strong monomorphismStrongMonomorphism2008-09-15 19:03:192008-10-15 17:05:35Definitionmonicstrongstrong epimorphism\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}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}
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\bibitem{fb} F. Borceux \emph{Basic Category Theory, Handbook of Categorical Algebra I}, Cambridge University Press, Cambridge (1994)
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