Changes for correction #8628 ('regular epimorphism'). |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb,amscd}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here |
Content:
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:
$$\xymatrix@1{X\ar[r]^d&A\ar[r]^f&B}=\xymatrix@1{X\ar[r]^d&A\ar[r]^g&B}$$
\par
An \emph{equalizer} of $f$ and $g$ is a morphism $d$ from an object $X \in \mathcal{C}$ to $A$, such that
\begin{enumerate}
\item $d$ \emph{equalizes} $f$ and $g$
\item $d$ 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:
$$\xymatrix@1{Y \ar[d]_h \ar[dr]^e \\ X \ar[r]_d & A}$$
\end{enumerate}
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 \emph{coequalizer} is a morphism $c$ from $B$ to an object $Z\in\mathcal{C}$ such that
\begin{enumerate}
\item $\xymatrix@1{A\ar[r]^f&B\ar[r]^c&Z}=\xymatrix@1{A\ar[r]^g&B\ar[r]^c&Z}$. Such a morphism is said to \emph{coequalize} $f$ and $g$.
\item $c$ 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:
$$\xymatrix@1{B \ar[dr]_e \ar[r]^c & Z \ar[d]^r \\ & Y}$$
\end{enumerate}
\par
\textbf{Remarks}
\begin{itemize}
\item An equalizer is a monomorphism (but not the other way around, a monomorphism that is also an equalizer is called a \emph{regular monomorphism}). A coequalizer is an epimorphism. This follows directly from the above definitions and definitions of monomorphisms and epimorphisms.
\item 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 $B$.
\item From the above discussion, we can safely say \emph{the} equalizer of $f$ and $g$ and \emph{the} coequalizer of $f$ and $g$.
\end{itemize}
\par
\textbf{Example} - Using equalizer and coequalizer to define kernel and cokernel.
\par
If a category $\mathcal{C}$ contains a zero object $O$, then given objects $A,B$, we can define a \emph{zero morphism}, or \emph{null morphism} to be the unique morphism $o$ of the composition of the two unique morphisms $A\to O$ and $O\to B$ in $\operatorname{Hom}(A,B)$:
$$\xymatrix@1{A\ar[r]^o&B}=\xymatrix@1{A\ar[r]&O\ar[r]&B}.$$
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.
\par
An equalizer is also known as a \emph{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. |
|
|
|
|
|
