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'orthogonal morphisms'
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| Title of object: |
orthogonal morphisms |
| Canonical Name: |
OrthogonalMorphisms |
| Type: |
Definition |
| Created on: |
2008-10-14 20:41:03 |
| Modified on: |
2008-10-15 17:45:26 |
| Classification: |
msc:18A32 |
| Defines: |
orthogonal |
| Synonyms: |
orthogonal morphisms=diagonally polar pair |
Preamble:
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Content:
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
$$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \\ C \ar[r]_g & D}$$
there is a unique morphism $h:B\to C$ such that the diagram
$$\xymatrix@+=3pc{A \ar[r]^f \ar[d] & B \ar[d] \ar@{.>}[dl]|h \\ C \ar[r]_g & D}$$
is commutative also. If $f\perp g$, we sometimes call the ordered pair $(f,g)$ a \emph{diagonally polar pair}.
\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{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 $h$.
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.
The proof of the dual statement is similar.
\end{proof}
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.
\begin{prop} If $f\perp g$ and $g\perp h$, then $f\perp h$. \end{prop}
In addition, $\perp$ preserves morphism composition, in the following sense:
\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}
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 $\mathcal{G}$.
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 $\mathcal{X}^*$.
Below are some properties of $^*$ and $_*$:
\begin{itemize}
\item $\mathcal{X} \subseteq \mathcal{Y}_*$ iff $\mathcal{Y}\subseteq \mathcal{X}^*$.
\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
If $f\in \mathcal{X}_*$, then the pullback of $f$ along any morphism is again in $\mathcal{X}_*$.
\end{itemize}
\begin{thebibliography}{9}
\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994)
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