| Version 8 |
Version 7 |
| 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 |
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 |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
| {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\\ |
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\\ |
| {A}\ar[r]_{f}&{B} |
{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: |
with $g$ an epimorphism, then there is a morphism $h: D\to A$ such that the following is another commutative diagram: |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
| {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]_{h} \\ |
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]_{h} \\ |
| {A}\ar[r]_{f}&{B} |
{A}\ar[r]_{f}&{B} |
| } |
} |
| $$ |
$$ |
|
|
| Note that the ``diagonal'' morphism $h$ is necessarily unique. |
Note that the ``diagonal'' morphism $h$ is necessarily unique. |
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| Dually, a \emph{strong epimorphism} is an epimorphism $f:A\to B$ such that, given a commutative diagram |
Dually, a \emph{strong epimorphism} is an epimorphism $f:A\to B$ such that, given a commutative diagram |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
| {A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\\ |
{A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\\ |
| {C}\ar[r]_{g}&{D} |
{C}\ar[r]_{g}&{D} |
| } |
} |
| $$ |
$$ |
| with $g$ a monomorphism, then there is a (unique) morphism $h:B\to C$ such that the following diagram is again commutative: |
with $g$ a monomorphism, then there is a (unique) morphism $h:B\to C$ such that the following diagram is again commutative: |
| $$\xymatrix@+=4pc{ |
$$\xymatrix@+=4pc{ |
| {A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\ar@{.>}[dl]_{h}\\ |
{A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\ar@{.>}[dl]_{h}\\ |
| {C}\ar[r]_{g}&{D} |
{C}\ar[r]_{g}&{D} |
| } |
} |
| $$ |
$$ |
|
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| \textbf{Remark}. Every regular monomorphism is strong (see proof \PMlinkname{here}{RegularMonomorphism}), and every strong monomorphism is \PMlinkname{extremal}{ExtremalMonomorphism}. |
\textbf{Remark}. Every regular monomorphism is strong (see proof \PMlinkname{here}{RegularMonomorphism}), and every strong monomorphism is \PMlinkname{extremal}{ExtremalMonomorphism}. |
| \begin{proof} |
\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 |
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 |
| $$\xymatrix@+=3pc{ |
$$\xymatrix@+=3pc{ |
| {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\\ |
{A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h}\\ |
| {A}\ar[r]_{f}&{B} |
{A}\ar[r]_{f}&{B} |
| } |
} |
| $$ |
$$ |
| Since $f$ is strong, there is a morphism $e:C\to A$ such that the diagram below is commutative |
Since $f$ is strong, there is a morphism $e:C\to A$ such that the diagram below is commutative |
| $$\xymatrix@+=3pc{ |
$$\xymatrix@+=3pc{ |
| {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]_{e} \\ |
{A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]_{e} \\ |
| {A}\ar[r]_{f}&{B} |
{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}). |
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} |
\end{proof} |
|
|
| \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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| \end{thebibliography} |
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