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| Title of object: |
strong monomorphism |
| Canonical Name: |
StrongMonomorphism |
| Type: |
Definition |
| Created on: |
2008-09-15 19:03:19 |
| Modified on: |
2008-10-15 01:39:00 |
| Classification: |
msc:18-00, msc:18A20 |
| Defines: |
strong, strong epimorphism |
| Synonyms: |
strong monomorphism=strong monic
strong monomorphism=strong epi
strong monomorphism=strong epic |
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Content:
Let $\mathcal{C}$ be a category. A monomorphism $f:A\to B$ in $\mathcal{C}$ is said to be a \emph{strong monomorphism} if, whenever we are given the following commutative diagram
$$\xymatrix@+=3pc{
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\\
{A}\ar[r]_{f}&{B}
}
$$
with $g$ an epimorphism, then there is a morphism $h: D\to A$ such that the following is another commutative diagram:
$$\xymatrix@+=3pc{
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]|{h} \\
{A}\ar[r]_{f}&{B}
}
$$
Note that the ``diagonal'' morphism $h$ is necessarily unique. In other words, a monomorphism is strong iff every epimorphism is orthogonal to it.
Dually, a \emph{strong epimorphism} is an epimorphism which is orthogonal to every monomorphism in the category.
\textbf{Remark}. Every regular monomorphism is strong (see proof \PMlinkname{here}{RegularMonomorphism}), and every strong monomorphism is \PMlinkname{extremal}{ExtremalMonomorphism}.
\begin{proof}
Suppose $f:A\to B$ is a strong monomorphism and that $f=h\circ g$ with $g:A\to C$ epimorphic. Then we have the following commutative diagram
$$\xymatrix@+=3pc{
{A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\\
{A}\ar[r]_{f}&{B}
}
$$
Since $f$ is strong, there is a morphism $e:C\to A$ such that the diagram below is commutative
$$\xymatrix@+=3pc{
{A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]_{e} \\
{A}\ar[r]_{f}&{B}
}
$$
This shows that $g$ is a split monomorphism, as $1_A=e\circ g$. But $g$ is epimorphic, we conclude that $g$ is an isomorphism (this fact is proved \PMlinkname{here}{PropertiesOfRegularAndExtremalMonomorphisms}).
\end{proof}
\begin{thebibliography}{9}
\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994)
\end{thebibliography} |
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