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| Title of object: |
strong monomorphism |
| Canonical Name: |
StrongMonomorphism |
| Type: |
Definition |
| Created on: |
2008-09-15 19:03:19 |
| Modified on: |
2008-09-16 16:54:04 |
| Classification: |
msc:18-00, msc:18A20 |
| Defines: |
strong, strong epimorphism |
| Synonyms: |
strong monomorphism=strong monic
strong monomorphism=strong epi
strong monomorphism=strong epic |
Preamble:
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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
$$\xymatrix@+=4pc{
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y}\\
{A}\ar[r]_{f}&{B}
}
$$
with $g$ an epimorphism, then there is a unique morphism $h: D\to A$ such that the following is another commutative diagram:
$$\xymatrix@+=4pc{
{C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]_{h} \\
{A}\ar[r]_{f}&{B}
}
$$
Dually, a \emph{strong epimorphism} is an epimorphism $f:A\to B$ such that, given a commutative diagram
$$\xymatrix@+=4pc{
{A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\\
{C}\ar[r]_{g}&{D}
}
$$
with $g$ a monomorphism, then there is a unique morphism $h:B\to C$ such that the following diagram is again commutative:
$$\xymatrix@+=4pc{
{A}\ar[r]^{f}\ar[d]_{x}&{B}\ar[d]^{y}\ar@{.>}[dl]_{h}\\
{C}\ar[r]_{g}&{D}
}
$$
\textbf{Remark}. Every regular monomorphism is strong, and every strong monomorphism is \PMlinkname{extremal}{ExtremalMonomorphism}.
More to come... |
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