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Version 14 |
| Let $\mathcal{C}$ be a category. A family of morphisms in $\mathcal{C}$ is said be \PMlinkescapetext{\emph{parallel}} if they belong to $\operatorname{Hom}(A,B)$, for some objects $A,B$ in $\mathcal{C}$. |
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: |
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| 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: |
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| $$\xymatrix@1{X\ar[r]^d&A\ar[r]^f&B}=\xymatrix@1{X\ar[r]^d&A\ar[r]^g&B}$$ |
$$\xymatrix@1{X\ar[r]^d&A\ar[r]^f&B}=\xymatrix@1{X\ar[r]^d&A\ar[r]^g&B}$$ |
| \par |
\par |
| An \emph{equalizer} of $f$ and $g$ is a morphism $d$ from an object $X \in \mathcal{C}$ to $A$, such that |
An \emph{equalizer} of $f$ and $g$ is a morphism $d$ from an object $X \in \mathcal{C}$ to $A$, such that |
| \begin{enumerate} |
\begin{enumerate} |
| \item $d$ \emph{equalizes} $f$ and $g$ |
\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: |
\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: |
| $$\xymatrix@1{Y \ar[d]_h \ar[dr]^e \\ X \ar[r]_d & A}$$ |
$$\xymatrix@1{Y \ar[d]_h \ar[dr]^e \\ X \ar[r]_d & A}$$ |
| \end{enumerate} |
\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 |
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} |
\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 $\xymatrix@1{A\ar[r]^f&B\ar[r]^c&Z}=\xymatrix@1{A\ar[r]^g&B\ar[r]^c&Z}$. 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: |
\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: |
| $$\xymatrix@1{B \ar[dr]_e \ar[r]^c & Z \ar[d]^r \\ & Y}$$ |
$$\xymatrix@1{B \ar[dr]_e \ar[r]^c & Z \ar[d]^r \\ & Y}$$ |
| \end{enumerate} |
\end{enumerate} |
| \par |
\par |
| \textbf{Remarks} |
\textbf{Remarks} |
| \begin{itemize} |
\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 (and conversely, an epimorphism that is also a coequalizer is called a \emph{regular epimorphism}). This follows directly from the above definitions and definitions of monomorphisms and epimorphisms. |
\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 (and conversely, an epimorphism that is also a coequalizer is called a \emph{regular 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 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 $g$. |
\item From the above discussion, we can safely say \emph{the} equalizer of $f$ and $g$ and \emph{the} coequalizer of $f$ and $g$. |
| \item The equalizer of a morphism $f:A\to B$ and itself is the identity morphism $1_A$ on $A$. |
\item The equalizer of a morphism $f:A\to B$ and itself is the identity morphism $1_A$ on $A$. |
| \end{itemize} |
\end{itemize} |
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| 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. |
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. |
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| \textbf{Remark}. An equalizer (coequalizer) is also known as a \emph{difference kernel} (\emph{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. |
\textbf{Remark}. An equalizer (coequalizer) is also known as a \emph{difference kernel} (\emph{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. |
