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Version 9 |
| A morphism $f:A\to B$ in a category $\mathcal{C}$ is said to be \emph{orthogonal} to a morphism $g:C\to D$ in $\mathcal{C}$, written $$f\perp g$$ if whenever we have a commutative diagram |
A morphism $f:A\to B$ in a category $\mathcal{C}$ is said to be \emph{orthogonal} to a morphism $g:C\to D$ in $\mathcal{C}$, written $$f\perp g$$ if whenever we have a commutative diagram |
| $$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \\ C \ar[r]_g & D}$$ |
$$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \\ C \ar[r]_g & D}$$ |
| there is a unique morphism $h:B\to C$ such that the diagram |
there is a unique morphism $h:B\to C$ such that the diagram |
| $$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \ar@{.>}[dl]|h \\ C \ar[r]_g & D}$$ |
$$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \ar@{.>}[dl]|h \\ C \ar[r]_g & D}$$ |
| is commutative also. If $f\perp g$, we sometimes call the ordered pair $(f,g)$ a \emph{diagonally polar pair}. |
is commutative also. If $f\perp g$, we sometimes call the ordered pair $(f,g)$ a \emph{diagonally polar pair}. |
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| \begin{prop} For an arbitrary $f$, $f\perp g$ for any isomorphism $g$. Dually, if $g$ is arbitrary, then $f\perp g$ for any isomorphism $f$. \end{prop} |
\begin{prop} For an arbitrary $f$, $f\perp g$ for any isomorphism $g$. Dually, if $g$ is arbitrary, then $f\perp g$ for any isomorphism $f$. \end{prop} |
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| \begin{proof} |
\begin{proof} |
| Suppose $x\circ f = g\circ y$, and $z\circ g=1_C$ and $g\circ z=1_D$. Then by defining $h:=z \circ x$, we get $h\circ f= z\circ x \circ f= z \circ g\circ y = 1_C \circ y = y$, and $g\circ h= g\circ z\circ x=1_D \circ x = x$. This shows the existence of $h$. |
Suppose $x\circ f = g\circ y$, and $z\circ g=1_C$ and $g\circ z=1_D$. Then by defining $h:=z \circ x$, we get $h\circ f= z\circ x \circ f= z \circ g\circ y = 1_C \circ y = y$, and $g\circ h= g\circ z\circ x=1_D \circ x = x$. This shows the existence of $h$. |
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| If $h':B\to C$ is another morphism such that $h'\circ f = y$ and $g\circ h'=x$. Then $h = z\circ x = z \circ g\circ h' = 1_C \circ h' = h'$, showing that $h$ is unique. |
If $h':B\to C$ is another morphism such that $h'\circ f = y$ and $g\circ h'=x$. Then $h = z\circ x = z \circ g\circ h' = 1_C \circ h' = h'$, showing that $h$ is unique. |
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| The proof of the dual statement is similar. |
The proof of the dual statement is similar. |
| \end{proof} |
\end{proof} |
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| Unlike the orthogonality relations on subspaces of a vector space, the orthogonality relation on morphisms is in general not reflexive nor symmetric. However, it is transitive. |
Unlike the orthogonality relations on subspaces of a vector space, the orthogonality relation on morphisms is in general not reflexive nor symmetric. However, it is transitive. |
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| \begin{prop} If $f\perp g$ and $g\perp h$, then $f\perp h$. \end{prop} |
\begin{prop} If $f\perp g$ and $g\perp h$, then $f\perp h$. \end{prop} |
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| In addition, $\perp$ preserves morphism composition, in the following sense: |
In addition, $\perp$ preserves morphism composition, in the following sense: |
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| \begin{prop} Suppose $(g,h)$ is a composable pair of morphisms ($h\circ g$ exists). If $f\perp g$ and $f\perp h$, then $f\perp (h\circ g)$. Similarly, if $g\perp f$ and $h\perp f$, then $(h\circ g)\perp f$. \end{prop} |
\begin{prop} Suppose $(g,h)$ is a composable pair of morphisms ($h\circ g$ exists). If $f\perp g$ and $f\perp h$, then $f\perp (h\circ g)$. Similarly, if $g\perp f$ and $h\perp f$, then $(h\circ g)\perp f$. \end{prop} |
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| More generally, if $\mathcal{F}$ and $\mathcal{G}$ are two classes of morphisms in a category $\mathcal{C}$, we say that $\mathcal{F}$ is \emph{orthogonal} to $\mathcal{G}$, or that $(\mathcal{F},\mathcal{G})$ is a \emph{diagonally polar pair}, written $\mathcal{F}\perp \mathcal{G}$, if $f\perp g$ for every $f$ in $\mathcal{F}$ and every $g$ in $\mathcal{G}$. |
More generally, if $\mathcal{F}$ and $\mathcal{G}$ are two classes of morphisms in a category $\mathcal{C}$, we say that $\mathcal{F}$ is \emph{orthogonal} to $\mathcal{G}$, or that $(\mathcal{F},\mathcal{G})$ is a \emph{diagonally polar pair}, written $\mathcal{F}\perp \mathcal{G}$, if $f\perp g$ for every $f$ in $\mathcal{F}$ and every $g$ in $\mathcal{G}$. |
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| For every class $\mathcal{X}$ of morphism, the largest class of morphisms in $\mathcal{C}$ such that $\mathcal{X}$ is orthogonal to is denoted by $\mathcal{X}_*$, and the largest class of morphisms that is orthogonal to $\mathcal{X}$ is denoted by $\mathcal{X}^*$. |
For every class $\mathcal{X}$ of morphism, the largest class of morphisms in $\mathcal{C}$ such that $\mathcal{X}$ is orthogonal to is denoted by $\mathcal{X}_*$, and the largest class of morphisms that is orthogonal to $\mathcal{X}$ is denoted by $\mathcal{X}^*$. |
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| Below are some properties of $^*$ and $_*$: |
Below are some properties of $^*$ and $_*$: |
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| \begin{itemize} |
\begin{itemize} |
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\item $\mathcal{X} \subseteq \mathcal{Y}_*$ iff $\mathcal{Y}\subseteq \mathcal{X}^*$. Equivalently, if $\mathscr{M}$ is the class of all subclasses of morphisms of $\mathcal{C}$, then $(-^*,-_*)$ is a Galois connection between $(\mathscr{M},\subseteq)$ and $(\mathscr{M},\supseteq)$.
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\item $\mathcal{X} \subseteq \mathcal{Y}_*$ iff $\mathcal{Y}\subseteq \mathcal{X}^*$.$.
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| \item A morphism is in both $\mathcal{X}^*$ and $\mathcal{X}_*$ iff it is an isomorphism. |
\item A morphism is in both $\mathcal{X}^*$ and $\mathcal{X}_*$ iff it is an isomorphism. |
| \item Given that $m=m_1\circ m_2$ exists in $\mathcal{C}$ and $m_2\in \mathcal{X}_*$, then $m \in \mathcal{X}_*$ iff $m_1\in \mathcal{X}_*$. |
\item Given that $m=m_1\circ m_2$ exists in $\mathcal{C}$ and $m_2\in \mathcal{X}_*$, then $m \in \mathcal{X}_*$ iff $m_1\in \mathcal{X}_*$. |
| \item |
\item |
| If $f\in \mathcal{X}_*$, then the pullback of $f$ along any morphism is again in $\mathcal{X}_*$. |
If $f\in \mathcal{X}_*$, then the pullback of $f$ along any morphism is again in $\mathcal{X}_*$. |
| \end{itemize} |
\end{itemize} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
| \bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994) |
\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994) |
| \end{thebibliography} |
\end{thebibliography} |
