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Homestrong monomorphism

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# strong monomorphism

Let $\mathcal{C}$ be a category. A monomorphism $f:A\to B$ in $\mathcal{C}$ is said to be a *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 *strong epimorphism* is an epimorphism which is orthogonal to every monomorphism in the category.

Remark. Every regular monomorphism is strong (see proof here), and every strong monomorphism is extremal.

###### 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 here). ∎

# References

- 1
F. Borceux
*Basic Category Theory, Handbook of Categorical Algebra I*, Cambridge University Press, Cambridge (1994)

## Mathematics Subject Classification

18A20*no label found*18-00

*no label found*

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