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[parent] strong monomorphism (Definition)

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

$\displaystyle \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:
$\displaystyle \xymatrix@+=3pc{ {C}\ar[r]^{g}\ar[d]_{x}&{D}\ar[d]^{y} \ar@{.>}[dl]\vert{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
$\displaystyle \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
$\displaystyle \xymatrix@+=3pc{ {A}\ar[r]^{g}\ar[d]_{1_A}&{C}\ar[d]^{h} \ar@{.>}[dl]\vert{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). $ \qedsymbol$

Bibliography

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



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See Also: properties of regular and extremal monomorphisms

Other names:  strong monic, strong epi, strong epic
Also defines:  strong, strong epimorphism

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Cross-references: isomorphism, split monomorphism, commutative, diagram, proof, regular monomorphism, orthogonal, iff, morphism, epimorphism, commutative diagram, monomorphism, category
There are 56 references to this entry.

This is version 11 of strong monomorphism, born on 2008-09-15, modified 2008-10-15.
Object id is 11038, canonical name is StrongMonomorphism.
Accessed 489 times total.

Classification:
AMS MSC18-00 (Category theory; homological algebra :: General reference works )
 18A20 (Category theory; homological algebra :: General theory of categories and functors :: Epimorphisms, monomorphisms, special classes of morphisms, null morphisms)

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